%------------------------------------------------------------------------------
% File     : ITP236_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_MinMax 00214_010433
% 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    : 0068_VEBT_MinMax_00214_010433 [Des22]

% Status   : Theorem
% Rating   : 0.00 v8.1.0
% Syntax   : Number of formulae    : 13392 (3532 unt;2885 typ;   0 def)
%            Number of atoms       : 26822 (8781 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 18150 (2737   ~; 546   |;1491   &)
%                                         (3027 <=>;10349  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   15 (   2 avg)
%            Number of FOOLs       : 1103 ( 619 fml; 484 var)
%            Number of X terms     :  770 (   0  []; 693 ite;  77 let)
%            Number of types       :  633 ( 632 usr)
%            Number of type conns  : 2723 (1800   >; 923   *;   0   +;   0  <<)
%            Number of predicates  :  150 ( 147 usr;   3 prp; 0-4 aty)
%            Number of functors    : 2117 (2117 usr; 463 con; 0-4 aty)
%            Number of variables   : 26332 (25495   !; 837   ?;26332   :)
% SPC      : TX0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 22:12:08.493
%------------------------------------------------------------------------------
% Could-be-implicit typings (632)
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tff(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Complex__Ocomplex_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Rat__Orat_J,type,
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tff(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Rat__Orat_J,type,
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tff(ty_n_t__VEBT____Definitions__OVEBT,type,
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tff(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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tff(ty_n_t__Code____Numeral__Ointeger,type,
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tff(ty_n_t__fun_It__Nat__Onat_M_Eo_J,type,
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tff(ty_n_t__fun_It__Int__Oint_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Int__Oint_J,type,
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tff(ty_n_t__Option__Ooption_I_Eo_J,type,
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tff(ty_n_t__List__Olist_I_Eo_J,type,
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tff(ty_n_t__Set__Oset_I_Eo_J,type,
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tff(ty_n_t__Real__Oreal,type,
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tff(ty_n_t__Rat__Orat,type,
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% Explicit typings (2253)
tff(sy_c_ATP_058Lamm__a____,type,
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tff(sy_c_ATP_058Lamm__aa____,type,
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tff(sy_c_ATP_058Lamm__aaa____,type,
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tff(sy_c_ATP_058Lamm__aae____,type,
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tff(sy_c_ATP_058Lamm__aah____,type,
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tff(sy_c_ATP_058Lamm__aaj____,type,
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tff(sy_c_ATP_058Lamm__aak____,type,
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tff(sy_c_ATP_058Lamm__aal____,type,
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tff(sy_c_ATP_058Lamm__aam____,type,
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tff(sy_c_ATP_058Lamm__aan____,type,
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tff(sy_c_ATP_058Lamm__aao____,type,
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tff(sy_c_ATP_058Lamm__aap____,type,
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tff(sy_c_ATP_058Lamm__aaq____,type,
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tff(sy_c_ATP_058Lamm__aar____,type,
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tff(sy_c_ATP_058Lamm__ab____,type,
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tff(sy_c_ATP_058Lamm__ac____,type,
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tff(sy_c_ATP_058Lamm__ad____,type,
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tff(sy_c_ATP_058Lamm__ae____,type,
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tff(sy_c_ATP_058Lamm__af____,type,
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tff(sy_c_ATP_058Lamm__ag____,type,
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tff(sy_c_ATP_058Lamm__ah____,type,
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tff(sy_c_ATP_058Lamm__ai____,type,
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tff(sy_c_ATP_058Lamm__aj____,type,
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tff(sy_c_ATP_058Lamm__ak____,type,
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tff(sy_c_ATP_058Lamm__al____,type,
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tff(sy_c_ATP_058Lamm__am____,type,
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tff(sy_c_ATP_058Lamm__an____,type,
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tff(sy_c_ATP_058Lamm__ao____,type,
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tff(sy_c_ATP_058Lamm__ap____,type,
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tff(sy_c_ATP_058Lamm__aq____,type,
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tff(sy_c_ATP_058Lamm__ar____,type,
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tff(sy_c_ATP_058Lamm__as____,type,
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tff(sy_c_ATP_058Lamm__at____,type,
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tff(sy_c_ATP_058Lamm__au____,type,
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tff(sy_c_ATP_058Lamm__av____,type,
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tff(sy_c_ATP_058Lamm__aw____,type,
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tff(sy_c_ATP_058Lamm__ax____,type,
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tff(sy_c_ATP_058Lamm__ay____,type,
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tff(sy_c_ATP_058Lamm__az____,type,
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tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: fun_co4998825143550818160real_o > fun_re7319081385389557872plex_o ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_nat_o > fun_fu4383628726152020410real_o ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: fun_real_fun_nat_o > fun_nat_fun_real_o ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_nat_o > fun_fu1148935994063782970_nat_o ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: fun_nat_fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: fun_nat_o > fun_fu1329347338547691834_int_o ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_int_fun_nat_o > fun_nat_fun_int_o ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_nat_o > fun_fu2079201134200759482plex_o ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: fun_co5676208753217392844_nat_o > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: fun_int_o > fun_fu8335073354567729686real_o ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: fun_real_fun_int_o > fun_int_fun_real_o ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: fun_int_o > fun_fu6560018175767245462_nat_o ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: fun_nat_fun_int_o > fun_int_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: set_complex > fun_set_complex_o ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: set_Pr1261947904930325089at_nat > fun_se4782206882984618170_nat_o ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: set_int > fun_set_int_o ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: produc3843707927480180839at_nat > fun_Pr3059359313727237556_nat_o ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: fun_Pr2028404973800837934_nat_o ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: int > fun_int_fun_int_o ).

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

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

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: int > fun_int_fun_int_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_set_nat_o ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: fun_real_o ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: fun_nat_o ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: fun_int_o ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: produc3843707927480180839at_nat > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: product_prod_nat_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: produc3843707927480180839at_nat > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: product_prod_nat_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: set_Pr958786334691620121nt_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: set_Pr1200682188532975830nteger > fun_fu3422731660242565741eger_o ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: set_Pr3486312109254157132nteger > fun_fu7695273699059480439eger_o ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: set_Pr6700895561065273913nt_int > fun_fu6865642474282033866_int_o ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: set_Pr4166680992137330533nt_int > fun_fu2978539155105838238_int_o ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: set_Pr1261947904930325089at_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: set_real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: set_list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: set_set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: set_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: set_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: fun_real_real ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: fun_int_int ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: fun_real_real ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: fun_int_int ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: fun_real_o > fun_real_o ).

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

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: fun_set_nat_o > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: fun_int_o > fun_int_o ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: set_Pr1261947904930325089at_nat > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: set_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: set_list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: set_set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: set_nat > fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: set_Pr1261947904930325089at_nat > fun_se374060482380105870_nat_o ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: set_real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: set_list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: produc3843707927480180839at_nat > fun_se5282722391638144526_nat_o ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: product_prod_nat_nat > fun_se374060482380105870_nat_o ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: product_prod_nat_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: produc3843707927480180839at_nat > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: set_real > fun_fu4757868175734369219real_o ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: set_real > fun_fu8602227297639178947real_o ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: set_real > fun_fu4472351043248340291real_o ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: set_nat > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: set_nat > fun_fu8304507701009772099_nat_o ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: set_nat > fun_fu3780232916559739075_nat_o ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: set_int > fun_fu7493154764834273859_int_o ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: set_int > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: set_int > fun_fu2620374121714387907_int_o ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: set_complex > fun_fu3074144567237359299plex_o ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_real_o > fun_fu8020951434679919326real_o ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: fun_real_o > fun_fu3654966910337544342_nat_o ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: fun_real_o > fun_fu3749727409370067934_int_o ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: fun_real_o > fun_fu8168279061277983450plex_o ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: fun_real_o > fun_fu7307196158820694110real_o ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_real_o > fun_fu2993764712125629974_nat_o ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: fun_real_o > fun_fu3088525211158153566_int_o ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: fun_real_o > fun_fu8433555683852769370plex_o ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: fun_real_o > fun_fu3317822389839199582real_o ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: fun_real_o > fun_fu6531550788665658134_nat_o ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: fun_real_o > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: fun_real_o > fun_fu2498659858537143873_nat_o ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_real_o > fun_fu2983119460465514377_int_o ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_real_o > fun_fu4846543092220166533plex_o ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_nat_o > fun_fu1031053159183118053real_o ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: fun_nat_o > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_nat_o > fun_fu5961977364416097381_int_o ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_nat_o > fun_fu80362980471826017plex_o ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_int_o > fun_fu1268630300833693065real_o ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: fun_int_o > fun_fu4050368934121317825_nat_o ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: list_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: list_list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: list_set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: list_VEBT_VEBT > fun_VEBT_VEBT_o ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: list_o > fun_o_o ).

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

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: list_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: set_Pr1261947904930325089at_nat > fun_na5060536216179792737_nat_o ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: set_Pr1261947904930325089at_nat > fun_na5060536216179792737_nat_o ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: fun_nat_fun_real_o ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: set_real > fun_fu7444430165986413063real_o ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: set_real > fun_fu9068519094125542729real_o ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: set_nat > fun_fu6176141739774794687_nat_o ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: set_nat > fun_fu6433034133342731997_nat_o ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: set_int > fun_fu7994259084669964039_int_o ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: set_int > fun_fu3930813892005118153_int_o ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: set_complex > fun_fu2445682883699234307plex_o ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: set_complex > fun_fu2142003765521078979plex_o ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: set_int > fun_fu4228744305898656589_int_o ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: set_complex > fun_fu806425790188207109plex_o ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: set_complex > fun_fu7523448966510524487plex_o ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: set_real > fun_fu6510078244439976541real_o ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: set_real > fun_fu5587483295140630005real_o ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: set_real > fun_fu7444430165986413063real_o ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: set_real > fun_fu9068519094125542729real_o ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: set_nat > fun_fu6176141739774794687_nat_o ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: set_nat > fun_fu6433034133342731997_nat_o ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: set_int > fun_fu7994259084669964039_int_o ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: set_int > fun_fu3930813892005118153_int_o ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: set_complex > fun_fu2445682883699234307plex_o ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: set_complex > fun_fu2142003765521078979plex_o ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: set_int > fun_fu4228744305898656589_int_o ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: set_complex > fun_fu806425790188207109plex_o ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: set_complex > fun_fu7523448966510524487plex_o ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: set_real > fun_fu6510078244439976541real_o ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: set_real > fun_fu5587483295140630005real_o ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: fun_nat_nat > fun_fu1217771548038759377at_nat ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: fun_o_o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: fun_Pr3238374608701176162_int_o > fun_fu4592178885959468770_int_o ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: fun_o_8988054265482332260nt_int > fun_fu8427683764667544036nt_int ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: fun_na3970208099681795815_nat_o > fun_fu484727072080742171_nat_o ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: fun_fu1115759055147092071_o_nat > fun_fu4228150600790569371at_nat ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_Pr2478283820680522243nt_int > fun_fu3898734532474705027nt_int ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: fun_na8713714802964096592at_nat > fun_fu1039311746440044420at_nat ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: fun_fu1413608398870722896at_nat > fun_fu7448223259553364356at_nat ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: fun_fu3275390434952296369_nat_o > fun_fu6384561554008368101_nat_o ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_Pr2602117396471556196at_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: fun_Pr2477239776880934807at_nat > fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: fun_Pr2028404973800837934_nat_o > fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: fun_Pr2478283820680522243nt_int > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_Pr3238374608701176162_int_o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: fun_complex_int > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: fun_nat_int > fun_nat_real ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: set_nat > fun_fu5289434066392675523_nat_o ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: set_nat > fun_fu8695033588447641837al_nat ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_real_fun_nat_nat > fun_nat_fun_real_nat ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: set_real > fun_fu5553606195750802321at_nat ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: set_nat > fun_fu927436805355353155_nat_o ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: set_nat > fun_fu6398909621167755501nt_nat ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_int_fun_nat_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: set_int > fun_fu3180206027003385489at_nat ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: set_nat > fun_fu1580565752866428867_nat_o ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: set_nat > fun_fu8020931516743756395ex_nat ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: fun_co636942507509095880at_nat > fun_na8457291176974909128ex_nat ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: set_complex > fun_fu3351286688164852109at_nat ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: set_complex > fun_fu3043630359712370243plex_o ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: set_complex > fun_fu7433039812794084273omplex ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: fun_re7208826350674946562omplex > fun_co286204877037288194omplex ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: set_real > fun_fu1853463631764337203omplex ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: set_complex > fun_fu5273811730930669507plex_o ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: set_complex > fun_fu6938252901338612493omplex ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: fun_na1591161823130732966omplex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: set_nat > fun_fu3656911019502402539omplex ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: set_complex > fun_fu4149201653251398851plex_o ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: set_complex > fun_fu7724386853639975089omplex ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_in1307110748760623106omplex > fun_co7673900812888889602omplex ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: set_int > fun_fu260611630652282675omplex ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: set_nat > fun_fu5253574275612187245l_real ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: set_real > fun_fu5321668254738230545t_real ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: set_nat > fun_fu1051927838966779501t_real ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: fun_int_fun_nat_real > fun_nat_fun_int_real ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: set_int > fun_fu5396388669021241873t_real ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: set_nat > fun_fu6287892695735310827x_real ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: fun_co8378349014452122532t_real > fun_na5723442673449079972x_real ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: set_complex > fun_fu55639602125797389t_real ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: set_real > fun_fu8783008166737851587real_o ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: set_real > fun_fu8018396694123807121at_nat ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: fun_nat_fun_real_nat > fun_real_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: set_nat > fun_fu91202020541382083_nat_o ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: set_nat > fun_fu1169599456156577517al_nat ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: fun_real_complex > fun_fu3532930458344041007omplex ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_nat_complex > fun_fu4302481349826706223omplex ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_int_complex > fun_fu3558165918884697647omplex ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: fun_real_real > fun_fu4478067837645853357l_real ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_int_real > fun_fu7030473000426307245t_real ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: set_complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: fun_complex_real > fun_fu3043978019632300333x_real ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: fun_real_rat > fun_fu5551332503302594393al_rat ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: fun_nat_rat > fun_fu3162257917499486297at_rat ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_int_rat > fun_fu3922030663826099033nt_rat ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: fun_complex_rat > fun_fu4396998534562958809ex_rat ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: fun_Pr2602117396471556196at_nat > fun_fu7732904369943091395at_nat ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: fun_real_rat > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: fun_int_rat > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: fun_real_int > fun_fu3737244727311182119al_int ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: fun_nat_nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_real_complex > fun_real_o ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: fun_int_complex > fun_int_o ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: fun_real_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: fun_int_real > fun_int_o ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: fun_complex_real > fun_complex_o ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: fun_real_rat > fun_real_o ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_int_rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: fun_complex_rat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: fun_real_nat > fun_real_o ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: fun_nat_real > fun_na7718645245721260691t_real ).

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

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

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: fun_nat_int > fun_nat_int ).

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: complex > fun_fu709861064203714431x_real ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: complex > fun_fu6638204120336560003ex_rat ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: complex > fun_fu2513798379308225899ex_nat ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: complex > fun_fu4692242605864086655ex_int ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: real > fun_fu9132278742410140537l_real ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: real > fun_fu7187499828406537085al_rat ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: real > fun_fu3768230810189712741al_nat ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: real > fun_fu5645595436833112697al_int ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: int > fun_fu4328765270694171897t_real ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: int > fun_fu1617438229336688381nt_rat ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: $o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: fun_complex_nat > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: fun_nat_nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: fun_nat_nat > fun_nat_real ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: extended_enat > fun_Ex2441982661509597004d_enat ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: code_integer > fun_Co6226381647958292740nteger ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: fun_na7542918316964655096d_enat ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: fun_nat_real > fun_fu6899955116716589935t_real ).

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

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

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_nat_complex ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: fun_nat_int ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: fun_nat_real ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: fun_nat_real ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: fun_nat_o > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: fun_nat_o > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: fun_nat_o > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: fun_nat_o > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: set_nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: set_nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: set_nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: fun_nat_complex > fun_co8378349014452122532t_real ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: fun_nat_complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: fun_nat_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: nat > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: nat > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: nat > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: fun_nat_real > fun_se2100927751953696370t_real ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: real > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: fun_nat_real ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: real > fun_fu3477889739401271834t_real ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: real > fun_fu4806068833880117316t_real ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: real > fun_fu2115079063411676732t_real ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: real > fun_fu4540635375566031512t_real ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: fun_int_int ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: fun_real_real ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: fun_re4238438383274954274t_real ).

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

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

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: real > fun_na6724076748692969250l_real ).

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

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: fun_int_nat > fun_int_int ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: fun_nat_int > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: fun_int_int > fun_int_real ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: fun_int_int > fun_int_rat ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: fun_real_fun_int_o ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: code_integer > fun_Co3544761417380347684eger_o ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: fun_nat_o ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__xz____,type,
    aTP_Lamm_xz: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__yb____,type,
    aTP_Lamm_yb: nat > fun_li6914387844523535256_nat_o ).

tff(sy_c_ATP_058Lamm__yc____,type,
    aTP_Lamm_yc: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__yd____,type,
    aTP_Lamm_yd: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__yf____,type,
    aTP_Lamm_yf: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__yh____,type,
    aTP_Lamm_yh: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__yi____,type,
    aTP_Lamm_yi: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__yj____,type,
    aTP_Lamm_yj: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__yk____,type,
    aTP_Lamm_yk: fun_real_o ).

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ym____,type,
    aTP_Lamm_ym: fun_int_nat ).

tff(sy_c_ATP_058Lamm__yn____,type,
    aTP_Lamm_yn: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yo____,type,
    aTP_Lamm_yo: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__yp____,type,
    aTP_Lamm_yp: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__yq____,type,
    aTP_Lamm_yq: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__yr____,type,
    aTP_Lamm_yr: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__ys____,type,
    aTP_Lamm_ys: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__yt____,type,
    aTP_Lamm_yt: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yu____,type,
    aTP_Lamm_yu: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__yv____,type,
    aTP_Lamm_yv: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__yw____,type,
    aTP_Lamm_yw: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__yx____,type,
    aTP_Lamm_yx: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__yy____,type,
    aTP_Lamm_yy: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__yz____,type,
    aTP_Lamm_yz: fun_real_real ).

tff(sy_c_ATP_058Lamm__za____,type,
    aTP_Lamm_za: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zb____,type,
    aTP_Lamm_zb: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zc____,type,
    aTP_Lamm_zc: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zd____,type,
    aTP_Lamm_zd: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__ze____,type,
    aTP_Lamm_ze: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__zf____,type,
    aTP_Lamm_zf: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__zg____,type,
    aTP_Lamm_zg: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__zh____,type,
    aTP_Lamm_zh: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__zi____,type,
    aTP_Lamm_zi: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__zj____,type,
    aTP_Lamm_zj: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zk____,type,
    aTP_Lamm_zk: fun_nat_real ).

tff(sy_c_ATP_058Lamm__zl____,type,
    aTP_Lamm_zl: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zm____,type,
    aTP_Lamm_zm: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zn____,type,
    aTP_Lamm_zn: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zo____,type,
    aTP_Lamm_zo: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zp____,type,
    aTP_Lamm_zp: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zq____,type,
    aTP_Lamm_zq: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__zr____,type,
    aTP_Lamm_zr: fun_nat_o > fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__zt____,type,
    aTP_Lamm_zt: real > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__zu____,type,
    aTP_Lamm_zu: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__zv____,type,
    aTP_Lamm_zv: fun_nat_set_nat ).

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

tff(sy_c_ATP_058Lamm__zx____,type,
    aTP_Lamm_zx: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__zy____,type,
    aTP_Lamm_zy: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__zz____,type,
    aTP_Lamm_zz: real > fun_co1148497455491383662plex_o ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
    archim3151403230148437115or_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
    archim6058952711729229775r_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
    archimedean_frac_rat: fun_rat_rat ).

tff(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
    archim2898591450579166408c_real: fun_real_real ).

tff(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
    archim7778729529865785530nd_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: fun_real_int ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: fun_nat_fun_nat_nat ).

tff(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
    gbinomial_complex: complex > fun_nat_complex ).

tff(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
    gbinomial_int: int > fun_nat_int ).

tff(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
    gbinomial_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
    gbinomial_rat: rat > fun_nat_rat ).

tff(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
    gbinomial_real: fun_re4238438383274954274t_real ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun_Pr9018117161326028078_int_o ).

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

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

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
    bit_ri7919022796975470100ot_int: fun_int_int ).

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

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

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Oremdups_001t__Nat__Onat,type,
    remdups_nat: list_nat > list_nat ).

tff(sy_c_List_Oreplicate_001_Eo,type,
    replicate_o: ( nat * $o ) > list_o ).

tff(sy_c_List_Oreplicate_001t__Int__Oint,type,
    replicate_int: ( nat * int ) > list_int ).

tff(sy_c_List_Oreplicate_001t__Nat__Onat,type,
    replicate_nat: ( nat * nat ) > list_nat ).

tff(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    replic4235873036481779905at_nat: ( nat * product_prod_nat_nat ) > list_P6011104703257516679at_nat ).

tff(sy_c_List_Oreplicate_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,
    replic2264142908078655527at_nat: ( nat * produc3843707927480180839at_nat ) > list_P5464809261938338413at_nat ).

tff(sy_c_List_Oreplicate_001t__Real__Oreal,type,
    replicate_real: ( nat * real ) > list_real ).

tff(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
    replicate_set_nat: ( nat * set_nat ) > list_set_nat ).

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

tff(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
    sorted_wrt_int: fun_int_fun_int_o > fun_list_int_o ).

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

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

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

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

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

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_List_Ozip_001_Eo_001_Eo,type,
    zip_o_o: ( list_o * list_o ) > list_P4002435161011370285od_o_o ).

tff(sy_c_List_Ozip_001_Eo_001t__Int__Oint,type,
    zip_o_int: ( list_o * list_int ) > list_P3795440434834930179_o_int ).

tff(sy_c_List_Ozip_001_Eo_001t__Nat__Onat,type,
    zip_o_nat: ( list_o * list_nat ) > list_P6285523579766656935_o_nat ).

tff(sy_c_List_Ozip_001_Eo_001t__Real__Oreal,type,
    zip_o_real: ( list_o * list_real ) > list_P5232166724548748803o_real ).

tff(sy_c_List_Ozip_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    zip_o_VEBT_VEBT: ( list_o * list_VEBT_VEBT ) > list_P7495141550334521929T_VEBT ).

tff(sy_c_List_Ozip_001t__Int__Oint_001t__Int__Oint,type,
    zip_int_int: ( list_int * list_int ) > list_P5707943133018811711nt_int ).

tff(sy_c_List_Ozip_001t__Int__Oint_001t__Real__Oreal,type,
    zip_int_real: ( list_int * list_real ) > list_P6863124054624500543t_real ).

tff(sy_c_List_Ozip_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    zip_int_VEBT_VEBT: ( list_int * list_VEBT_VEBT ) > list_P7524865323317820941T_VEBT ).

tff(sy_c_List_Ozip_001t__Nat__Onat_001_Eo,type,
    zip_nat_o: ( list_nat * list_o ) > list_P7333126701944960589_nat_o ).

tff(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
    zip_nat_nat: ( list_nat * list_nat ) > list_P6011104703257516679at_nat ).

tff(sy_c_List_Ozip_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    zip_nat_VEBT_VEBT: ( list_nat * list_VEBT_VEBT ) > list_P5647936690300460905T_VEBT ).

tff(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    zip_Pr4664179122662387191at_nat: ( list_P6011104703257516679at_nat * list_P6011104703257516679at_nat ) > list_P8469869581646625389at_nat ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001_Eo,type,
    zip_real_o: ( list_real * list_o ) > list_P3595434254542482545real_o ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001t__Int__Oint,type,
    zip_real_int: ( list_real * list_int ) > list_P4344331454722006975al_int ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001t__Nat__Onat,type,
    zip_real_nat: ( list_real * list_nat ) > list_P6834414599653733731al_nat ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001t__Real__Oreal,type,
    zip_real_real: ( list_real * list_real ) > list_P8689742595348180415l_real ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    zip_real_VEBT_VEBT: ( list_real * list_VEBT_VEBT ) > list_P877281246627933069T_VEBT ).

tff(sy_c_List_Ozip_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    zip_set_nat_set_nat: ( list_set_nat * list_set_nat ) > list_P6254988961118846195et_nat ).

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    zip_VEBT_VEBT_o: ( list_VEBT_VEBT * list_o ) > list_P3126845725202233233VEBT_o ).

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    zip_VEBT_VEBT_int: ( list_VEBT_VEBT * list_int ) > list_P4547456442757143711BT_int ).

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    zip_VEBT_VEBT_nat: ( list_VEBT_VEBT * list_nat ) > list_P7037539587688870467BT_nat ).

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    zip_VEBT_VEBT_real: ( list_VEBT_VEBT * list_real ) > list_P2623026923184700063T_real ).

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    zip_VE537291747668921783T_VEBT: ( list_VEBT_VEBT * list_VEBT_VEBT ) > list_P7413028617227757229T_VEBT ).

tff(sy_c_List_Ozip_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    zip_fu7084642285130276018nteger: ( list_f8851413139013959296e_term * list_P5578671422887162913nteger ) > list_P1054633106646461488nteger ).

tff(sy_c_List_Ozip_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    zip_fu2910578966527623617nt_int: ( list_f628289730585369423e_term * list_P5707943133018811711nt_int ) > list_P932735248015874751nt_int ).

tff(sy_c_List_Ozip_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    zip_fu4451626411769283112nteger: ( list_f4306079564415238026e_term * list_P5578671422887162913nteger ) > list_P2959130015091922598nteger ).

tff(sy_c_List_Ozip_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    zip_fu5048005673116055189nt_int: ( list_f280343417242313083e_term * list_P5707943133018811711nt_int ) > list_P2843741347166045587nt_int ).

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

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

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

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

tff(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
    case_nat_option_num: ( option_num * fun_nat_option_num * nat ) > option_num ).

tff(sy_c_Nat_Onat_Opred,type,
    pred: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
    semiring_1_Nats_int: set_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: fun_nat_Code_integer ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
    semiri8010041392384452111omplex: fun_nat_complex ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
    semiri4216267220026989637d_enat: fun_na8797796813480665199d_enat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: fun_nat_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: fun_nat_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: fun_nat_real ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
    semiri2816024913162550771omplex: fun_complex_complex > fun_na1591161823130732966omplex ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
    semiri8420488043553186161ux_int: ( fun_int_int * nat ) > fun_int_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
    semiri8422978514062236437ux_nat: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
    semiri7787848453975740701ux_rat: ( fun_rat_rat * nat ) > fun_rat_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
    semiri7260567687927622513x_real: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
    size_size_list_o: fun_list_o_nat ).

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
    size_size_list_int: fun_list_int_nat ).

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
    size_size_list_nat: fun_list_nat_nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
    size_size_list_num: list_num > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
    size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
    size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
    size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    size_s2970893825323803983at_int: list_P3521021558325789923at_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
    size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
    size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
    size_size_list_real: list_real > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    size_s3254054031482475050et_nat: list_set_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    size_s6755466524823107622T_VEBT: fun_li1940556759920501533BT_nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
    size_size_num: fun_num_nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
    size_size_option_nat: option_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
    size_size_option_num: option_num > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
    size_size_VEBT_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: list_nat > nat ).

tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: fun_li1025265955262268170_nat_o ).

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

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun_Pr2028404973800837934_nat_o ).

tff(sy_c_Nat__Bijection_Oprod__encode,type,
    nat_prod_encode: fun_Pr2602117396471556196at_nat ).

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

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun_set_nat_nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: fun_nat_nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: fun_na6724076748692969250l_real ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun_real_real ).

tff(sy_c_Num_OBitM,type,
    bitM: fun_num_num ).

tff(sy_c_Num_Oinc,type,
    inc: fun_num_num ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
    neg_nu7757733837767384882nteger: code_integer > code_integer ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
    neg_nu6511756317524482435omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
    neg_nu3811975205180677377ec_int: fun_int_int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
    neg_nu3179335615603231917ec_rat: fun_rat_rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
    neg_nu6075765906172075777c_real: fun_real_real ).

tff(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
    neg_numeral_sub_int: num > fun_num_int ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: fun_num_num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun_num_num ).

tff(sy_c_Num_Onum_OOne,type,
    one: num ).

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

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: fun_num_nat ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: fun_nat_num ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
    numera6620942414471956472nteger: num > code_integer ).

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

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

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
    numeral_numeral_int: fun_num_int ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: fun_num_nat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
    numeral_numeral_rat: fun_num_rat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
    numeral_numeral_real: fun_num_real ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: fun_num_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Int__Oint,type,
    none_int: option_int ).

tff(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Real__Oreal,type,
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tff(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_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__fun_It__Int__Oint_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Real__Oreal_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_Omin_001t__Int__Oint,type,
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tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
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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_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_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_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
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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,
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tff(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
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tff(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_OPair_001_Eo_001t__Real__Oreal,type,
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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__Int__Oint_001t__Real__Oreal,type,
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tff(sy_c_Product__Type_OPair_001t__List__Olist_I_Eo_J_001t__List__Olist_I_Eo_J,type,
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tff(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001_Eo,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
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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__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
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tff(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
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tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001_Eo,type,
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tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Int__Oint,type,
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tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_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_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Rat_OFract,type,
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tff(sy_c_Rat_OFrct,type,
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tff(sy_c_Rat_ORep__Rat,type,
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tff(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
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tff(sy_c_Rat_Onormalize,type,
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tff(sy_c_Rat_Oof__int,type,
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tff(sy_c_Rat_Opositive,type,
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tff(sy_c_Rat_Oquotient__of,type,
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tff(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
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tff(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
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tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
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tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
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tff(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
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tff(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
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tff(sy_c_Relation_OField_001t__Nat__Onat,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
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tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
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tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
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tff(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
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tff(sy_c_Series_Osuminf_001t__Int__Oint,type,
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tff(sy_c_Series_Osuminf_001t__Nat__Onat,type,
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tff(sy_c_Series_Osuminf_001t__Real__Oreal,type,
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tff(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
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tff(sy_c_Series_Osummable_001t__Int__Oint,type,
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tff(sy_c_Series_Osummable_001t__Nat__Onat,type,
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tff(sy_c_Series_Osummable_001t__Real__Oreal,type,
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tff(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
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tff(sy_c_Series_Osums_001t__Int__Oint,type,
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tff(sy_c_Series_Osums_001t__Nat__Onat,type,
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tff(sy_c_Series_Osums_001t__Real__Oreal,type,
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tff(sy_c_Set_OCollect_001_Eo,type,
    collect_o: fun_o_o > set_o ).

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tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,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__Real__Oreal,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_Othe__elem_001t__Real__Oreal,type,
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tff(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
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tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
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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,
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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_OatLeast_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Num__Onum,type,
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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Rat__Orat,type,
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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
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    topolo5044208981011980120l_real: ( set_real * fun_real_real ) > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
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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,
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tff(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
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tff(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
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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_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
    cos_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__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: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
    tan_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
    tanh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

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

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

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

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

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_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,
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tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
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tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
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tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
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tff(sy_c_VEBT__Definitions_Oset__vebt,type,
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tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
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tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
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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,
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tff(sy_c_VEBT__Member_Ovebt__member,type,
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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,
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tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
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tff(sy_c_VEBT__MinMax_Ovebt__maxt,type,
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tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
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tff(sy_c_VEBT__MinMax_Ovebt__mint,type,
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tff(sy_c_VEBT__MinMax_Ovebt__mint__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_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_Wellfounded_Opred__nat,type,
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tff(sy_c_Wellfounded_Owf_001t__Int__Oint,type,
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tff(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
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    member568628332442017744T_VEBT: ( produc8243902056947475879T_VEBT * set_Pr6192946355708809607T_VEBT ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    member6901806533512420791nteger: ( produc7509114972854644512nteger * set_Pr1200682188532975830nteger ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member5890090881338784710nt_int: ( produc4788131091073775919nt_int * set_Pr4166680992137330533nt_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    member8978266922497061677nteger: ( produc2453742332425399446nteger * set_Pr3486312109254157132nteger ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member753276645541486746nt_int: ( produc3043004577601930755nt_int * set_Pr6700895561065273913nt_int ) > $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__Int__Oint_J,type,
    member_set_int: ( set_int * set_set_int ) > $o ).

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

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

tff(sy_c_member_001t__fun_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    member7516538326899768071e_term: ( fun_Co3460408610320954864e_term * set_fu1527098227522870310e_term ) > $o ).

tff(sy_c_member_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    member4871225538671630166e_term: ( fun_in5730379987311621695e_term * set_fu3629540756087427317e_term ) > $o ).

tff(sy_c_member_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    member8464590939221061905e_term: ( fun_Pr5098669352780829690e_term * set_fu8280898353253485872e_term ) > $o ).

tff(sy_c_member_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    member1495369047980753794e_term: ( fun_Pr7763680713840462443e_term * set_fu5653557704672872225e_term ) > $o ).

tff(sy_v_deg,type,
    deg: nat ).

tff(sy_v_m____,type,
    m: nat ).

tff(sy_v_ma,type,
    ma: nat ).

tff(sy_v_maxs____,type,
    maxs: nat ).

tff(sy_v_mi,type,
    mi: nat ).

tff(sy_v_n____,type,
    n: nat ).

tff(sy_v_summary,type,
    summary: vEBT_VEBT ).

tff(sy_v_thesis____,type,
    thesis: $o ).

tff(sy_v_treeList,type,
    treeList: list_VEBT_VEBT ).

% Relevant facts (10490)
tff(fact_0__092_060open_062_092_060exists_062x_O_Aboth__member__options_A_ItreeList_A_B_Amaxs_J_Ax_092_060close_062,axiom,
    ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,maxs)),X_1) ).

% \<open>\<exists>x. both_member_options (treeList ! maxs) x\<close>
tff(fact_1_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
     => ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(T),X_1) ) ).

% not_min_Null_member
tff(fact_2__092_060open_062Some_Amaxs_A_061_Avebt__maxt_Asummary_092_060close_062,axiom,
    some_nat(maxs) = vEBT_vebt_maxt(summary) ).

% \<open>Some maxs = vebt_maxt summary\<close>
tff(fact_3_both__member__options__def,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( aa_nat_o(vEBT_V8194947554948674370ptions(T),X)
    <=> ( aa_nat_o(vEBT_V5719532721284313246member(T),X)
        | aa_nat_o(vEBT_VEBT_membermima(T),X) ) ) ).

% both_member_options_def
tff(fact_4__C5_C_I7_J,axiom,
    ( ( mi = ma )
   => ! [X2: vEBT_VEBT] :
        ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(treeList))
       => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_12) ) ) ).

% "5"(7)
tff(fact_5__C5_C_I1_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(treeList))
     => aa_nat_o(vEBT_invar_vebt(X2),n) ) ).

% "5"(1)
tff(fact_6_in__children__def,axiom,
    ! [Na: nat,TreeLista: list_VEBT_VEBT,X: nat] :
      ( aa_nat_o(vEBT_V5917875025757280293ildren(Na,TreeLista),X)
    <=> aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),Na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),Na)) ) ).

% in_children_def
tff(fact_7_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,Na: nat,X: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(T),X)
      <=> aa_nat_o(vEBT_vebt_member(T),X) ) ) ).

% both_member_options_equiv_member
tff(fact_8_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,Na: nat,X: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(T),X)
       => aa_nat_o(vEBT_vebt_member(T),X) ) ) ).

% valid_member_both_member_options
tff(fact_9_intind,axiom,
    ! [I: nat,Na: nat,P: fun_nat_o,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( aa_nat_o(P,X)
       => aa_nat_o(P,aa_nat_nat(nth_nat(replicate_nat(Na,X)),I)) ) ) ).

% intind
tff(fact_10_intind,axiom,
    ! [I: nat,Na: nat,P: fun_int_o,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( aa_int_o(P,X)
       => aa_int_o(P,aa_nat_int(nth_int(replicate_int(Na,X)),I)) ) ) ).

% intind
tff(fact_11_intind,axiom,
    ! [I: nat,Na: nat,P: fun_VEBT_VEBT_o,X: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( aa_VEBT_VEBT_o(P,X)
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(replicate_VEBT_VEBT(Na,X),I)) ) ) ).

% intind
tff(fact_12_assms_I2_J,axiom,
    mi != ma ).

% assms(2)
tff(fact_13_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat] : list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,I)) = Xs ).

% list_update_id
tff(fact_14_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] : list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),I)) = Xs ).

% list_update_id
tff(fact_15_list__update__id,axiom,
    ! [Xs: list_int,I: nat] : list_update_int(Xs,I,aa_nat_int(nth_int(Xs),I)) = Xs ).

% list_update_id
tff(fact_16_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( I != J )
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),J) = nth_VEBT_VEBT(Xs,J) ) ) ).

% nth_list_update_neq
tff(fact_17_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_nat,X: nat] :
      ( ( I != J )
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_18_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_int,X: int] :
      ( ( I != J )
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),J) = aa_nat_int(nth_int(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_19_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
     => ~ aa_nat_o(vEBT_vebt_member(T),X) ) ).

% min_Null_member
tff(fact_20__C5_C_I2_J,axiom,
    aa_nat_o(vEBT_invar_vebt(summary),m) ).

% "5"(2)
tff(fact_21_bit__split__inv,axiom,
    ! [X: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),D)),D) = X ).

% bit_split_inv
tff(fact_22__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062maxs_O_ASome_Amaxs_A_061_Avebt__maxt_Asummary_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Maxs: nat] : some_nat(Maxs) != vEBT_vebt_maxt(summary) ).

% \<open>\<And>thesis. (\<And>maxs. Some maxs = vebt_maxt summary \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_23_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Na: nat,X: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),Na)
     => ( aa_nat_o(vEBT_vebt_member(Tree),X)
       => ( aa_nat_o(vEBT_V5719532721284313246member(Tree),X)
          | aa_nat_o(vEBT_VEBT_membermima(Tree),X) ) ) ) ).

% member_valid_both_member_options
tff(fact_24_maxt__member,axiom,
    ! [T: vEBT_VEBT,Na: nat,Maxi: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => aa_nat_o(vEBT_vebt_member(T),Maxi) ) ) ).

% maxt_member
tff(fact_25_member__correct,axiom,
    ! [T: vEBT_VEBT,Na: nat,X: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(vEBT_vebt_member(T),X)
      <=> aa_set_nat_o(member_nat(X),vEBT_set_vebt(T)) ) ) ).

% member_correct
tff(fact_26__092_060open_062both__member__options_Asummary_A_Ihigh_Ama_An_J_092_060close_062,axiom,
    aa_nat_o(vEBT_V8194947554948674370ptions(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),n)) ).

% \<open>both_member_options summary (high ma n)\<close>
tff(fact_27_nth__replicate,axiom,
    ! [I: nat,Na: nat,X: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( nth_VEBT_VEBT(replicate_VEBT_VEBT(Na,X),I) = X ) ) ).

% nth_replicate
tff(fact_28_nth__replicate,axiom,
    ! [I: nat,Na: nat,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( aa_nat_int(nth_int(replicate_int(Na,X)),I) = X ) ) ).

% nth_replicate
tff(fact_29_nth__replicate,axiom,
    ! [I: nat,Na: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( aa_nat_nat(nth_nat(replicate_nat(Na,X)),I) = X ) ) ).

% nth_replicate
tff(fact_30__C5_C_I8_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,mi),ma) ).

% "5"(8)
tff(fact_31_greater__shift,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(X)),some_nat(Y)) ) ).

% greater_shift
tff(fact_32_less__shift,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(X)),some_nat(Y)) ) ).

% less_shift
tff(fact_33__092_060open_062high_Ama_An_A_060_Athe_A_Ivebt__maxt_Asummary_J_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),n)),the_nat(vEBT_vebt_maxt(summary))) ).

% \<open>high ma n < the (vebt_maxt summary)\<close>
tff(fact_34_mint__member,axiom,
    ! [T: vEBT_VEBT,Na: nat,Maxi: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_mint(T) = some_nat(Maxi) )
       => aa_nat_o(vEBT_vebt_member(T),Maxi) ) ) ).

% mint_member
tff(fact_35_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,Na: nat,Maxi: nat,X: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => ( aa_nat_o(vEBT_vebt_member(T),X)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Maxi) ) ) ) ).

% maxt_corr_help
tff(fact_36_maxt__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,X: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_maxt(T) = some_nat(X) )
       => aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T)),X) ) ) ).

% maxt_corr
tff(fact_37_maxt__sound,axiom,
    ! [T: vEBT_VEBT,Na: nat,X: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T)),X)
       => ( vEBT_vebt_maxt(T) = some_nat(X) ) ) ) ).

% maxt_sound
tff(fact_38_buildup__nothing__in__leaf,axiom,
    ! [Na: nat,X: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_vebt_buildup(Na)),X) ).

% buildup_nothing_in_leaf
tff(fact_39_buildup__nothing__in__min__max,axiom,
    ! [Na: nat,X: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_vebt_buildup(Na)),X) ).

% buildup_nothing_in_min_max
tff(fact_40_option_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( some_nat(X22) = some_nat(Y2) )
    <=> ( X22 = Y2 ) ) ).

% option.inject
tff(fact_41_option_Oinject,axiom,
    ! [X22: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( some_P7363390416028606310at_nat(X22) = some_P7363390416028606310at_nat(Y2) )
    <=> ( X22 = Y2 ) ) ).

% option.inject
tff(fact_42_option_Oinject,axiom,
    ! [X22: num,Y2: num] :
      ( ( some_num(X22) = some_num(Y2) )
    <=> ( X22 = Y2 ) ) ).

% option.inject
tff(fact_43_valid__eq,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(T),D)
    <=> aa_nat_o(vEBT_invar_vebt(T),D) ) ).

% valid_eq
tff(fact_44_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),D)
     => aa_nat_o(vEBT_VEBT_valid(T),D) ) ).

% valid_eq1
tff(fact_45_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(T),D)
     => aa_nat_o(vEBT_invar_vebt(T),D) ) ).

% valid_eq2
tff(fact_46_inthall,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: fun_Pr4796943543331071514_nat_o,Na: nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),set_Pr5648618587558075414at_nat(Xs))
         => aa_Pro6150637604445964947_nat_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s5460976970255530739at_nat(Xs))
       => aa_Pro6150637604445964947_nat_o(P,nth_Pr7617993195940197384at_nat(Xs,Na)) ) ) ).

% inthall
tff(fact_47_inthall,axiom,
    ! [Xs: list_real,P: fun_real_o,Na: nat] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),set_real2(Xs))
         => aa_real_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_real(Xs))
       => aa_real_o(P,aa_nat_real(nth_real(Xs),Na)) ) ) ).

% inthall
tff(fact_48_inthall,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_o,Na: nat] :
      ( ! [X3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X3),set_set_nat2(Xs))
         => aa_set_nat_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s3254054031482475050et_nat(Xs))
       => aa_set_nat_o(P,nth_set_nat(Xs,Na)) ) ) ).

% inthall
tff(fact_49_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Na: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Na)) ) ) ).

% inthall
tff(fact_50_inthall,axiom,
    ! [Xs: list_o,P: fun_o_o,Na: nat] :
      ( ! [X3: $o] :
          ( member_o((X3),set_o2(Xs))
         => aa_o_o(P,(X3)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_o_nat(size_size_list_o,Xs))
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Na)) ) ) ).

% inthall
tff(fact_51_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Na: nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => aa_nat_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_nat_nat(size_size_list_nat,Xs))
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Na)) ) ) ).

% inthall
tff(fact_52_inthall,axiom,
    ! [Xs: list_int,P: fun_int_o,Na: nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),set_int2(Xs))
         => aa_int_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_int_nat(size_size_list_int,Xs))
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Na)) ) ) ).

% inthall
tff(fact_53_deg__not__0,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% deg_not_0
tff(fact_54__092_060open_062high_Ama_An_A_092_060le_062_Athe_A_Ivebt__maxt_Asummary_J_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),n)),the_nat(vEBT_vebt_maxt(summary))) ).

% \<open>high ma n \<le> the (vebt_maxt summary)\<close>
tff(fact_55_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(T),zero_zero_nat) ).

% valid_tree_deg_neq_0
tff(fact_56_valid__0__not,axiom,
    ! [T: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(T),zero_zero_nat) ).

% valid_0_not
tff(fact_57_min__in__set__def,axiom,
    ! [Xs: set_nat,X: nat] :
      ( aa_nat_o(vEBT_VEBT_min_in_set(Xs),X)
    <=> ( aa_set_nat_o(member_nat(X),Xs)
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),X4) ) ) ) ).

% min_in_set_def
tff(fact_58_max__in__set__def,axiom,
    ! [Xs: set_nat,X: nat] :
      ( aa_nat_o(vEBT_VEBT_max_in_set(Xs),X)
    <=> ( aa_set_nat_o(member_nat(X),Xs)
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),X) ) ) ) ).

% max_in_set_def
tff(fact_59_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_60_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_61_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_62_mem__Collect__eq,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( aa_set_set_nat_o(member_set_nat(A),collect_set_nat(P))
    <=> aa_set_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_63_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_64_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_65_Collect__mem__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : collec3392354462482085612at_nat(aTP_Lamm_a(A2)) = A2 ).

% Collect_mem_eq
tff(fact_66_Collect__mem__eq,axiom,
    ! [A2: set_real] : collect_real(aTP_Lamm_aa(A2)) = A2 ).

% Collect_mem_eq
tff(fact_67_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] : collect_list_nat(aTP_Lamm_ab(A2)) = A2 ).

% Collect_mem_eq
tff(fact_68_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] : collect_set_nat(aTP_Lamm_ac(A2)) = A2 ).

% Collect_mem_eq
tff(fact_69_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aTP_Lamm_ad(A2)) = A2 ).

% Collect_mem_eq
tff(fact_70_Collect__mem__eq,axiom,
    ! [A2: set_int] : collect_int(aTP_Lamm_ae(A2)) = A2 ).

% Collect_mem_eq
tff(fact_71_Collect__cong,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( ! [X3: real] :
          ( aa_real_o(P,X3)
        <=> aa_real_o(Q,X3) )
     => ( collect_real(P) = collect_real(Q) ) ) ).

% Collect_cong
tff(fact_72_Collect__cong,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ! [X3: list_nat] :
          ( aa_list_nat_o(P,X3)
        <=> aa_list_nat_o(Q,X3) )
     => ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_73_Collect__cong,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X3: set_nat] :
          ( aa_set_nat_o(P,X3)
        <=> aa_set_nat_o(Q,X3) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_74_Collect__cong,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(P,X3)
        <=> aa_nat_o(Q,X3) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_75_Collect__cong,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( aa_int_o(P,X3)
        <=> aa_int_o(Q,X3) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_76_buildup__gives__valid,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(vEBT_invar_vebt(vEBT_vebt_buildup(Na)),Na) ) ).

% buildup_gives_valid
tff(fact_77_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( vEBT_set_vebt(T) = vEBT_VEBT_set_vebt(T) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_78_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,Na: nat,Mini: nat,X: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_mint(T) = some_nat(Mini) )
       => ( aa_nat_o(vEBT_vebt_member(T),X)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mini),X) ) ) ) ).

% mint_corr_help
tff(fact_79_replicate__eq__replicate,axiom,
    ! [Mb: nat,X: vEBT_VEBT,Na: nat,Y: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(Mb,X) = replicate_VEBT_VEBT(Na,Y) )
    <=> ( ( Mb = Na )
        & ( ( Mb != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_80_replicate__eq__replicate,axiom,
    ! [Mb: nat,X: int,Na: nat,Y: int] :
      ( ( replicate_int(Mb,X) = replicate_int(Na,Y) )
    <=> ( ( Mb = Na )
        & ( ( Mb != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_81_replicate__eq__replicate,axiom,
    ! [Mb: nat,X: nat,Na: nat,Y: nat] :
      ( ( replicate_nat(Mb,X) = replicate_nat(Na,Y) )
    <=> ( ( Mb = Na )
        & ( ( Mb != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_82_length__replicate,axiom,
    ! [Na: nat,X: vEBT_VEBT] : aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,replicate_VEBT_VEBT(Na,X)) = Na ).

% length_replicate
tff(fact_83_length__replicate,axiom,
    ! [Na: nat,X: $o] : aa_list_o_nat(size_size_list_o,replicate_o(Na,(X))) = Na ).

% length_replicate
tff(fact_84_length__replicate,axiom,
    ! [Na: nat,X: nat] : aa_list_nat_nat(size_size_list_nat,replicate_nat(Na,X)) = Na ).

% length_replicate
tff(fact_85_length__replicate,axiom,
    ! [Na: nat,X: int] : aa_list_int_nat(size_size_list_int,replicate_int(Na,X)) = Na ).

% length_replicate
tff(fact_86_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,list_u1324408373059187874T_VEBT(Xs,I,X)) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) ).

% length_list_update
tff(fact_87_length__list__update,axiom,
    ! [Xs: list_o,I: nat,X: $o] : aa_list_o_nat(size_size_list_o,list_update_o(Xs,I,(X))) = aa_list_o_nat(size_size_list_o,Xs) ).

% length_list_update
tff(fact_88_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,X: nat] : aa_list_nat_nat(size_size_list_nat,list_update_nat(Xs,I,X)) = aa_list_nat_nat(size_size_list_nat,Xs) ).

% length_list_update
tff(fact_89_length__list__update,axiom,
    ! [Xs: list_int,I: nat,X: int] : aa_list_int_nat(size_size_list_int,list_update_int(Xs,I,X)) = aa_list_int_nat(size_size_list_int,Xs) ).

% length_list_update
tff(fact_90_mint__sound,axiom,
    ! [T: vEBT_VEBT,Na: nat,X: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T)),X)
       => ( vEBT_vebt_mint(T) = some_nat(X) ) ) ) ).

% mint_sound
tff(fact_91_mint__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,X: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_mint(T) = some_nat(X) )
       => aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T)),X) ) ) ).

% mint_corr
tff(fact_92_Ball__set__replicate,axiom,
    ! [Na: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(replicate_VEBT_VEBT(Na,A)))
         => aa_VEBT_VEBT_o(P,X4) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        | ( Na = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_93_Ball__set__replicate,axiom,
    ! [Na: nat,A: int,P: fun_int_o] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int(X4),set_int2(replicate_int(Na,A)))
         => aa_int_o(P,X4) )
    <=> ( aa_int_o(P,A)
        | ( Na = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_94_Ball__set__replicate,axiom,
    ! [Na: nat,A: nat,P: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(replicate_nat(Na,A)))
         => aa_nat_o(P,X4) )
    <=> ( aa_nat_o(P,A)
        | ( Na = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_95_Bex__set__replicate,axiom,
    ! [Na: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ? [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(replicate_VEBT_VEBT(Na,A)))
          & aa_VEBT_VEBT_o(P,X4) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        & ( Na != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_96_Bex__set__replicate,axiom,
    ! [Na: nat,A: int,P: fun_int_o] :
      ( ? [X4: int] :
          ( aa_set_int_o(member_int(X4),set_int2(replicate_int(Na,A)))
          & aa_int_o(P,X4) )
    <=> ( aa_int_o(P,A)
        & ( Na != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_97_Bex__set__replicate,axiom,
    ! [Na: nat,A: nat,P: fun_nat_o] :
      ( ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(replicate_nat(Na,A)))
          & aa_nat_o(P,X4) )
    <=> ( aa_nat_o(P,A)
        & ( Na != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_98_in__set__replicate,axiom,
    ! [X: product_prod_nat_nat,Na: nat,Y: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(Na,Y)))
    <=> ( ( X = Y )
        & ( Na != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_99_in__set__replicate,axiom,
    ! [X: real,Na: nat,Y: real] :
      ( aa_set_real_o(member_real(X),set_real2(replicate_real(Na,Y)))
    <=> ( ( X = Y )
        & ( Na != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_100_in__set__replicate,axiom,
    ! [X: set_nat,Na: nat,Y: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(X),set_set_nat2(replicate_set_nat(Na,Y)))
    <=> ( ( X = Y )
        & ( Na != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_101_in__set__replicate,axiom,
    ! [X: vEBT_VEBT,Na: nat,Y: vEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(replicate_VEBT_VEBT(Na,Y)))
    <=> ( ( X = Y )
        & ( Na != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_102_in__set__replicate,axiom,
    ! [X: int,Na: nat,Y: int] :
      ( aa_set_int_o(member_int(X),set_int2(replicate_int(Na,Y)))
    <=> ( ( X = Y )
        & ( Na != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_103_in__set__replicate,axiom,
    ! [X: nat,Na: nat,Y: nat] :
      ( aa_set_nat_o(member_nat(X),set_nat2(replicate_nat(Na,Y)))
    <=> ( ( X = Y )
        & ( Na != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_104_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),I)
     => ( list_u1324408373059187874T_VEBT(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_105_list__update__beyond,axiom,
    ! [Xs: list_o,I: nat,X: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_o_nat(size_size_list_o,Xs)),I)
     => ( list_update_o(Xs,I,(X)) = Xs ) ) ).

% list_update_beyond
tff(fact_106_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),I)
     => ( list_update_nat(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_107_list__update__beyond,axiom,
    ! [Xs: list_int,I: nat,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Xs)),I)
     => ( list_update_int(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_108__C5_C_I4_J,axiom,
    m = aa_nat_nat(suc,n) ).

% "5"(4)
tff(fact_109_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),I) = X ) ) ).

% nth_list_update_eq
tff(fact_110_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_o,X: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_o_nat(size_size_list_o,Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,I,(X))),I)
      <=> (X) ) ) ).

% nth_list_update_eq
tff(fact_111_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),I) = X ) ) ).

% nth_list_update_eq
tff(fact_112_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),I) = X ) ) ).

% nth_list_update_eq
tff(fact_113_lesseq__shift,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_lesseq,some_nat(X)),some_nat(Y)) ) ).

% lesseq_shift
tff(fact_114_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_lis7044684121217598542BT_nat(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_115_set__swap,axiom,
    ! [I: nat,Xs: list_o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_o_nat(size_size_list_o,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_list_o_nat(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_116_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_list_nat_nat(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_117_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_list_int_nat(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_118_Ex__list__of__length,axiom,
    ! [Na: nat] :
    ? [Xs2: list_VEBT_VEBT] : aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs2) = Na ).

% Ex_list_of_length
tff(fact_119_Ex__list__of__length,axiom,
    ! [Na: nat] :
    ? [Xs2: list_o] : aa_list_o_nat(size_size_list_o,Xs2) = Na ).

% Ex_list_of_length
tff(fact_120_Ex__list__of__length,axiom,
    ! [Na: nat] :
    ? [Xs2: list_nat] : aa_list_nat_nat(size_size_list_nat,Xs2) = Na ).

% Ex_list_of_length
tff(fact_121_Ex__list__of__length,axiom,
    ! [Na: nat] :
    ? [Xs2: list_int] : aa_list_int_nat(size_size_list_int,Xs2) = Na ).

% Ex_list_of_length
tff(fact_122_neq__if__length__neq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) != aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_123_neq__if__length__neq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( aa_list_o_nat(size_size_list_o,Xs) != aa_list_o_nat(size_size_list_o,Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_124_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) != aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_125_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) != aa_list_int_nat(size_size_list_int,Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_126_option_Osel,axiom,
    ! [X22: nat] : the_nat(some_nat(X22)) = X22 ).

% option.sel
tff(fact_127_option_Osel,axiom,
    ! [X22: product_prod_nat_nat] : the_Pr8591224930841456533at_nat(some_P7363390416028606310at_nat(X22)) = X22 ).

% option.sel
tff(fact_128_option_Osel,axiom,
    ! [X22: num] : the_num(some_num(X22)) = X22 ).

% option.sel
tff(fact_129_length__pos__if__in__set,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s5460976970255530739at_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_130_length__pos__if__in__set,axiom,
    ! [X: real,Xs: list_real] :
      ( aa_set_real_o(member_real(X),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_131_length__pos__if__in__set,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(X),set_set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3254054031482475050et_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_132_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)) ) ).

% length_pos_if_in_set
tff(fact_133_length__pos__if__in__set,axiom,
    ! [X: $o,Xs: list_o] :
      ( member_o((X),set_o2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_list_o_nat(size_size_list_o,Xs)) ) ).

% length_pos_if_in_set
tff(fact_134_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_list_nat_nat(size_size_list_nat,Xs)) ) ).

% length_pos_if_in_set
tff(fact_135_length__pos__if__in__set,axiom,
    ! [X: int,Xs: list_int] :
      ( aa_set_int_o(member_int(X),set_int2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_list_int_nat(size_size_list_int,Xs)) ) ).

% length_pos_if_in_set
tff(fact_136_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,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys2)),aa_lis7044684121217598542BT_nat(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_137_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,aa_list_o_nat(size_size_list_o,Ys2)),aa_list_o_nat(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_138_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,aa_list_nat_nat(size_size_list_nat,Ys2)),aa_list_nat_nat(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_139_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,aa_list_int_nat(size_size_list_int,Ys2)),aa_list_int_nat(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_140_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Xs = Ys )
    <=> ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
           => ( nth_VEBT_VEBT(Xs,I2) = nth_VEBT_VEBT(Ys,I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_141_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( Xs = Ys )
    <=> ( ( aa_list_o_nat(size_size_list_o,Xs) = aa_list_o_nat(size_size_list_o,Ys) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_o_nat(size_size_list_o,Xs))
           => ( aa_nat_o(nth_o(Xs),I2)
            <=> aa_nat_o(nth_o(Ys),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_142_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs))
           => ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Ys),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_143_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
    <=> ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_int_nat(size_size_list_int,Xs))
           => ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Ys),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_144_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na6887940067898170868VEBT_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_13: vEBT_VEBT] : aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I2),X_13) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs3) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I2),nth_VEBT_VEBT(Xs3,I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_145_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_o_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_13: $o] : aa_o_o(aa_nat_fun_o_o(P,I2),(X_13)) )
    <=> ? [Xs3: list_o] :
          ( ( aa_list_o_nat(size_size_list_o,Xs3) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_o_o(aa_nat_fun_o_o(P,I2),aa_nat_o(nth_o(Xs3),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_146_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I2),X_13) )
    <=> ? [Xs3: list_nat] :
          ( ( aa_list_nat_nat(size_size_list_nat,Xs3) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_nat_o(aa_nat_fun_nat_o(P,I2),aa_nat_nat(nth_nat(Xs3),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_147_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_int_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_13: int] : aa_int_o(aa_nat_fun_int_o(P,I2),X_13) )
    <=> ? [Xs3: list_int] :
          ( ( aa_list_int_nat(size_size_list_int,Xs3) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_int_o(aa_nat_fun_int_o(P,I2),aa_nat_int(nth_int(Xs3),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_148_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
           => ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Ys,I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_149_nth__equalityI,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( aa_list_o_nat(size_size_list_o,Xs) = aa_list_o_nat(size_size_list_o,Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_o_nat(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_150_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_151_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_int_nat(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_152_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))
         => ( X3 = X ) )
     => ( replicate_VEBT_VEBT(aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_153_replicate__length__same,axiom,
    ! [Xs: list_o,X: $o] :
      ( ! [X3: $o] :
          ( member_o((X3),set_o2(Xs))
         => ( (X3)
          <=> (X) ) )
     => ( replicate_o(aa_list_o_nat(size_size_list_o,Xs),(X)) = Xs ) ) ).

% replicate_length_same
tff(fact_154_replicate__length__same,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => ( X3 = X ) )
     => ( replicate_nat(aa_list_nat_nat(size_size_list_nat,Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_155_replicate__length__same,axiom,
    ! [Xs: list_int,X: int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),set_int2(Xs))
         => ( X3 = X ) )
     => ( replicate_int(aa_list_int_nat(size_size_list_int,Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_156_replicate__eqI,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Na: nat,X: product_prod_nat_nat] :
      ( ( size_s5460976970255530739at_nat(Xs) = Na )
     => ( ! [Y3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y3),set_Pr5648618587558075414at_nat(Xs))
           => ( Y3 = X ) )
       => ( Xs = replic4235873036481779905at_nat(Na,X) ) ) ) ).

% replicate_eqI
tff(fact_157_replicate__eqI,axiom,
    ! [Xs: list_real,Na: nat,X: real] :
      ( ( size_size_list_real(Xs) = Na )
     => ( ! [Y3: real] :
            ( aa_set_real_o(member_real(Y3),set_real2(Xs))
           => ( Y3 = X ) )
       => ( Xs = replicate_real(Na,X) ) ) ) ).

% replicate_eqI
tff(fact_158_replicate__eqI,axiom,
    ! [Xs: list_set_nat,Na: nat,X: set_nat] :
      ( ( size_s3254054031482475050et_nat(Xs) = Na )
     => ( ! [Y3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(Y3),set_set_nat2(Xs))
           => ( Y3 = X ) )
       => ( Xs = replicate_set_nat(Na,X) ) ) ) ).

% replicate_eqI
tff(fact_159_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,Na: nat,X: vEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = Na )
     => ( ! [Y3: vEBT_VEBT] :
            ( member_VEBT_VEBT(Y3,set_VEBT_VEBT2(Xs))
           => ( Y3 = X ) )
       => ( Xs = replicate_VEBT_VEBT(Na,X) ) ) ) ).

% replicate_eqI
tff(fact_160_replicate__eqI,axiom,
    ! [Xs: list_o,Na: nat,X: $o] :
      ( ( aa_list_o_nat(size_size_list_o,Xs) = Na )
     => ( ! [Y3: $o] :
            ( member_o((Y3),set_o2(Xs))
           => ( (Y3)
            <=> (X) ) )
       => ( Xs = replicate_o(Na,(X)) ) ) ) ).

% replicate_eqI
tff(fact_161_replicate__eqI,axiom,
    ! [Xs: list_nat,Na: nat,X: nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = Na )
     => ( ! [Y3: nat] :
            ( aa_set_nat_o(member_nat(Y3),set_nat2(Xs))
           => ( Y3 = X ) )
       => ( Xs = replicate_nat(Na,X) ) ) ) ).

% replicate_eqI
tff(fact_162_replicate__eqI,axiom,
    ! [Xs: list_int,Na: nat,X: int] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = Na )
     => ( ! [Y3: int] :
            ( aa_set_int_o(member_int(Y3),set_int2(Xs))
           => ( Y3 = X ) )
       => ( Xs = replicate_int(Na,X) ) ) ) ).

% replicate_eqI
tff(fact_163_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) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I2)) ) ) ).

% all_set_conv_all_nth
tff(fact_164_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P: fun_o_o] :
      ( ! [X4: $o] :
          ( member_o((X4),set_o2(Xs))
         => aa_o_o(P,(X4)) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_o_nat(size_size_list_o,Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I2)) ) ) ).

% all_set_conv_all_nth
tff(fact_165_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) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I2)) ) ) ).

% all_set_conv_all_nth
tff(fact_166_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) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_int_nat(size_size_list_int,Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I2)) ) ) ).

% all_set_conv_all_nth
tff(fact_167_all__nth__imp__all__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: fun_Pr4796943543331071514_nat_o,X: product_prod_nat_nat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s5460976970255530739at_nat(Xs))
         => aa_Pro6150637604445964947_nat_o(P,nth_Pr7617993195940197384at_nat(Xs,I3)) )
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs))
       => aa_Pro6150637604445964947_nat_o(P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_168_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_o,X: 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(X),set_real2(Xs))
       => aa_real_o(P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_169_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_o,X: set_nat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3254054031482475050et_nat(Xs))
         => aa_set_nat_o(P,nth_set_nat(Xs,I3)) )
     => ( aa_set_set_nat_o(member_set_nat(X),set_set_nat2(Xs))
       => aa_set_nat_o(P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_170_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,X: vEBT_VEBT] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I3)) )
     => ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
       => aa_VEBT_VEBT_o(P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_171_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: fun_o_o,X: $o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_o_nat(size_size_list_o,Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I3)) )
     => ( member_o((X),set_o2(Xs))
       => aa_o_o(P,(X)) ) ) ).

% all_nth_imp_all_set
tff(fact_172_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_o,X: nat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I3)) )
     => ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
       => aa_nat_o(P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_173_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_o,X: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_int_nat(size_size_list_int,Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I3)) )
     => ( aa_set_int_o(member_int(X),set_int2(Xs))
       => aa_int_o(P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_174_in__set__conv__nth,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s5460976970255530739at_nat(Xs))
          & ( nth_Pr7617993195940197384at_nat(Xs,I2) = X ) ) ) ).

% in_set_conv_nth
tff(fact_175_in__set__conv__nth,axiom,
    ! [X: real,Xs: list_real] :
      ( aa_set_real_o(member_real(X),set_real2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Xs))
          & ( aa_nat_real(nth_real(Xs),I2) = X ) ) ) ).

% in_set_conv_nth
tff(fact_176_in__set__conv__nth,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(X),set_set_nat2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3254054031482475050et_nat(Xs))
          & ( nth_set_nat(Xs,I2) = X ) ) ) ).

% in_set_conv_nth
tff(fact_177_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
          & ( nth_VEBT_VEBT(Xs,I2) = X ) ) ) ).

% in_set_conv_nth
tff(fact_178_in__set__conv__nth,axiom,
    ! [X: $o,Xs: list_o] :
      ( member_o((X),set_o2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_o_nat(size_size_list_o,Xs))
          & ( aa_nat_o(nth_o(Xs),I2)
          <=> (X) ) ) ) ).

% in_set_conv_nth
tff(fact_179_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs))
          & ( aa_nat_nat(nth_nat(Xs),I2) = X ) ) ) ).

% in_set_conv_nth
tff(fact_180_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( aa_set_int_o(member_int(X),set_int2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_int_nat(size_size_list_int,Xs))
          & ( aa_nat_int(nth_int(Xs),I2) = X ) ) ) ).

% in_set_conv_nth
tff(fact_181_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => ( ! [X3: vEBT_VEBT] :
            ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))
           => aa_VEBT_VEBT_o(P,X3) )
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Na)) ) ) ).

% list_ball_nth
tff(fact_182_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_o,P: fun_o_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_o_nat(size_size_list_o,Xs))
     => ( ! [X3: $o] :
            ( member_o((X3),set_o2(Xs))
           => aa_o_o(P,(X3)) )
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_183_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
           => aa_nat_o(P,X3) )
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_184_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_int,P: fun_int_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_int_nat(size_size_list_int,Xs))
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),set_int2(Xs))
           => aa_int_o(P,X3) )
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_185_nth__mem,axiom,
    ! [Na: nat,Xs: list_P6011104703257516679at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s5460976970255530739at_nat(Xs))
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(nth_Pr7617993195940197384at_nat(Xs,Na)),set_Pr5648618587558075414at_nat(Xs)) ) ).

% nth_mem
tff(fact_186_nth__mem,axiom,
    ! [Na: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_real(Xs))
     => aa_set_real_o(member_real(aa_nat_real(nth_real(Xs),Na)),set_real2(Xs)) ) ).

% nth_mem
tff(fact_187_nth__mem,axiom,
    ! [Na: nat,Xs: list_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s3254054031482475050et_nat(Xs))
     => aa_set_set_nat_o(member_set_nat(nth_set_nat(Xs,Na)),set_set_nat2(Xs)) ) ).

% nth_mem
tff(fact_188_nth__mem,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => member_VEBT_VEBT(nth_VEBT_VEBT(Xs,Na),set_VEBT_VEBT2(Xs)) ) ).

% nth_mem
tff(fact_189_nth__mem,axiom,
    ! [Na: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_o_nat(size_size_list_o,Xs))
     => member_o(aa_nat_o(nth_o(Xs),Na),set_o2(Xs)) ) ).

% nth_mem
tff(fact_190_nth__mem,axiom,
    ! [Na: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_nat_nat(size_size_list_nat,Xs))
     => aa_set_nat_o(member_nat(aa_nat_nat(nth_nat(Xs),Na)),set_nat2(Xs)) ) ).

% nth_mem
tff(fact_191_nth__mem,axiom,
    ! [Na: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_int_nat(size_size_list_int,Xs))
     => aa_set_int_o(member_int(aa_nat_int(nth_int(Xs),Na)),set_int2(Xs)) ) ).

% nth_mem
tff(fact_192_set__update__memI,axiom,
    ! [Na: nat,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s5460976970255530739at_nat(Xs))
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,Na,X))) ) ).

% set_update_memI
tff(fact_193_set__update__memI,axiom,
    ! [Na: nat,Xs: list_real,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_real(Xs))
     => aa_set_real_o(member_real(X),set_real2(list_update_real(Xs,Na,X))) ) ).

% set_update_memI
tff(fact_194_set__update__memI,axiom,
    ! [Na: nat,Xs: list_set_nat,X: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s3254054031482475050et_nat(Xs))
     => aa_set_set_nat_o(member_set_nat(X),set_set_nat2(list_update_set_nat(Xs,Na,X))) ) ).

% set_update_memI
tff(fact_195_set__update__memI,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => member_VEBT_VEBT(X,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,Na,X))) ) ).

% set_update_memI
tff(fact_196_set__update__memI,axiom,
    ! [Na: nat,Xs: list_o,X: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_o_nat(size_size_list_o,Xs))
     => member_o((X),set_o2(list_update_o(Xs,Na,(X)))) ) ).

% set_update_memI
tff(fact_197_set__update__memI,axiom,
    ! [Na: nat,Xs: list_nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_nat_nat(size_size_list_nat,Xs))
     => aa_set_nat_o(member_nat(X),set_nat2(list_update_nat(Xs,Na,X))) ) ).

% set_update_memI
tff(fact_198_set__update__memI,axiom,
    ! [Na: nat,Xs: list_int,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_int_nat(size_size_list_int,Xs))
     => aa_set_int_o(member_int(X),set_int2(list_update_int(Xs,Na,X))) ) ).

% set_update_memI
tff(fact_199_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I,X) = Xs )
      <=> ( nth_VEBT_VEBT(Xs,I) = X ) ) ) ).

% list_update_same_conv
tff(fact_200_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_o,X: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_o_nat(size_size_list_o,Xs))
     => ( ( list_update_o(Xs,I,(X)) = Xs )
      <=> ( aa_nat_o(nth_o(Xs),I)
        <=> (X) ) ) ) ).

% list_update_same_conv
tff(fact_201_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( ( list_update_nat(Xs,I,X) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I) = X ) ) ) ).

% list_update_same_conv
tff(fact_202_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs))
     => ( ( list_update_int(Xs,I,X) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),I) = X ) ) ) ).

% list_update_same_conv
tff(fact_203_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),J) = $ite(I = J,X,nth_VEBT_VEBT(Xs,J)) ) ) ).

% nth_list_update
tff(fact_204_nth__list__update,axiom,
    ! [I: nat,Xs: list_o,X: $o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_o_nat(size_size_list_o,Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,I,(X))),J)
      <=> $ite(I = J,(X),aa_nat_o(nth_o(Xs),J)) ) ) ).

% nth_list_update
tff(fact_205_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,X: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J) = $ite(I = J,X,aa_nat_nat(nth_nat(Xs),J)) ) ) ).

% nth_list_update
tff(fact_206_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,X: int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),J) = $ite(I = J,X,aa_nat_int(nth_int(Xs),J)) ) ) ).

% nth_list_update
tff(fact_207_le0,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Na) ).

% le0
tff(fact_208_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_209_less__nat__zero__code,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_210_neq0__conv,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% neq0_conv
tff(fact_211_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_212_not__gr__zero,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_213_le__zero__eq,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),zero_zero_nat)
    <=> ( Na = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_214__C5_C_I5_J,axiom,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,n),m) ).

% "5"(5)
tff(fact_215_ex__least__nat__le,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,Na)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Na)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K2)
               => ~ aa_nat_o(P,I4) )
            & aa_nat_o(P,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_216_buildup__gives__empty,axiom,
    ! [Na: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Na)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_217_dual__order_Orefl,axiom,
    ! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(A),A) ).

% dual_order.refl
tff(fact_218_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_219_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_220_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_221_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_222_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)
     => ~ ! [N: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(suc,N)) ) ).

% even_odd_cases
tff(fact_223_add__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,some_nat(X)),some_nat(Y)) = some_nat(Z) ) ) ).

% add_shift
tff(fact_224_order__refl,axiom,
    ! [X: set_int] : aa_set_int_o(ord_less_eq_set_int(X),X) ).

% order_refl
tff(fact_225_order__refl,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),X) ).

% order_refl
tff(fact_226_order__refl,axiom,
    ! [X: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),X) ).

% order_refl
tff(fact_227_order__refl,axiom,
    ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),X) ).

% order_refl
tff(fact_228_order__refl,axiom,
    ! [X: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),X) ).

% order_refl
tff(fact_229_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_230_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_231_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_232_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_233_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_234_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_235_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_236_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_237_nat_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X22) = aa_nat_nat(suc,Y2) )
    <=> ( X22 = Y2 ) ) ).

% nat.inject
tff(fact_238_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_239_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(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_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_240_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(plus_plus_rat(C),A)),aa_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_241_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_242_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_243_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(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_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_244_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(plus_plus_rat(A),C)),aa_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_245_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_246_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_247_add_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.right_neutral
tff(fact_248_add_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% add.right_neutral
tff(fact_249_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_250_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_251_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_252_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) )
    <=> ( A = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_253_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(plus_plus_rat(A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_254_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_255_add__cancel__left__left,axiom,
    ! [B: complex,A: complex] :
      ( ( aa_complex_complex(plus_plus_complex(B),A) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_left
tff(fact_256_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_257_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_258_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_259_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_260_add__cancel__left__right,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_right
tff(fact_261_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_262_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_263_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_264_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_265_add__cancel__right__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(B),A) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_left
tff(fact_266_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_267_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(B),A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_268_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_269_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_270_add__cancel__right__right,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(A),B) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_right
tff(fact_271_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_272_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(A),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_273_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_274_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_275_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_276_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_277_add__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add_0
tff(fact_278_add__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add_0
tff(fact_279_add__0,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add_0
tff(fact_280_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_281_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_282_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(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_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_283_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(plus_plus_rat(C),A)),aa_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_284_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_285_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_286_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(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_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_287_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(plus_plus_rat(A),C)),aa_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_288_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_289_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_290_lessI,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,Na)) ).

% lessI
tff(fact_291_Suc__mono,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Na)) ) ).

% Suc_mono
tff(fact_292_Suc__less__eq,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ).

% Suc_less_eq
tff(fact_293_Suc__le__mono,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),aa_nat_nat(suc,Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb) ) ).

% Suc_le_mono
tff(fact_294_add__Suc__right,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)) ).

% add_Suc_right
tff(fact_295_add__is__0,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na) = zero_zero_nat )
    <=> ( ( Mb = zero_zero_nat )
        & ( Na = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_296_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_297_nat__add__left__cancel__less,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ).

% nat_add_left_cancel_less
tff(fact_298_nat__add__left__cancel__le,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% nat_add_left_cancel_le
tff(fact_299_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_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_300_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_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_301_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_302_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_303_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_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_304_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_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_305_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_306_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_307_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(aa_rea7340605660300375871l_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_308_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(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_309_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_310_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_311_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(aa_rea7340605660300375871l_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_312_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(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_313_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_314_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_315_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_316_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(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_317_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_318_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_319_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(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_320_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_321_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_322_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(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_323_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_324_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_325_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(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_326_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_327_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_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_328_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_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_329_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_330_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_331_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_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_332_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_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_333_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_334_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_335_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_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_336_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_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_337_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_338_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_339_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_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_340_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_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_341_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_342_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_343_zero__less__Suc,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,Na)) ).

% zero_less_Suc
tff(fact_344_less__Suc0,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Na = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_345_add__gr__0,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na))
    <=> ( 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),Na) ) ) ).

% add_gr_0
tff(fact_346_one__is__add,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na) )
    <=> ( ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
          & ( Na = zero_zero_nat ) )
        | ( ( Mb = zero_zero_nat )
          & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_347_add__is__1,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
          & ( Na = zero_zero_nat ) )
        | ( ( Mb = zero_zero_nat )
          & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_348_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_349_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_350_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_351_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_352_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_353_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_rat_rat(plus_plus_rat(I),K) = aa_rat_rat(plus_plus_rat(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_354_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_355_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_356_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(suc,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_357_add__Suc,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Mb)),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)) ).

% add_Suc
tff(fact_358_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add1
tff(fact_359_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(plus_plus_rat(A2),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add1
tff(fact_360_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_361_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add1
tff(fact_362_group__cancel_Oadd2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add2
tff(fact_363_group__cancel_Oadd2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = aa_rat_rat(plus_plus_rat(K),B) )
     => ( aa_rat_rat(plus_plus_rat(A),B2) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add2
tff(fact_364_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_365_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add2
tff(fact_366_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% add.assoc
tff(fact_367_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% add.assoc
tff(fact_368_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_369_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_370_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_371_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_372_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_373_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_374_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_375_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_376_add_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) ).

% add.commute
tff(fact_377_add_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(B),A) ).

% add.commute
tff(fact_378_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_379_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_380_add_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% add.left_commute
tff(fact_381_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(plus_plus_rat(B),aa_rat_rat(plus_plus_rat(A),C)) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% add.left_commute
tff(fact_382_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_383_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_384_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_385_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_386_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_387_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_388_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_389_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_390_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_391_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_392_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(suc,X) = aa_nat_nat(suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_393_n__not__Suc__n,axiom,
    ! [Na: nat] : Na != aa_nat_nat(suc,Na) ).

% n_not_Suc_n
tff(fact_394_add__Suc__shift,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Mb)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(suc,Na)) ).

% add_Suc_shift
tff(fact_395_less__imp__Suc__add,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
     => ? [K2: nat] : Na = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)) ) ).

% less_imp_Suc_add
tff(fact_396_less__iff__Suc__add,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
    <=> ? [K3: nat] : Na = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K3)) ) ).

% less_iff_Suc_add
tff(fact_397_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_398_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_399_less__natE,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
     => ~ ! [Q2: nat] : Na != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q2)) ) ).

% less_natE
tff(fact_400_bot_Oextremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(bot_bo2099793752762293965at_nat),A) ).

% bot.extremum
tff(fact_401_bot_Oextremum,axiom,
    ! [A: set_real] : aa_set_real_o(ord_less_eq_set_real(bot_bot_set_real),A) ).

% bot.extremum
tff(fact_402_bot_Oextremum,axiom,
    ! [A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,bot_bot_set_nat),A) ).

% bot.extremum
tff(fact_403_bot_Oextremum,axiom,
    ! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A) ).

% bot.extremum
tff(fact_404_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_405_bot_Oextremum__unique,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A),bot_bo2099793752762293965at_nat)
    <=> ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_unique
tff(fact_406_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),bot_bot_set_real)
    <=> ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
tff(fact_407_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),bot_bot_set_nat)
    <=> ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_408_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_409_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_410_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A),bot_bo2099793752762293965at_nat)
     => ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_411_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),bot_bot_set_real)
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
tff(fact_412_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),bot_bot_set_nat)
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_413_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_414_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_415_bot_Oextremum__strict,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ~ aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(A),bot_bo2099793752762293965at_nat) ).

% bot.extremum_strict
tff(fact_416_bot_Oextremum__strict,axiom,
    ! [A: set_real] : ~ aa_set_real_o(ord_less_set_real(A),bot_bot_set_real) ).

% bot.extremum_strict
tff(fact_417_bot_Oextremum__strict,axiom,
    ! [A: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),bot_bot_set_nat) ).

% bot.extremum_strict
tff(fact_418_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_419_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_420_bot_Onot__eq__extremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( A != bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(bot_bo2099793752762293965at_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_421_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
    <=> aa_set_real_o(ord_less_set_real(bot_bot_set_real),A) ) ).

% bot.not_eq_extremum
tff(fact_422_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,bot_bot_set_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_423_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_424_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_425_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_426_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_427_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_428_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_429_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_430_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_431_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_432_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_433_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_434_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_435_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_436_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_437_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(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_mono
tff(fact_438_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(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_mono
tff(fact_439_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_440_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_441_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(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% add_left_mono
tff(fact_442_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(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% add_left_mono
tff(fact_443_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_444_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_445_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ~ ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ) ).

% less_eqE
tff(fact_446_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(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ).

% add_right_mono
tff(fact_447_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(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) ) ).

% add_right_mono
tff(fact_448_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_449_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_450_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ? [C3: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).

% le_iff_add
tff(fact_451_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(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_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_452_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(plus_plus_rat(C),A)),aa_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_453_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_454_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_455_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(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_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_456_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(plus_plus_rat(A),C)),aa_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_457_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_458_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_459_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_460_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_461_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_462_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_463_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_464_add_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.comm_neutral
tff(fact_465_add_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% add.comm_neutral
tff(fact_466_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_467_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_468_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_469_add_Ogroup__left__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add.group_left_neutral
tff(fact_470_add_Ogroup__left__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add.group_left_neutral
tff(fact_471_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_472_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_473_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_474_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_475_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_476_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_477_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_478_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_479_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_480_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_481_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_482_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_483_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_484_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_485_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(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_strict_mono
tff(fact_486_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(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_strict_mono
tff(fact_487_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_488_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_489_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(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% add_strict_left_mono
tff(fact_490_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(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% add_strict_left_mono
tff(fact_491_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_492_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_493_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(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ).

% add_strict_right_mono
tff(fact_494_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(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) ) ).

% add_strict_right_mono
tff(fact_495_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_496_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_497_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(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_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_498_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(plus_plus_rat(C),A)),aa_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_499_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_500_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_501_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(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_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_502_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(plus_plus_rat(A),C)),aa_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_503_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_504_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_505_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero_nat != aa_nat_nat(suc,X22) ).

% nat.distinct(1)
tff(fact_506_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_507_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_508_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa_nat_nat(suc,X22) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_509_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_510_nat__induct,axiom,
    ! [P: fun_nat_o,Na: 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,Na) ) ) ).

% nat_induct
tff(fact_511_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Mb: nat,Na: nat] :
      ( ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,X3),zero_zero_nat)
     => ( ! [Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,zero_zero_nat),aa_nat_nat(suc,Y3))
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(suc,X3)),aa_nat_nat(suc,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,Mb),Na) ) ) ) ).

% diff_induct
tff(fact_512_zero__induct,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,zero_zero_nat) ) ) ).

% zero_induct
tff(fact_513_Suc__neq__Zero,axiom,
    ! [Mb: nat] : aa_nat_nat(suc,Mb) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_514_Zero__neq__Suc,axiom,
    ! [Mb: nat] : zero_zero_nat != aa_nat_nat(suc,Mb) ).

% Zero_neq_Suc
tff(fact_515_Zero__not__Suc,axiom,
    ! [Mb: nat] : zero_zero_nat != aa_nat_nat(suc,Mb) ).

% Zero_not_Suc
tff(fact_516_not0__implies__Suc,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => ? [M: nat] : Na = aa_nat_nat(suc,M) ) ).

% not0_implies_Suc
tff(fact_517_plus__nat_Oadd__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Na) = Na ).

% plus_nat.add_0
tff(fact_518_add__eq__self__zero,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na) = Mb )
     => ( Na = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_519_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K)
     => ( ( K != aa_nat_nat(suc,I) )
       => ~ ! [J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_520_Suc__lessD,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ).

% Suc_lessD
tff(fact_521_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K)
     => ~ ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_522_Suc__lessI,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
     => ( ( aa_nat_nat(suc,Mb) != Na )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),Na) ) ) ).

% Suc_lessI
tff(fact_523_less__SucE,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Na))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
       => ( Mb = Na ) ) ) ).

% less_SucE
tff(fact_524_less__SucI,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Na)) ) ).

% less_SucI
tff(fact_525_Ex__less__Suc,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Na))
          & aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,Na)
        | ? [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
            & aa_nat_o(P,I2) ) ) ) ).

% Ex_less_Suc
tff(fact_526_less__Suc__eq,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
        | ( Mb = Na ) ) ) ).

% less_Suc_eq
tff(fact_527_not__less__eq,axiom,
    ! [Mb: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,Mb)) ) ).

% not_less_eq
tff(fact_528_All__less__Suc,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Na))
         => aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,Na)
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
           => aa_nat_o(P,I2) ) ) ) ).

% All_less_Suc
tff(fact_529_Suc__less__eq2,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),Mb)
    <=> ? [M2: nat] :
          ( ( Mb = aa_nat_nat(suc,M2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M2) ) ) ).

% Suc_less_eq2
tff(fact_530_less__antisym,axiom,
    ! [Na: nat,Mb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,Mb))
       => ( Mb = Na ) ) ) ).

% less_antisym
tff(fact_531_Suc__less__SucD,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ).

% Suc_less_SucD
tff(fact_532_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K) ) ) ).

% less_trans_Suc
tff(fact_533_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,K2: 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),K2)
               => ( aa_nat_o(aa_nat_fun_nat_o(P,I3),J2)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P,J2),K2)
                   => aa_nat_o(aa_nat_fun_nat_o(P,I3),K2) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_534_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_535_not__less__less__Suc__eq,axiom,
    ! [Na: nat,Mb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,Mb))
      <=> ( Na = Mb ) ) ) ).

% not_less_less_Suc_eq
tff(fact_536_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K) ) ).

% add_lessD1
tff(fact_537_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_less_mono
tff(fact_538_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_539_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_540_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_less_mono1
tff(fact_541_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_542_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_543_less__add__eq__less,axiom,
    ! [K: nat,L: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ) ).

% less_add_eq_less
tff(fact_544_Suc__leD,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% Suc_leD
tff(fact_545_le__SucE,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Na))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
       => ( Mb = aa_nat_nat(suc,Na) ) ) ) ).

% le_SucE
tff(fact_546_le__SucI,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Na)) ) ).

% le_SucI
tff(fact_547_Suc__le__D,axiom,
    ! [Na: nat,M3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),M3)
     => ? [M: nat] : M3 = aa_nat_nat(suc,M) ) ).

% Suc_le_D
tff(fact_548_le__Suc__eq,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
        | ( Mb = aa_nat_nat(suc,Na) ) ) ) ).

% le_Suc_eq
tff(fact_549_Suc__n__not__le__n,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),Na) ).

% Suc_n_not_le_n
tff(fact_550_not__less__eq__eq,axiom,
    ! [Mb: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),Mb) ) ).

% not_less_eq_eq
tff(fact_551_full__nat__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( ! [N: nat] :
          ( ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M4)),N)
             => aa_nat_o(P,M4) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Na) ) ).

% full_nat_induct
tff(fact_552_nat__induct__at__least,axiom,
    ! [Mb: nat,Na: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( 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,Na) ) ) ) ).

% nat_induct_at_least
tff(fact_553_transitive__stepwise__le,axiom,
    ! [Mb: nat,Na: nat,R: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(R,X3),X3)
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X3),Y3)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y3),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(R,X3),Z2) ) )
         => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(R,N),aa_nat_nat(suc,N))
           => aa_nat_o(aa_nat_fun_nat_o(R,Mb),Na) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_554_add__leE,axiom,
    ! [Mb: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),Na)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ) ).

% add_leE
tff(fact_555_le__add1,axiom,
    ! [Na: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Mb)) ).

% le_add1
tff(fact_556_le__add2,axiom,
    ! [Na: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)) ).

% le_add2
tff(fact_557_add__leD1,axiom,
    ! [Mb: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% add_leD1
tff(fact_558_add__leD2,axiom,
    ! [Mb: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ).

% add_leD2
tff(fact_559_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
     => ? [N: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N) ) ).

% le_Suc_ex
tff(fact_560_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_le_mono
tff(fact_561_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_le_mono1
tff(fact_562_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_563_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_564_nat__le__iff__add,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
    <=> ? [K3: nat] : Na = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K3) ) ).

% nat_le_iff_add
tff(fact_565_subset__code_I1_J,axiom,
    ! [Xs: list_P6011104703257516679at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Xs)),B2)
    <=> ! [X4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),set_Pr5648618587558075414at_nat(Xs))
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),B2) ) ) ).

% subset_code(1)
tff(fact_566_subset__code_I1_J,axiom,
    ! [Xs: list_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_real2(Xs)),B2)
    <=> ! [X4: real] :
          ( aa_set_real_o(member_real(X4),set_real2(Xs))
         => aa_set_real_o(member_real(X4),B2) ) ) ).

% subset_code(1)
tff(fact_567_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),B2)
    <=> ! [X4: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X4),set_set_nat2(Xs))
         => aa_set_set_nat_o(member_set_nat(X4),B2) ) ) ).

% subset_code(1)
tff(fact_568_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),B2)
    <=> ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
         => member_VEBT_VEBT(X4,B2) ) ) ).

% subset_code(1)
tff(fact_569_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Xs)),B2)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
         => aa_set_nat_o(member_nat(X4),B2) ) ) ).

% subset_code(1)
tff(fact_570_subset__code_I1_J,axiom,
    ! [Xs: list_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),B2)
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),set_int2(Xs))
         => aa_set_int_o(member_int(X4),B2) ) ) ).

% subset_code(1)
tff(fact_571_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(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ) ) ).

% add_decreasing
tff(fact_572_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(plus_plus_rat(A),C)),B) ) ) ).

% add_decreasing
tff(fact_573_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_574_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_575_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(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_increasing
tff(fact_576_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(plus_plus_rat(A),C)) ) ) ).

% add_increasing
tff(fact_577_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_578_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_579_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(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_580_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(plus_plus_rat(A),C)),B) ) ) ).

% add_decreasing2
tff(fact_581_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_582_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_583_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(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_increasing2
tff(fact_584_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(plus_plus_rat(A),C)) ) ) ).

% add_increasing2
tff(fact_585_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_586_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_587_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(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_588_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(plus_plus_rat(A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_589_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_590_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_591_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(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_nonpos_nonpos
tff(fact_592_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(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_593_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_594_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_595_add__nonneg__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y) = zero_zero_real )
        <=> ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_596_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_rat_rat(plus_plus_rat(X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_597_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X)
     => ( 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,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_598_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( 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,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_599_add__nonpos__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y) = zero_zero_real )
        <=> ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_600_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => ( ( aa_rat_rat(plus_plus_rat(X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_601_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),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,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_602_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),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,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_603_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(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_less_le_mono
tff(fact_604_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(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_less_le_mono
tff(fact_605_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_606_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_607_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(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_le_less_mono
tff(fact_608_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(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_le_less_mono
tff(fact_609_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_610_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_611_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_612_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_613_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_614_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_615_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_616_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_617_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_618_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_619_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(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% pos_add_strict
tff(fact_620_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(plus_plus_rat(A),C)) ) ) ).

% pos_add_strict
tff(fact_621_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_622_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_623_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ ! [C2: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_624_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(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_pos_pos
tff(fact_625_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(plus_plus_rat(A),B)) ) ) ).

% add_pos_pos
tff(fact_626_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_627_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_628_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(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_neg_neg
tff(fact_629_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(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_630_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_631_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_632_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,Na: 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,Na),N2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Na)),aa_nat_real(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_633_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,Na: 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,Na),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Na)),aa_nat_rat(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_634_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,Na: 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,Na),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Na)),aa_nat_num(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_635_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,Na: 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,Na),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Na)),aa_nat_nat(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_636_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,Na: 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,Na),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Na)),aa_nat_int(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_637_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,Na: 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,Na)),aa_nat_real(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_638_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,Na: 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,Na)),aa_nat_rat(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_639_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,Na: 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,Na)),aa_nat_num(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_640_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,Na: 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,Na)),aa_nat_nat(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_641_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,Na: 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,Na)),aa_nat_int(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_642_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_int,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,N)),aa_nat_set_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,Na)),aa_nat_set_int(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_643_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,Na: 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,Na),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Na)),aa_nat_rat(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_644_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,Na: 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,Na),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Na)),aa_nat_num(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_645_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,Na: 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,Na),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Na)),aa_nat_nat(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_646_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,Na: 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,Na),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Na)),aa_nat_int(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_647_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_int,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,aa_nat_nat(suc,N))),aa_nat_set_int(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,N2)),aa_nat_set_int(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_648_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,Na: 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,Na),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_649_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,Na: 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,Na),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_650_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,Na: 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,Na),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_651_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,Na: 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,Na),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_652_less__Suc__eq__0__disj,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Na))
    <=> ( ( Mb = zero_zero_nat )
        | ? [J3: nat] :
            ( ( Mb = aa_nat_nat(suc,J3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Na) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_653_gr0__implies__Suc,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ? [M: nat] : Na = aa_nat_nat(suc,M) ) ).

% gr0_implies_Suc
tff(fact_654_All__less__Suc2,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Na))
         => aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
           => aa_nat_o(P,aa_nat_nat(suc,I2)) ) ) ) ).

% All_less_Suc2
tff(fact_655_gr0__conv__Suc,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ? [M5: nat] : Na = aa_nat_nat(suc,M5) ) ).

% gr0_conv_Suc
tff(fact_656_Ex__less__Suc2,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Na))
          & aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        | ? [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
            & aa_nat_o(P,aa_nat_nat(suc,I2)) ) ) ) ).

% Ex_less_Suc2
tff(fact_657_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ? [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_658_le__imp__less__Suc,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Na)) ) ).

% le_imp_less_Suc
tff(fact_659_less__eq__Suc__le,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),Mb) ) ).

% less_eq_Suc_le
tff(fact_660_less__Suc__eq__le,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% less_Suc_eq_le
tff(fact_661_le__less__Suc__eq,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,Mb))
      <=> ( Na = Mb ) ) ) ).

% le_less_Suc_eq
tff(fact_662_Suc__le__lessD,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ).

% Suc_le_lessD
tff(fact_663_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_664_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_665_Suc__le__eq,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ).

% Suc_le_eq
tff(fact_666_Suc__leI,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Na) ) ).

% Suc_leI
tff(fact_667_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,Mb: nat,K: 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)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K))) ) ).

% mono_nat_linear_lb
tff(fact_668_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(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_669_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(plus_plus_rat(A),C)) ) ) ).

% add_strict_increasing2
tff(fact_670_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_671_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_672_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(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_strict_increasing
tff(fact_673_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(plus_plus_rat(A),C)) ) ) ).

% add_strict_increasing
tff(fact_674_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_675_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_676_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(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_677_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(plus_plus_rat(A),B)) ) ) ).

% add_pos_nonneg
tff(fact_678_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_679_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_680_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(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_nonpos_neg
tff(fact_681_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(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_682_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_683_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_684_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(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_685_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(plus_plus_rat(A),B)) ) ) ).

% add_nonneg_pos
tff(fact_686_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_687_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_688_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(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_neg_nonpos
tff(fact_689_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(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_690_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_691_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_692_ex__least__nat__less,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,Na)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),Na)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),K2)
               => ~ aa_nat_o(P,I4) )
            & aa_nat_o(P,aa_nat_nat(suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_693_set__update__subsetI,axiom,
    ! [Xs: list_P6011104703257516679at_nat,A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,I: nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Xs)),A2)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A2)
       => aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,I,X))),A2) ) ) ).

% set_update_subsetI
tff(fact_694_set__update__subsetI,axiom,
    ! [Xs: list_real,A2: set_real,X: real,I: nat] :
      ( aa_set_real_o(ord_less_eq_set_real(set_real2(Xs)),A2)
     => ( aa_set_real_o(member_real(X),A2)
       => aa_set_real_o(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,X))),A2) ) ) ).

% set_update_subsetI
tff(fact_695_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A2: set_set_nat,X: set_nat,I: nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),A2)
     => ( aa_set_set_nat_o(member_set_nat(X),A2)
       => aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_set_nat2(list_update_set_nat(Xs,I,X))),A2) ) ) ).

% set_update_subsetI
tff(fact_696_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),A2)
     => ( member_VEBT_VEBT(X,A2)
       => ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,X)),A2) ) ) ).

% set_update_subsetI
tff(fact_697_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,X: nat,I: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Xs)),A2)
     => ( aa_set_nat_o(member_nat(X),A2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(list_update_nat(Xs,I,X))),A2) ) ) ).

% set_update_subsetI
tff(fact_698_set__update__subsetI,axiom,
    ! [Xs: list_int,A2: set_int,X: int,I: nat] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),A2)
     => ( aa_set_int_o(member_int(X),A2)
       => aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X))),A2) ) ) ).

% set_update_subsetI
tff(fact_699_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_700_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_701_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_702_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_703_le__cases3,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),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),X)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z) )
       => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),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),X) )
           => ( ( 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),X) )
             => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),X)
                 => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_704_le__cases3,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),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),X)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Z) )
       => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),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),X) )
           => ( ( 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),X) )
             => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),X)
                 => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_705_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),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),X)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z) )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),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),X) )
           => ( ( 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),X) )
             => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),X)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_706_le__cases3,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),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),X)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),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),X) )
           => ( ( 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),X) )
             => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),X)
                 => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_707_order__class_Oorder__eq__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
    <=> ( aa_set_int_o(ord_less_eq_set_int(X),Y)
        & aa_set_int_o(ord_less_eq_set_int(Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_708_order__class_Oorder__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_709_order__class_Oorder__eq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_710_order__class_Oorder__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_711_order__class_Oorder__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_712_ord__eq__le__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( A = B )
     => ( aa_set_int_o(ord_less_eq_set_int(B),C)
       => aa_set_int_o(ord_less_eq_set_int(A),C) ) ) ).

% ord_eq_le_trans
tff(fact_713_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_714_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_715_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_716_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_717_ord__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
     => ( ( B = C )
       => aa_set_int_o(ord_less_eq_set_int(A),C) ) ) ).

% ord_le_eq_trans
tff(fact_718_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_719_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_720_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_721_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_722_order__antisym,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(X),Y)
     => ( aa_set_int_o(ord_less_eq_set_int(Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_723_order__antisym,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_724_order__antisym,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_725_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_726_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_727_order_Otrans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
     => ( aa_set_int_o(ord_less_eq_set_int(B),C)
       => aa_set_int_o(ord_less_eq_set_int(A),C) ) ) ).

% order.trans
tff(fact_728_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_729_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_730_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_731_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_732_order__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(X),Y)
     => ( aa_set_int_o(ord_less_eq_set_int(Y),Z)
       => aa_set_int_o(ord_less_eq_set_int(X),Z) ) ) ).

% order_trans
tff(fact_733_order__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),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,X),Z) ) ) ).

% order_trans
tff(fact_734_order__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),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,X),Z) ) ) ).

% order_trans
tff(fact_735_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),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,X),Z) ) ) ).

% order_trans
tff(fact_736_order__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),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,X),Z) ) ) ).

% order_trans
tff(fact_737_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_738_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_739_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_740_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_741_dual__order_Oeq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
    <=> ( aa_set_int_o(ord_less_eq_set_int(B),A)
        & aa_set_int_o(ord_less_eq_set_int(A),B) ) ) ).

% dual_order.eq_iff
tff(fact_742_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_743_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_744_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_745_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_746_dual__order_Oantisym,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B),A)
     => ( aa_set_int_o(ord_less_eq_set_int(A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_747_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_748_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_749_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_750_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_751_dual__order_Otrans,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B),A)
     => ( aa_set_int_o(ord_less_eq_set_int(C),B)
       => aa_set_int_o(ord_less_eq_set_int(C),A) ) ) ).

% dual_order.trans
tff(fact_752_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_753_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_754_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_755_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_756_antisym,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
     => ( aa_set_int_o(ord_less_eq_set_int(B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_757_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_758_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_759_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_760_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_761_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
    <=> ( aa_set_int_o(ord_less_eq_set_int(A),B)
        & aa_set_int_o(ord_less_eq_set_int(B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_762_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_763_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_764_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_765_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_766_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),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_767_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),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_768_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)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),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_769_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)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),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_770_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),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_771_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),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_772_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)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X3)),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_773_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)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,X3)),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_774_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),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_775_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),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_776_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),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_777_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),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_778_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),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_779_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),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_780_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),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_781_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),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_782_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),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_783_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),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_784_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)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),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_785_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)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X3)),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_786_order__eq__refl,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
     => aa_set_int_o(ord_less_eq_set_int(X),Y) ) ).

% order_eq_refl
tff(fact_787_order__eq__refl,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% order_eq_refl
tff(fact_788_order__eq__refl,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y) ) ).

% order_eq_refl
tff(fact_789_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ).

% order_eq_refl
tff(fact_790_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ).

% order_eq_refl
tff(fact_791_linorder__linear,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) ) ).

% linorder_linear
tff(fact_792_linorder__linear,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
      | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X) ) ).

% linorder_linear
tff(fact_793_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ).

% linorder_linear
tff(fact_794_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ).

% linorder_linear
tff(fact_795_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),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_796_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),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_797_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),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_798_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),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_799_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),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_800_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),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_801_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),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_802_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),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_803_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)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),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_804_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)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X3)),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_805_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 )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),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_806_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 )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),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_807_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 )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),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_808_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 )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),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_809_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 )
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),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_810_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 )
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),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_811_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 )
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),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_812_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 )
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),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_813_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 )
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),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_814_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 )
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X3)),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_815_linorder__le__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) ) ).

% linorder_le_cases
tff(fact_816_linorder__le__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X) ) ).

% linorder_le_cases
tff(fact_817_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ).

% linorder_le_cases
tff(fact_818_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ).

% linorder_le_cases
tff(fact_819_order__antisym__conv,axiom,
    ! [Y: set_int,X: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Y),X)
     => ( aa_set_int_o(ord_less_eq_set_int(X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_820_order__antisym__conv,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_821_order__antisym__conv,axiom,
    ! [Y: num,X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_822_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_823_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_824_zero__reorient,axiom,
    ! [X: complex] :
      ( ( zero_zero_complex = X )
    <=> ( X = zero_zero_complex ) ) ).

% zero_reorient
tff(fact_825_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
    <=> ( X = zero_zero_real ) ) ).

% zero_reorient
tff(fact_826_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
    <=> ( X = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_827_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
    <=> ( X = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_828_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
    <=> ( X = zero_zero_int ) ) ).

% zero_reorient
tff(fact_829_lt__ex,axiom,
    ! [X: real] :
    ? [Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),X) ).

% lt_ex
tff(fact_830_lt__ex,axiom,
    ! [X: rat] :
    ? [Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),X) ).

% lt_ex
tff(fact_831_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),X) ).

% lt_ex
tff(fact_832_gt__ex,axiom,
    ! [X: real] :
    ? [X_1: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X),X_1) ).

% gt_ex
tff(fact_833_gt__ex,axiom,
    ! [X: rat] :
    ? [X_1: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),X_1) ).

% gt_ex
tff(fact_834_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),X_1) ).

% gt_ex
tff(fact_835_gt__ex,axiom,
    ! [X: int] :
    ? [X_1: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,X),X_1) ).

% gt_ex
tff(fact_836_dense,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => ? [Z2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),Y) ) ) ).

% dense
tff(fact_837_dense,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ? [Z2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),Y) ) ) ).

% dense
tff(fact_838_less__imp__neq,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_839_less__imp__neq,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_840_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_841_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_842_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_843_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_844_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_845_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_846_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_847_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_848_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_849_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_850_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_851_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_852_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_853_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_854_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_855_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_856_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_857_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_858_less__induct,axiom,
    ! [P: fun_nat_o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),X3)
             => aa_nat_o(P,Y4) )
         => aa_nat_o(P,X3) )
     => aa_nat_o(P,A) ) ).

% less_induct
tff(fact_859_antisym__conv3,axiom,
    ! [Y: real,X: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_860_antisym__conv3,axiom,
    ! [Y: rat,X: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_861_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_862_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_863_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_864_linorder__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => ( ( X != Y )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ) ) ).

% linorder_cases
tff(fact_865_linorder__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( ( X != Y )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ) ).

% linorder_cases
tff(fact_866_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( ( X != Y )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ) ).

% linorder_cases
tff(fact_867_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( ( X != Y )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ) ).

% linorder_cases
tff(fact_868_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( ( X != Y )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ) ).

% linorder_cases
tff(fact_869_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_870_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_871_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_872_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_873_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_874_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% dual_order.irrefl
tff(fact_875_dual__order_Oirrefl,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% dual_order.irrefl
tff(fact_876_dual__order_Oirrefl,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% dual_order.irrefl
tff(fact_877_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% dual_order.irrefl
tff(fact_878_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% dual_order.irrefl
tff(fact_879_exists__least__iff,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_13: nat] : aa_nat_o(P,X_13)
    <=> ? [N3: nat] :
          ( aa_nat_o(P,N3)
          & ! [M5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),N3)
             => ~ aa_nat_o(P,M5) ) ) ) ).

% exists_least_iff
tff(fact_880_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_881_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_882_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_883_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_884_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_885_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_886_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_887_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_888_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_889_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_890_not__less__iff__gr__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X)
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_891_not__less__iff__gr__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X)
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_892_not__less__iff__gr__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X)
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_893_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_894_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X)
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_895_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_896_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_897_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_898_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_899_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_900_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_901_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_902_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_903_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_904_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_905_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_906_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_907_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_908_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_909_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_910_linorder__neqE,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ) ) ).

% linorder_neqE
tff(fact_911_linorder__neqE,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ) ).

% linorder_neqE
tff(fact_912_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ) ).

% linorder_neqE
tff(fact_913_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ) ).

% linorder_neqE
tff(fact_914_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ) ).

% linorder_neqE
tff(fact_915_order__less__asym,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ) ).

% order_less_asym
tff(fact_916_order__less__asym,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ).

% order_less_asym
tff(fact_917_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ).

% order_less_asym
tff(fact_918_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% order_less_asym
tff(fact_919_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% order_less_asym
tff(fact_920_linorder__neq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ) ) ).

% linorder_neq_iff
tff(fact_921_linorder__neq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ) ).

% linorder_neq_iff
tff(fact_922_linorder__neq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ) ).

% linorder_neq_iff
tff(fact_923_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ) ).

% linorder_neq_iff
tff(fact_924_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ) ).

% linorder_neq_iff
tff(fact_925_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_926_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_927_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_928_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_929_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_930_order__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),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,X),Z) ) ) ).

% order_less_trans
tff(fact_931_order__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),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,X),Z) ) ) ).

% order_less_trans
tff(fact_932_order__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),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,X),Z) ) ) ).

% order_less_trans
tff(fact_933_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),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,X),Z) ) ) ).

% order_less_trans
tff(fact_934_order__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),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,X),Z) ) ) ).

% order_less_trans
tff(fact_935_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)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),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_936_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)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),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_937_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)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X3)),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_938_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)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X3)),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_939_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)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X3)),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_940_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),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_941_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),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_942_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X3)),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_943_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X3)),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_944_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X3)),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_945_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 )
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),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_946_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 )
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),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_947_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 )
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X3)),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_948_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 )
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X3)),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_949_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 )
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X3)),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_950_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 )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),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_951_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 )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),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_952_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 )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X3)),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_953_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 )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X3)),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_954_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 )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X3)),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_955_order__less__irrefl,axiom,
    ! [X: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),X) ).

% order_less_irrefl
tff(fact_956_order__less__irrefl,axiom,
    ! [X: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),X) ).

% order_less_irrefl
tff(fact_957_order__less__irrefl,axiom,
    ! [X: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),X) ).

% order_less_irrefl
tff(fact_958_order__less__irrefl,axiom,
    ! [X: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),X) ).

% order_less_irrefl
tff(fact_959_order__less__irrefl,axiom,
    ! [X: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),X) ).

% order_less_irrefl
tff(fact_960_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)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),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_961_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),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_962_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X3)),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_963_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)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X3)),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_964_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)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),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_965_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)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),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_966_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),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_967_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X3)),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_968_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)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),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_969_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)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),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_970_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)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),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_971_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)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),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_972_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)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X3)),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_973_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)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X3)),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_974_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)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X3)),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_975_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),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_976_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),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_977_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X3)),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_978_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X3)),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_979_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X3)),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_980_order__less__not__sym,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ) ).

% order_less_not_sym
tff(fact_981_order__less__not__sym,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ).

% order_less_not_sym
tff(fact_982_order__less__not__sym,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ).

% order_less_not_sym
tff(fact_983_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% order_less_not_sym
tff(fact_984_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% order_less_not_sym
tff(fact_985_order__less__imp__triv,axiom,
    ! [X: real,Y: real,P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_986_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_987_order__less__imp__triv,axiom,
    ! [X: num,Y: num,P: $o] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_988_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_989_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_990_linorder__less__linear,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
      | ( X = Y )
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ) ).

% linorder_less_linear
tff(fact_991_linorder__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
      | ( X = Y )
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ).

% linorder_less_linear
tff(fact_992_linorder__less__linear,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
      | ( X = Y )
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ).

% linorder_less_linear
tff(fact_993_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
      | ( X = Y )
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% linorder_less_linear
tff(fact_994_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
      | ( X = Y )
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% linorder_less_linear
tff(fact_995_order__less__imp__not__eq,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_996_order__less__imp__not__eq,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_997_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_998_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_999_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1000_order__less__imp__not__eq2,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_1001_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_1002_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_1003_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_1004_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_1005_order__less__imp__not__less,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ) ).

% order_less_imp_not_less
tff(fact_1006_order__less__imp__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ).

% order_less_imp_not_less
tff(fact_1007_order__less__imp__not__less,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ).

% order_less_imp_not_less
tff(fact_1008_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% order_less_imp_not_less
tff(fact_1009_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% order_less_imp_not_less
tff(fact_1010_nat__neq__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( Mb != Na )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb) ) ) ).

% nat_neq_iff
tff(fact_1011_less__not__refl,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Na) ).

% less_not_refl
tff(fact_1012_less__not__refl2,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb)
     => ( Mb != Na ) ) ).

% less_not_refl2
tff(fact_1013_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S),T)
     => ( S != T ) ) ).

% less_not_refl3
tff(fact_1014_less__irrefl__nat,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Na) ).

% less_irrefl_nat
tff(fact_1015_nat__less__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( ! [N: nat] :
          ( ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N)
             => aa_nat_o(P,M4) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Na) ) ).

% nat_less_induct
tff(fact_1016_infinite__descent,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(P,N)
         => ? [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N)
              & ~ aa_nat_o(P,M4) ) )
     => aa_nat_o(P,Na) ) ).

% infinite_descent
tff(fact_1017_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ) ).

% linorder_neqE_nat
tff(fact_1018_le__refl,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Na) ).

% le_refl
tff(fact_1019_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K) ) ) ).

% le_trans
tff(fact_1020_eq__imp__le,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( Mb = Na )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% eq_imp_le
tff(fact_1021_le__antisym,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
       => ( Mb = Na ) ) ) ).

% le_antisym
tff(fact_1022_nat__le__linear,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb) ) ).

% nat_le_linear
tff(fact_1023_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y3: nat] :
            ( aa_nat_o(P,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => ? [X3: nat] :
            ( aa_nat_o(P,X3)
            & ! [Y4: nat] :
                ( aa_nat_o(P,Y4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),X3) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_1024_size__neq__size__imp__neq,axiom,
    ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,X) != aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1025_size__neq__size__imp__neq,axiom,
    ! [X: list_o,Y: list_o] :
      ( ( aa_list_o_nat(size_size_list_o,X) != aa_list_o_nat(size_size_list_o,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1026_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,X) != aa_list_nat_nat(size_size_list_nat,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1027_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( aa_list_int_nat(size_size_list_int,X) != aa_list_int_nat(size_size_list_int,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1028_size__neq__size__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( aa_num_nat(size_size_num,X) != aa_num_nat(size_size_num,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1029_zero__le,axiom,
    ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X) ).

% zero_le
tff(fact_1030_leD,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ).

% leD
tff(fact_1031_leD,axiom,
    ! [Y: set_int,X: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Y),X)
     => ~ aa_set_int_o(ord_less_set_int(X),Y) ) ).

% leD
tff(fact_1032_leD,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ).

% leD
tff(fact_1033_leD,axiom,
    ! [Y: num,X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y) ) ).

% leD
tff(fact_1034_leD,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ).

% leD
tff(fact_1035_leD,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ).

% leD
tff(fact_1036_leI,axiom,
    ! [X: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X) ) ).

% leI
tff(fact_1037_leI,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) ) ).

% leI
tff(fact_1038_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X) ) ).

% leI
tff(fact_1039_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ).

% leI
tff(fact_1040_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ).

% leI
tff(fact_1041_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_1042_nless__le,axiom,
    ! [A: set_int,B: set_int] :
      ( ~ aa_set_int_o(ord_less_set_int(A),B)
    <=> ( ~ aa_set_int_o(ord_less_eq_set_int(A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1043_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_1044_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_1045_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_1046_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_1047_antisym__conv1,axiom,
    ! [X: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_1048_antisym__conv1,axiom,
    ! [X: set_int,Y: set_int] :
      ( ~ aa_set_int_o(ord_less_set_int(X),Y)
     => ( aa_set_int_o(ord_less_eq_set_int(X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_1049_antisym__conv1,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_1050_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_1051_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_1052_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_1053_antisym__conv2,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_1054_antisym__conv2,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(X),Y)
     => ( ~ aa_set_int_o(ord_less_set_int(X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_1055_antisym__conv2,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_1056_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_1057_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_1058_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_1059_dense__ge,axiom,
    ! [Z: real,Y: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X3) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ).

% dense_ge
tff(fact_1060_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).

% dense_ge
tff(fact_1061_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X3: 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),Z) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ).

% dense_le
tff(fact_1062_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X3: 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),Z) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).

% dense_le
tff(fact_1063_less__le__not__le,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X) ) ) ).

% less_le_not_le
tff(fact_1064_less__le__not__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_set_int(X),Y)
    <=> ( aa_set_int_o(ord_less_eq_set_int(X),Y)
        & ~ aa_set_int_o(ord_less_eq_set_int(Y),X) ) ) ).

% less_le_not_le
tff(fact_1065_less__le__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) ) ) ).

% less_le_not_le
tff(fact_1066_less__le__not__le,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X) ) ) ).

% less_le_not_le
tff(fact_1067_less__le__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ) ).

% less_le_not_le
tff(fact_1068_less__le__not__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ) ).

% less_le_not_le
tff(fact_1069_not__le__imp__less,axiom,
    ! [Y: real,X: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ).

% not_le_imp_less
tff(fact_1070_not__le__imp__less,axiom,
    ! [Y: rat,X: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ).

% not_le_imp_less
tff(fact_1071_not__le__imp__less,axiom,
    ! [Y: num,X: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y) ) ).

% not_le_imp_less
tff(fact_1072_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ).

% not_le_imp_less
tff(fact_1073_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ).

% not_le_imp_less
tff(fact_1074_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_1075_order_Oorder__iff__strict,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
    <=> ( aa_set_int_o(ord_less_set_int(A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1076_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_1077_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_1078_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_1079_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_1080_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_1081_order_Ostrict__iff__order,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1082_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_1083_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_1084_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_1085_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_1086_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_1087_order_Ostrict__trans1,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
     => ( aa_set_int_o(ord_less_set_int(B),C)
       => aa_set_int_o(ord_less_set_int(A),C) ) ) ).

% order.strict_trans1
tff(fact_1088_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_1089_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_1090_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_1091_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_1092_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_1093_order_Ostrict__trans2,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B)
     => ( aa_set_int_o(ord_less_eq_set_int(B),C)
       => aa_set_int_o(ord_less_set_int(A),C) ) ) ).

% order.strict_trans2
tff(fact_1094_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_1095_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_1096_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_1097_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_1098_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_1099_order_Ostrict__iff__not,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A),B)
        & ~ aa_set_int_o(ord_less_eq_set_int(B),A) ) ) ).

% order.strict_iff_not
tff(fact_1100_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_1101_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_1102_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_1103_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_1104_dense__ge__bounded,axiom,
    ! [Z: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X)
     => ( ! [W: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),W)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),X)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),W) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ) ).

% dense_ge_bounded
tff(fact_1105_dense__ge__bounded,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X)
     => ( ! [W: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),W)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),X)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),W) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).

% dense_ge_bounded
tff(fact_1106_dense__le__bounded,axiom,
    ! [X: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => ( ! [W: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),W)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Y)
             => 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,Y),Z) ) ) ).

% dense_le_bounded
tff(fact_1107_dense__le__bounded,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( ! [W: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),W)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Y)
             => 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,Y),Z) ) ) ).

% dense_le_bounded
tff(fact_1108_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_1109_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B),A)
    <=> ( aa_set_int_o(ord_less_set_int(B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1110_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_1111_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_1112_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_1113_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_1114_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_1115_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_set_int(B),A)
    <=> ( aa_set_int_o(ord_less_eq_set_int(B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1116_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_1117_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_1118_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_1119_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_1120_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_1121_dual__order_Ostrict__trans1,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B),A)
     => ( aa_set_int_o(ord_less_set_int(C),B)
       => aa_set_int_o(ord_less_set_int(C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1122_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_1123_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_1124_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_1125_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_1126_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_1127_dual__order_Ostrict__trans2,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_set_int(B),A)
     => ( aa_set_int_o(ord_less_eq_set_int(C),B)
       => aa_set_int_o(ord_less_set_int(C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1128_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_1129_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_1130_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_1131_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_1132_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_1133_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_set_int(B),A)
    <=> ( aa_set_int_o(ord_less_eq_set_int(B),A)
        & ~ aa_set_int_o(ord_less_eq_set_int(A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1134_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_1135_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_1136_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_1137_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_1138_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_1139_order_Ostrict__implies__order,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B)
     => aa_set_int_o(ord_less_eq_set_int(A),B) ) ).

% order.strict_implies_order
tff(fact_1140_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_1141_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_1142_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_1143_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_1144_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_1145_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_set_int(B),A)
     => aa_set_int_o(ord_less_eq_set_int(B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1146_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_1147_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_1148_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_1149_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_1150_order__le__less,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_1151_order__le__less,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(X),Y)
    <=> ( aa_set_int_o(ord_less_set_int(X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_1152_order__le__less,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_1153_order__le__less,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_1154_order__le__less,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_1155_order__le__less,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_1156_order__less__le,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_1157_order__less__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_set_int(X),Y)
    <=> ( aa_set_int_o(ord_less_eq_set_int(X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_1158_order__less__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_1159_order__less__le,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_1160_order__less__le,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_1161_order__less__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_1162_linorder__not__le,axiom,
    ! [X: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ) ).

% linorder_not_le
tff(fact_1163_linorder__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ).

% linorder_not_le
tff(fact_1164_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ).

% linorder_not_le
tff(fact_1165_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% linorder_not_le
tff(fact_1166_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% linorder_not_le
tff(fact_1167_linorder__not__less,axiom,
    ! [X: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X) ) ).

% linorder_not_less
tff(fact_1168_linorder__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) ) ).

% linorder_not_less
tff(fact_1169_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X) ) ).

% linorder_not_less
tff(fact_1170_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ).

% linorder_not_less
tff(fact_1171_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ).

% linorder_not_less
tff(fact_1172_order__less__imp__le,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ).

% order_less_imp_le
tff(fact_1173_order__less__imp__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_set_int(X),Y)
     => aa_set_int_o(ord_less_eq_set_int(X),Y) ) ).

% order_less_imp_le
tff(fact_1174_order__less__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% order_less_imp_le
tff(fact_1175_order__less__imp__le,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y) ) ).

% order_less_imp_le
tff(fact_1176_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ).

% order_less_imp_le
tff(fact_1177_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ).

% order_less_imp_le
tff(fact_1178_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_1179_order__le__neq__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
     => ( ( A != B )
       => aa_set_int_o(ord_less_set_int(A),B) ) ) ).

% order_le_neq_trans
tff(fact_1180_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_1181_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_1182_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_1183_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_1184_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_1185_order__neq__le__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A != B )
     => ( aa_set_int_o(ord_less_eq_set_int(A),B)
       => aa_set_int_o(ord_less_set_int(A),B) ) ) ).

% order_neq_le_trans
tff(fact_1186_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_1187_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_1188_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_1189_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_1190_order__le__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),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,X),Z) ) ) ).

% order_le_less_trans
tff(fact_1191_order__le__less__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(X),Y)
     => ( aa_set_int_o(ord_less_set_int(Y),Z)
       => aa_set_int_o(ord_less_set_int(X),Z) ) ) ).

% order_le_less_trans
tff(fact_1192_order__le__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),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,X),Z) ) ) ).

% order_le_less_trans
tff(fact_1193_order__le__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),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,X),Z) ) ) ).

% order_le_less_trans
tff(fact_1194_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),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,X),Z) ) ) ).

% order_le_less_trans
tff(fact_1195_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),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,X),Z) ) ) ).

% order_le_less_trans
tff(fact_1196_order__less__le__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),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,X),Z) ) ) ).

% order_less_le_trans
tff(fact_1197_order__less__le__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_set_int(X),Y)
     => ( aa_set_int_o(ord_less_eq_set_int(Y),Z)
       => aa_set_int_o(ord_less_set_int(X),Z) ) ) ).

% order_less_le_trans
tff(fact_1198_order__less__le__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),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,X),Z) ) ) ).

% order_less_le_trans
tff(fact_1199_order__less__le__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),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,X),Z) ) ) ).

% order_less_le_trans
tff(fact_1200_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),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,X),Z) ) ) ).

% order_less_le_trans
tff(fact_1201_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),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,X),Z) ) ) ).

% order_less_le_trans
tff(fact_1202_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)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),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_1203_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),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_1204_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X3)),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_1205_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)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X3)),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_1206_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)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),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_1207_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)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),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_1208_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),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_1209_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X3)),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_1210_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)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),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_1211_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)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),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_1212_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X3)),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_1213_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),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_1214_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),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_1215_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),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_1216_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),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_1217_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X3)),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_1218_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),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_1219_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),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_1220_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),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_1221_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),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_1222_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X3)),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_1223_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),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_1224_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),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_1225_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),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_1226_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),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_1227_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X3)),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_1228_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),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_1229_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),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_1230_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),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_1231_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),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_1232_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)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),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_1233_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),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_1234_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X3)),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_1235_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)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X3)),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_1236_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)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),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_1237_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)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),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_1238_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)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),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_1239_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)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X3)),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_1240_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)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),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_1241_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)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),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_1242_linorder__le__less__linear,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ) ).

% linorder_le_less_linear
tff(fact_1243_linorder__le__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ).

% linorder_le_less_linear
tff(fact_1244_linorder__le__less__linear,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ).

% linorder_le_less_linear
tff(fact_1245_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% linorder_le_less_linear
tff(fact_1246_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% linorder_le_less_linear
tff(fact_1247_order__le__imp__less__or__eq,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1248_order__le__imp__less__or__eq,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(X),Y)
     => ( aa_set_int_o(ord_less_set_int(X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1249_order__le__imp__less__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1250_order__le__imp__less__or__eq,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1251_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1252_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1253_gr__zeroI,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% gr_zeroI
tff(fact_1254_not__less__zero,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% not_less_zero
tff(fact_1255_gr__implies__not__zero,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
     => ( Na != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_1256_zero__less__iff__neq__zero,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ( Na != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_1257_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_1258_gr0I,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% gr0I
tff(fact_1259_not__gr0,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% not_gr0
tff(fact_1260_not__less0,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% not_less0
tff(fact_1261_less__zeroE,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% less_zeroE
tff(fact_1262_gr__implies__not0,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
     => ( Na != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_1263_infinite__descent0,axiom,
    ! [P: fun_nat_o,Na: 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)
             => ? [M4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N)
                  & ~ aa_nat_o(P,M4) ) ) )
       => aa_nat_o(P,Na) ) ) ).

% infinite_descent0
tff(fact_1264_less__eq__nat_Osimps_I1_J,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Na) ).

% less_eq_nat.simps(1)
tff(fact_1265_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_1266_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_1267_le__0__eq,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),zero_zero_nat)
    <=> ( Na = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_1268_nat__less__le,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
        & ( Mb != Na ) ) ) ).

% nat_less_le
tff(fact_1269_less__imp__le__nat,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% less_imp_le_nat
tff(fact_1270_le__eq__less__or__eq,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
        | ( Mb = Na ) ) ) ).

% le_eq_less_or_eq
tff(fact_1271_less__or__eq__imp__le,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
        | ( Mb = Na ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% less_or_eq_imp_le
tff(fact_1272_le__neq__implies__less,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( ( Mb != Na )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ) ).

% le_neq_implies_less
tff(fact_1273_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_1274_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_1275_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_1276_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_1277_assms_I1_J,axiom,
    aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary)),deg) ).

% assms(1)
tff(fact_1278_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_mint(T) = none_nat )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
tff(fact_1279_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( vEBT_vebt_maxt(T) = none_nat )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
tff(fact_1280_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ).

% field_le_epsilon
tff(fact_1281_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(plus_plus_rat(Y),E)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% field_le_epsilon
tff(fact_1282_option_Osize__gen_I2_J,axiom,
    ! [X: fun_nat_nat,X22: nat] : size_option_nat(X,some_nat(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(X,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_1283_option_Osize__gen_I2_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat,X22: product_prod_nat_nat] : size_o8335143837870341156at_nat(X,some_P7363390416028606310at_nat(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(X,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_1284_option_Osize__gen_I2_J,axiom,
    ! [X: fun_num_nat,X22: num] : size_option_num(X,some_num(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(X,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_1285_add__less__zeroD,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real) ) ) ).

% add_less_zeroD
tff(fact_1286_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(X),Y)),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_1287_add__less__zeroD,axiom,
    ! [X: 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,X),Y)),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_1288_empty__subsetI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(bot_bo2099793752762293965at_nat),A2) ).

% empty_subsetI
tff(fact_1289_empty__subsetI,axiom,
    ! [A2: set_real] : aa_set_real_o(ord_less_eq_set_real(bot_bot_set_real),A2) ).

% empty_subsetI
tff(fact_1290_empty__subsetI,axiom,
    ! [A2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,bot_bot_set_nat),A2) ).

% empty_subsetI
tff(fact_1291_empty__subsetI,axiom,
    ! [A2: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A2) ).

% empty_subsetI
tff(fact_1292_subset__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),bot_bo2099793752762293965at_nat)
    <=> ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% subset_empty
tff(fact_1293_subset__empty,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),bot_bot_set_real)
    <=> ( A2 = bot_bot_set_real ) ) ).

% subset_empty
tff(fact_1294_subset__empty,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),bot_bot_set_nat)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_1295_subset__empty,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),bot_bot_set_int)
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_1296_empty__iff,axiom,
    ! [C: set_nat] : ~ aa_set_set_nat_o(member_set_nat(C),bot_bot_set_set_nat) ).

% empty_iff
tff(fact_1297_empty__iff,axiom,
    ! [C: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),bot_bo2099793752762293965at_nat) ).

% empty_iff
tff(fact_1298_empty__iff,axiom,
    ! [C: real] : ~ aa_set_real_o(member_real(C),bot_bot_set_real) ).

% empty_iff
tff(fact_1299_empty__iff,axiom,
    ! [C: nat] : ~ aa_set_nat_o(member_nat(C),bot_bot_set_nat) ).

% empty_iff
tff(fact_1300_empty__iff,axiom,
    ! [C: int] : ~ aa_set_int_o(member_int(C),bot_bot_set_int) ).

% empty_iff
tff(fact_1301_all__not__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ! [X4: set_nat] : ~ aa_set_set_nat_o(member_set_nat(X4),A2)
    <=> ( A2 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
tff(fact_1302_all__not__in__conv,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ! [X4: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A2)
    <=> ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% all_not_in_conv
tff(fact_1303_all__not__in__conv,axiom,
    ! [A2: set_real] :
      ( ! [X4: real] : ~ aa_set_real_o(member_real(X4),A2)
    <=> ( A2 = bot_bot_set_real ) ) ).

% all_not_in_conv
tff(fact_1304_all__not__in__conv,axiom,
    ! [A2: set_nat] :
      ( ! [X4: nat] : ~ aa_set_nat_o(member_nat(X4),A2)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_1305_all__not__in__conv,axiom,
    ! [A2: set_int] :
      ( ! [X4: int] : ~ aa_set_int_o(member_int(X4),A2)
    <=> ( A2 = bot_bot_set_int ) ) ).

% all_not_in_conv
tff(fact_1306_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya)),Na)
     => ( Dega = Na ) ) ).

% deg_deg_n
tff(fact_1307_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,Na)))
     => ? [Info2: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,Na)),TreeList,S2) ) ).

% deg_SUcn_Node
tff(fact_1308_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_1309_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_1310_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_1311_empty__Collect__eq,axiom,
    ! [P: fun_real_o] :
      ( ( bot_bot_set_real = collect_real(P) )
    <=> ! [X4: real] : ~ aa_real_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_1312_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_1313_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_1314_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_1315_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_1316_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_1317_Collect__empty__eq,axiom,
    ! [P: fun_real_o] :
      ( ( collect_real(P) = bot_bot_set_real )
    <=> ! [X4: real] : ~ aa_real_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_1318_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_1319_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_1320_psubsetI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( ( A2 != B2 )
       => aa_set_int_o(ord_less_set_int(A2),B2) ) ) ).

% psubsetI
tff(fact_1321_subset__antisym,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),A2)
       => ( A2 = B2 ) ) ) ).

% subset_antisym
tff(fact_1322_subsetI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A2)
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),B2) )
     => aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),B2) ) ).

% subsetI
tff(fact_1323_subsetI,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),A2)
         => aa_set_real_o(member_real(X3),B2) )
     => aa_set_real_o(ord_less_eq_set_real(A2),B2) ) ).

% subsetI
tff(fact_1324_subsetI,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ! [X3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X3),A2)
         => aa_set_set_nat_o(member_set_nat(X3),B2) )
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2) ) ).

% subsetI
tff(fact_1325_subsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A2)
         => aa_set_nat_o(member_nat(X3),B2) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2) ) ).

% subsetI
tff(fact_1326_subsetI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A2)
         => aa_set_int_o(member_int(X3),B2) )
     => aa_set_int_o(ord_less_eq_set_int(A2),B2) ) ).

% subsetI
tff(fact_1327_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 )
       => ( ! [X2: vEBT_VEBT] :
              ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeLista))
             => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_12) )
          & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_12) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_1328_not__Some__eq,axiom,
    ! [X: option_nat] :
      ( ! [Y5: nat] : X != some_nat(Y5)
    <=> ( X = none_nat ) ) ).

% not_Some_eq
tff(fact_1329_not__Some__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ! [Y5: product_prod_nat_nat] : X != some_P7363390416028606310at_nat(Y5)
    <=> ( X = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_1330_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ! [Y5: num] : X != some_num(Y5)
    <=> ( X = none_num ) ) ).

% not_Some_eq
tff(fact_1331_not__None__eq,axiom,
    ! [X: option_nat] :
      ( ( X != none_nat )
    <=> ? [Y5: nat] : X = some_nat(Y5) ) ).

% not_None_eq
tff(fact_1332_not__None__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( X != none_P5556105721700978146at_nat )
    <=> ? [Y5: product_prod_nat_nat] : X = some_P7363390416028606310at_nat(Y5) ) ).

% not_None_eq
tff(fact_1333_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
    <=> ? [Y5: num] : X = some_num(Y5) ) ).

% not_None_eq
tff(fact_1334_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( some_nat(the_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_1335_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_1336_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( some_num(the_num(Option)) = Option ) ) ).

% option.collapse
tff(fact_1337_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_1338_psubsetE,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B2)
     => ~ ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
         => aa_set_int_o(ord_less_eq_set_int(B2),A2) ) ) ).

% psubsetE
tff(fact_1339_psubset__eq,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B2)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
        & ( A2 != B2 ) ) ) ).

% psubset_eq
tff(fact_1340_psubset__imp__subset,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B2)
     => aa_set_int_o(ord_less_eq_set_int(A2),B2) ) ).

% psubset_imp_subset
tff(fact_1341_psubset__subset__trans,axiom,
    ! [A2: set_int,B2: set_int,C4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B2)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),C4)
       => aa_set_int_o(ord_less_set_int(A2),C4) ) ) ).

% psubset_subset_trans
tff(fact_1342_subset__not__subset__eq,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B2)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
        & ~ aa_set_int_o(ord_less_eq_set_int(B2),A2) ) ) ).

% subset_not_subset_eq
tff(fact_1343_subset__psubset__trans,axiom,
    ! [A2: set_int,B2: set_int,C4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( aa_set_int_o(ord_less_set_int(B2),C4)
       => aa_set_int_o(ord_less_set_int(A2),C4) ) ) ).

% subset_psubset_trans
tff(fact_1344_subset__iff__psubset__eq,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
    <=> ( aa_set_int_o(ord_less_set_int(A2),B2)
        | ( A2 = B2 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1345_Collect__mono__iff,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( aa_set_real_o(ord_less_eq_set_real(collect_real(P)),collect_real(Q))
    <=> ! [X4: real] :
          ( aa_real_o(P,X4)
         => aa_real_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_1346_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_1347_Collect__mono__iff,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(P)),collect_set_nat(Q))
    <=> ! [X4: set_nat] :
          ( aa_set_nat_o(P,X4)
         => aa_set_nat_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_1348_Collect__mono__iff,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(P)),collect_nat(Q))
    <=> ! [X4: nat] :
          ( aa_nat_o(P,X4)
         => aa_nat_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_1349_Collect__mono__iff,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(collect_int(P)),collect_int(Q))
    <=> ! [X4: int] :
          ( aa_int_o(P,X4)
         => aa_int_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_1350_set__eq__subset,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( A2 = B2 )
    <=> ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
        & aa_set_int_o(ord_less_eq_set_int(B2),A2) ) ) ).

% set_eq_subset
tff(fact_1351_subset__trans,axiom,
    ! [A2: set_int,B2: set_int,C4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),C4)
       => aa_set_int_o(ord_less_eq_set_int(A2),C4) ) ) ).

% subset_trans
tff(fact_1352_Collect__mono,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( ! [X3: real] :
          ( aa_real_o(P,X3)
         => aa_real_o(Q,X3) )
     => aa_set_real_o(ord_less_eq_set_real(collect_real(P)),collect_real(Q)) ) ).

% Collect_mono
tff(fact_1353_Collect__mono,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ! [X3: list_nat] :
          ( aa_list_nat_o(P,X3)
         => aa_list_nat_o(Q,X3) )
     => ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q)) ) ).

% Collect_mono
tff(fact_1354_Collect__mono,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X3: set_nat] :
          ( aa_set_nat_o(P,X3)
         => aa_set_nat_o(Q,X3) )
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(P)),collect_set_nat(Q)) ) ).

% Collect_mono
tff(fact_1355_Collect__mono,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(P,X3)
         => aa_nat_o(Q,X3) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(P)),collect_nat(Q)) ) ).

% Collect_mono
tff(fact_1356_Collect__mono,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( aa_int_o(P,X3)
         => aa_int_o(Q,X3) )
     => aa_set_int_o(ord_less_eq_set_int(collect_int(P)),collect_int(Q)) ) ).

% Collect_mono
tff(fact_1357_subset__refl,axiom,
    ! [A2: set_int] : aa_set_int_o(ord_less_eq_set_int(A2),A2) ).

% subset_refl
tff(fact_1358_subset__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),B2)
    <=> ! [T2: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(T2),A2)
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(T2),B2) ) ) ).

% subset_iff
tff(fact_1359_subset__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
    <=> ! [T2: real] :
          ( aa_set_real_o(member_real(T2),A2)
         => aa_set_real_o(member_real(T2),B2) ) ) ).

% subset_iff
tff(fact_1360_subset__iff,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
    <=> ! [T2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(T2),A2)
         => aa_set_set_nat_o(member_set_nat(T2),B2) ) ) ).

% subset_iff
tff(fact_1361_subset__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
    <=> ! [T2: nat] :
          ( aa_set_nat_o(member_nat(T2),A2)
         => aa_set_nat_o(member_nat(T2),B2) ) ) ).

% subset_iff
tff(fact_1362_subset__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
    <=> ! [T2: int] :
          ( aa_set_int_o(member_int(T2),A2)
         => aa_set_int_o(member_int(T2),B2) ) ) ).

% subset_iff
tff(fact_1363_equalityD2,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( A2 = B2 )
     => aa_set_int_o(ord_less_eq_set_int(B2),A2) ) ).

% equalityD2
tff(fact_1364_equalityD1,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( A2 = B2 )
     => aa_set_int_o(ord_less_eq_set_int(A2),B2) ) ).

% equalityD1
tff(fact_1365_subset__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),B2)
    <=> ! [X4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A2)
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),B2) ) ) ).

% subset_eq
tff(fact_1366_subset__eq,axiom,
    ! [A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
    <=> ! [X4: real] :
          ( aa_set_real_o(member_real(X4),A2)
         => aa_set_real_o(member_real(X4),B2) ) ) ).

% subset_eq
tff(fact_1367_subset__eq,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
    <=> ! [X4: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X4),A2)
         => aa_set_set_nat_o(member_set_nat(X4),B2) ) ) ).

% subset_eq
tff(fact_1368_subset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A2)
         => aa_set_nat_o(member_nat(X4),B2) ) ) ).

% subset_eq
tff(fact_1369_subset__eq,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A2)
         => aa_set_int_o(member_int(X4),B2) ) ) ).

% subset_eq
tff(fact_1370_equalityE,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( A2 = B2 )
     => ~ ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
         => ~ aa_set_int_o(ord_less_eq_set_int(B2),A2) ) ) ).

% equalityE
tff(fact_1371_subsetD,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),B2)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),A2)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),B2) ) ) ).

% subsetD
tff(fact_1372_subsetD,axiom,
    ! [A2: set_real,B2: set_real,C: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
     => ( aa_set_real_o(member_real(C),A2)
       => aa_set_real_o(member_real(C),B2) ) ) ).

% subsetD
tff(fact_1373_subsetD,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
     => ( aa_set_set_nat_o(member_set_nat(C),A2)
       => aa_set_set_nat_o(member_set_nat(C),B2) ) ) ).

% subsetD
tff(fact_1374_subsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( aa_set_nat_o(member_nat(C),A2)
       => aa_set_nat_o(member_nat(C),B2) ) ) ).

% subsetD
tff(fact_1375_subsetD,axiom,
    ! [A2: set_int,B2: set_int,C: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( aa_set_int_o(member_int(C),A2)
       => aa_set_int_o(member_int(C),B2) ) ) ).

% subsetD
tff(fact_1376_in__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),B2)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A2)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),B2) ) ) ).

% in_mono
tff(fact_1377_in__mono,axiom,
    ! [A2: set_real,B2: set_real,X: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
     => ( aa_set_real_o(member_real(X),A2)
       => aa_set_real_o(member_real(X),B2) ) ) ).

% in_mono
tff(fact_1378_in__mono,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,X: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
     => ( aa_set_set_nat_o(member_set_nat(X),A2)
       => aa_set_set_nat_o(member_set_nat(X),B2) ) ) ).

% in_mono
tff(fact_1379_in__mono,axiom,
    ! [A2: set_nat,B2: set_nat,X: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( aa_set_nat_o(member_nat(X),A2)
       => aa_set_nat_o(member_nat(X),B2) ) ) ).

% in_mono
tff(fact_1380_in__mono,axiom,
    ! [A2: set_int,B2: set_int,X: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( aa_set_int_o(member_int(X),A2)
       => aa_set_int_o(member_int(X),B2) ) ) ).

% in_mono
tff(fact_1381_bot__set__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(bot_bo8828579575001852285_nat_o) ).

% bot_set_def
tff(fact_1382_bot__set__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(bot_bo3967814874310145751_nat_o) ).

% bot_set_def
tff(fact_1383_bot__set__def,axiom,
    bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(bot_bo5074786513356115630_nat_o) ).

% bot_set_def
tff(fact_1384_bot__set__def,axiom,
    bot_bot_set_real = collect_real(bot_bot_fun_real_o) ).

% bot_set_def
tff(fact_1385_bot__set__def,axiom,
    bot_bot_set_nat = collect_nat(bot_bot_fun_nat_o) ).

% bot_set_def
tff(fact_1386_bot__set__def,axiom,
    bot_bot_set_int = collect_int(bot_bot_fun_int_o) ).

% bot_set_def
tff(fact_1387_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_1388_combine__options__cases,axiom,
    ! [X: option_nat,P: fun_op5794009454290973322_nat_o,Y: option_nat] :
      ( ( ( X = none_nat )
       => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,X),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,X),Y) )
       => ( ! [A3: nat,B3: nat] :
              ( ( X = some_nat(A3) )
             => ( ( Y = some_nat(B3) )
               => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,X),Y) ) )
         => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_1389_combine__options__cases,axiom,
    ! [X: option_nat,P: fun_op3832135881594366065_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,X),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,X),Y) )
       => ( ! [A3: nat,B3: product_prod_nat_nat] :
              ( ( X = some_nat(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,X),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_1390_combine__options__cases,axiom,
    ! [X: option_nat,P: fun_op6120599277751319488_num_o,Y: option_num] :
      ( ( ( X = none_nat )
       => aa_option_num_o(aa_opt5077314168511351209_num_o(P,X),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt5077314168511351209_num_o(P,X),Y) )
       => ( ! [A3: nat,B3: num] :
              ( ( X = some_nat(A3) )
             => ( ( Y = some_num(B3) )
               => aa_option_num_o(aa_opt5077314168511351209_num_o(P,X),Y) ) )
         => aa_option_num_o(aa_opt5077314168511351209_num_o(P,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_1391_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: fun_op7326680082016885773_nat_o,Y: option_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,X),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,X),Y) )
       => ( ! [A3: product_prod_nat_nat,B3: nat] :
              ( ( X = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_nat(B3) )
               => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,X),Y) ) )
         => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_1392_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: fun_op1221328669798608686_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,X),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,X),Y) )
       => ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( ( X = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,X),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_1393_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: fun_op7653269905477231939_num_o,Y: option_num] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => aa_option_num_o(aa_opt7868583491048818492_num_o(P,X),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,X),Y) )
       => ( ! [A3: product_prod_nat_nat,B3: num] :
              ( ( X = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_num(B3) )
               => aa_option_num_o(aa_opt7868583491048818492_num_o(P,X),Y) ) )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_1394_combine__options__cases,axiom,
    ! [X: option_num,P: fun_op167334234089630144_nat_o,Y: option_nat] :
      ( ( ( X = none_num )
       => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,X),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,X),Y) )
       => ( ! [A3: num,B3: nat] :
              ( ( X = some_num(A3) )
             => ( ( Y = some_nat(B3) )
               => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,X),Y) ) )
         => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_1395_combine__options__cases,axiom,
    ! [X: option_num,P: fun_op5373868528577402299_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_num )
       => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,X),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,X),Y) )
       => ( ! [A3: num,B3: product_prod_nat_nat] :
              ( ( X = some_num(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,X),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_1396_combine__options__cases,axiom,
    ! [X: option_num,P: fun_op493924057549976310_num_o,Y: option_num] :
      ( ( ( X = none_num )
       => aa_option_num_o(aa_opt8674010985164783839_num_o(P,X),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,X),Y) )
       => ( ! [A3: num,B3: num] :
              ( ( X = some_num(A3) )
             => ( ( Y = some_num(B3) )
               => aa_option_num_o(aa_opt8674010985164783839_num_o(P,X),Y) ) )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_1397_split__option__all,axiom,
    ! [P: fun_option_nat_o] :
      ( ! [X_13: option_nat] : aa_option_nat_o(P,X_13)
    <=> ( aa_option_nat_o(P,none_nat)
        & ! [X4: nat] : aa_option_nat_o(P,some_nat(X4)) ) ) ).

% split_option_all
tff(fact_1398_split__option__all,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ! [X_13: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_13)
    <=> ( 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_1399_split__option__all,axiom,
    ! [P: fun_option_num_o] :
      ( ! [X_13: option_num] : aa_option_num_o(P,X_13)
    <=> ( aa_option_num_o(P,none_num)
        & ! [X4: num] : aa_option_num_o(P,some_num(X4)) ) ) ).

% split_option_all
tff(fact_1400_split__option__ex,axiom,
    ! [P: fun_option_nat_o] :
      ( ? [X_13: option_nat] : aa_option_nat_o(P,X_13)
    <=> ( aa_option_nat_o(P,none_nat)
        | ? [X4: nat] : aa_option_nat_o(P,some_nat(X4)) ) ) ).

% split_option_ex
tff(fact_1401_split__option__ex,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ? [X_13: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_13)
    <=> ( 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_1402_split__option__ex,axiom,
    ! [P: fun_option_num_o] :
      ( ? [X_13: option_num] : aa_option_num_o(P,X_13)
    <=> ( aa_option_num_o(P,none_num)
        | ? [X4: num] : aa_option_num_o(P,some_num(X4)) ) ) ).

% split_option_ex
tff(fact_1403_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X23: nat] : Y != some_nat(X23) ) ).

% option.exhaust
tff(fact_1404_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] : Y != some_P7363390416028606310at_nat(X23) ) ).

% option.exhaust
tff(fact_1405_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X23: num] : Y != some_num(X23) ) ).

% option.exhaust
tff(fact_1406_option_OdiscI,axiom,
    ! [Option: option_nat,X22: nat] :
      ( ( Option = some_nat(X22) )
     => ( Option != none_nat ) ) ).

% option.discI
tff(fact_1407_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X22) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_1408_option_OdiscI,axiom,
    ! [Option: option_num,X22: num] :
      ( ( Option = some_num(X22) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_1409_option_Odistinct_I1_J,axiom,
    ! [X22: nat] : none_nat != some_nat(X22) ).

% option.distinct(1)
tff(fact_1410_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X22) ).

% option.distinct(1)
tff(fact_1411_option_Odistinct_I1_J,axiom,
    ! [X22: num] : none_num != some_num(X22) ).

% option.distinct(1)
tff(fact_1412_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_1413_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_1414_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_1415_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_1416_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_1417_option_Osize_I3_J,axiom,
    size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1418_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1419_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1420_option_Osize__gen_I1_J,axiom,
    ! [X: fun_nat_nat] : size_option_nat(X,none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_1421_option_Osize__gen_I1_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(X,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_1422_option_Osize__gen_I1_J,axiom,
    ! [X: fun_num_nat] : size_option_num(X,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_1423_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option = some_nat(the_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_1424_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_1425_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option = some_num(the_num(Option)) ) ) ).

% option.exhaust_sel
tff(fact_1426_linorder__neqE__linordered__idom,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1427_linorder__neqE__linordered__idom,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1428_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1429_linordered__field__no__ub,axiom,
    ! [X2: real] :
    ? [X_1: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X2),X_1) ).

% linordered_field_no_ub
tff(fact_1430_linordered__field__no__ub,axiom,
    ! [X2: rat] :
    ? [X_1: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),X_1) ).

% linordered_field_no_ub
tff(fact_1431_linordered__field__no__lb,axiom,
    ! [X2: real] :
    ? [Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),X2) ).

% linordered_field_no_lb
tff(fact_1432_linordered__field__no__lb,axiom,
    ! [X2: rat] :
    ? [Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),X2) ).

% linordered_field_no_lb
tff(fact_1433_not__psubset__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ~ aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(A2),bot_bo2099793752762293965at_nat) ).

% not_psubset_empty
tff(fact_1434_not__psubset__empty,axiom,
    ! [A2: set_real] : ~ aa_set_real_o(ord_less_set_real(A2),bot_bot_set_real) ).

% not_psubset_empty
tff(fact_1435_not__psubset__empty,axiom,
    ! [A2: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),bot_bot_set_nat) ).

% not_psubset_empty
tff(fact_1436_not__psubset__empty,axiom,
    ! [A2: set_int] : ~ aa_set_int_o(ord_less_set_int(A2),bot_bot_set_int) ).

% not_psubset_empty
tff(fact_1437_ex__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ? [X4: set_nat] : aa_set_set_nat_o(member_set_nat(X4),A2)
    <=> ( A2 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
tff(fact_1438_ex__in__conv,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ? [X4: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A2)
    <=> ( A2 != bot_bo2099793752762293965at_nat ) ) ).

% ex_in_conv
tff(fact_1439_ex__in__conv,axiom,
    ! [A2: set_real] :
      ( ? [X4: real] : aa_set_real_o(member_real(X4),A2)
    <=> ( A2 != bot_bot_set_real ) ) ).

% ex_in_conv
tff(fact_1440_ex__in__conv,axiom,
    ! [A2: set_nat] :
      ( ? [X4: nat] : aa_set_nat_o(member_nat(X4),A2)
    <=> ( A2 != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_1441_ex__in__conv,axiom,
    ! [A2: set_int] :
      ( ? [X4: int] : aa_set_int_o(member_int(X4),A2)
    <=> ( A2 != bot_bot_set_int ) ) ).

% ex_in_conv
tff(fact_1442_equals0I,axiom,
    ! [A2: set_set_nat] :
      ( ! [Y3: set_nat] : ~ aa_set_set_nat_o(member_set_nat(Y3),A2)
     => ( A2 = bot_bot_set_set_nat ) ) ).

% equals0I
tff(fact_1443_equals0I,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ! [Y3: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y3),A2)
     => ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% equals0I
tff(fact_1444_equals0I,axiom,
    ! [A2: set_real] :
      ( ! [Y3: real] : ~ aa_set_real_o(member_real(Y3),A2)
     => ( A2 = bot_bot_set_real ) ) ).

% equals0I
tff(fact_1445_equals0I,axiom,
    ! [A2: set_nat] :
      ( ! [Y3: nat] : ~ aa_set_nat_o(member_nat(Y3),A2)
     => ( A2 = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_1446_equals0I,axiom,
    ! [A2: set_int] :
      ( ! [Y3: int] : ~ aa_set_int_o(member_int(Y3),A2)
     => ( A2 = bot_bot_set_int ) ) ).

% equals0I
tff(fact_1447_equals0D,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( A2 = bot_bot_set_set_nat )
     => ~ aa_set_set_nat_o(member_set_nat(A),A2) ) ).

% equals0D
tff(fact_1448_equals0D,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( A2 = bot_bo2099793752762293965at_nat )
     => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A2) ) ).

% equals0D
tff(fact_1449_equals0D,axiom,
    ! [A2: set_real,A: real] :
      ( ( A2 = bot_bot_set_real )
     => ~ aa_set_real_o(member_real(A),A2) ) ).

% equals0D
tff(fact_1450_equals0D,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( A2 = bot_bot_set_nat )
     => ~ aa_set_nat_o(member_nat(A),A2) ) ).

% equals0D
tff(fact_1451_equals0D,axiom,
    ! [A2: set_int,A: int] :
      ( ( A2 = bot_bot_set_int )
     => ~ aa_set_int_o(member_int(A),A2) ) ).

% equals0D
tff(fact_1452_emptyE,axiom,
    ! [A: set_nat] : ~ aa_set_set_nat_o(member_set_nat(A),bot_bot_set_set_nat) ).

% emptyE
tff(fact_1453_emptyE,axiom,
    ! [A: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),bot_bo2099793752762293965at_nat) ).

% emptyE
tff(fact_1454_emptyE,axiom,
    ! [A: real] : ~ aa_set_real_o(member_real(A),bot_bot_set_real) ).

% emptyE
tff(fact_1455_emptyE,axiom,
    ! [A: nat] : ~ aa_set_nat_o(member_nat(A),bot_bot_set_nat) ).

% emptyE
tff(fact_1456_emptyE,axiom,
    ! [A: int] : ~ aa_set_int_o(member_int(A),bot_bot_set_int) ).

% emptyE
tff(fact_1457_option_Osize_I4_J,axiom,
    ! [X22: nat] : size_size_option_nat(some_nat(X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_1458_option_Osize_I4_J,axiom,
    ! [X22: product_prod_nat_nat] : size_s170228958280169651at_nat(some_P7363390416028606310at_nat(X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_1459_option_Osize_I4_J,axiom,
    ! [X22: num] : size_size_option_num(some_num(X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_1460_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X: 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)),X)
    <=> ( ( X = Mia )
        | ( X = Maa ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_1461_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),X) ).

% vebt_member.simps(4)
tff(fact_1462_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),X) ).

% vebt_member.simps(3)
tff(fact_1463_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_1464_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uu,zero_zero_nat,Uv,Uw)),Ux) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_1465_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_1466_old_Oprod_Oinject,axiom,
    ! [A: int,B: int,A4: int,B4: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A4),B4) )
    <=> ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
tff(fact_1467_old_Oprod_Oinject,axiom,
    ! [A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger,A4: fun_Co3460408610320954864e_term,B4: produc8923325533196201883nteger] :
      ( ( aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A),B) = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A4),B4) )
    <=> ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
tff(fact_1468_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,A4: fun_Pr5098669352780829690e_term,B4: produc8923325533196201883nteger] :
      ( ( aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A),B) = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A4),B4) )
    <=> ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
tff(fact_1469_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,A4: fun_Pr7763680713840462443e_term,B4: product_prod_int_int] :
      ( ( aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A4),B4) )
    <=> ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
tff(fact_1470_old_Oprod_Oinject,axiom,
    ! [A: fun_in5730379987311621695e_term,B: product_prod_int_int,A4: fun_in5730379987311621695e_term,B4: product_prod_int_int] :
      ( ( aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),B4) )
    <=> ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
tff(fact_1471_prod_Oinject,axiom,
    ! [X1: int,X22: int,Y1: int,Y2: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
tff(fact_1472_prod_Oinject,axiom,
    ! [X1: fun_Co3460408610320954864e_term,X22: produc8923325533196201883nteger,Y1: fun_Co3460408610320954864e_term,Y2: produc8923325533196201883nteger] :
      ( ( aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X1),X22) = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
tff(fact_1473_prod_Oinject,axiom,
    ! [X1: fun_Pr5098669352780829690e_term,X22: produc8923325533196201883nteger,Y1: fun_Pr5098669352780829690e_term,Y2: produc8923325533196201883nteger] :
      ( ( aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X1),X22) = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
tff(fact_1474_prod_Oinject,axiom,
    ! [X1: fun_Pr7763680713840462443e_term,X22: product_prod_int_int,Y1: fun_Pr7763680713840462443e_term,Y2: product_prod_int_int] :
      ( ( aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X1),X22) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
tff(fact_1475_prod_Oinject,axiom,
    ! [X1: fun_in5730379987311621695e_term,X22: product_prod_int_int,Y1: fun_in5730379987311621695e_term,Y2: product_prod_int_int] :
      ( ( aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X1),X22) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
tff(fact_1476_nth__enumerate__eq,axiom,
    ! [Mb: nat,Xs: list_VEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => ( nth_Pr744662078594809490T_VEBT(enumerate_VEBT_VEBT(Na,Xs),Mb) = produc599794634098209291T_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Mb),nth_VEBT_VEBT(Xs,Mb)) ) ) ).

% nth_enumerate_eq
tff(fact_1477_nth__enumerate__eq,axiom,
    ! [Mb: nat,Xs: list_o,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_list_o_nat(size_size_list_o,Xs))
     => ( nth_Pr112076138515278198_nat_o(enumerate_o(Na,Xs),Mb) = product_Pair_nat_o(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Mb),aa_nat_o(nth_o(Xs),Mb)) ) ) ).

% nth_enumerate_eq
tff(fact_1478_nth__enumerate__eq,axiom,
    ! [Mb: nat,Xs: list_nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( nth_Pr7617993195940197384at_nat(enumerate_nat(Na,Xs),Mb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Mb)),aa_nat_nat(nth_nat(Xs),Mb)) ) ) ).

% nth_enumerate_eq
tff(fact_1479_nth__enumerate__eq,axiom,
    ! [Mb: nat,Xs: list_int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_list_int_nat(size_size_list_int,Xs))
     => ( nth_Pr3440142176431000676at_int(enumerate_int(Na,Xs),Mb) = product_Pair_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Mb),aa_nat_int(nth_int(Xs),Mb)) ) ) ).

% nth_enumerate_eq
tff(fact_1480_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_set_nat_o(finite_finite_nat,vEBT_VEBT_set_vebt(T)) ) ).

% set_vebt_finite
tff(fact_1481_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_1482_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : finite5795047828879050333T_VEBT(set_VEBT_VEBT2(Xs)) ).

% List.finite_set
tff(fact_1483_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : aa_set_nat_o(finite_finite_nat,set_nat2(Xs)) ).

% List.finite_set
tff(fact_1484_List_Ofinite__set,axiom,
    ! [Xs: list_int] : aa_set_int_o(finite_finite_int,set_int2(Xs)) ).

% List.finite_set
tff(fact_1485_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : aa_set_complex_o(finite3207457112153483333omplex,set_complex2(Xs)) ).

% List.finite_set
tff(fact_1486_List_Ofinite__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat] : aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,set_Pr5648618587558075414at_nat(Xs)) ).

% List.finite_set
tff(fact_1487_length__enumerate,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT] : size_s4762443039079500285T_VEBT(enumerate_VEBT_VEBT(Na,Xs)) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) ).

% length_enumerate
tff(fact_1488_length__enumerate,axiom,
    ! [Na: nat,Xs: list_o] : size_s6491369823275344609_nat_o(enumerate_o(Na,Xs)) = aa_list_o_nat(size_size_list_o,Xs) ).

% length_enumerate
tff(fact_1489_length__enumerate,axiom,
    ! [Na: nat,Xs: list_nat] : size_s5460976970255530739at_nat(enumerate_nat(Na,Xs)) = aa_list_nat_nat(size_size_list_nat,Xs) ).

% length_enumerate
tff(fact_1490_length__enumerate,axiom,
    ! [Na: nat,Xs: list_int] : size_s2970893825323803983at_int(enumerate_int(Na,Xs)) = aa_list_int_nat(size_size_list_int,Xs) ).

% length_enumerate
tff(fact_1491_psubsetD,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(A2),B2)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),A2)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),B2) ) ) ).

% psubsetD
tff(fact_1492_psubsetD,axiom,
    ! [A2: set_real,B2: set_real,C: real] :
      ( aa_set_real_o(ord_less_set_real(A2),B2)
     => ( aa_set_real_o(member_real(C),A2)
       => aa_set_real_o(member_real(C),B2) ) ) ).

% psubsetD
tff(fact_1493_psubsetD,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A2),B2)
     => ( aa_set_set_nat_o(member_set_nat(C),A2)
       => aa_set_set_nat_o(member_set_nat(C),B2) ) ) ).

% psubsetD
tff(fact_1494_psubsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2)
     => ( aa_set_nat_o(member_nat(C),A2)
       => aa_set_nat_o(member_nat(C),B2) ) ) ).

% psubsetD
tff(fact_1495_psubsetD,axiom,
    ! [A2: set_int,B2: set_int,C: int] :
      ( aa_set_int_o(ord_less_set_int(A2),B2)
     => ( aa_set_int_o(member_int(C),A2)
       => aa_set_int_o(member_int(C),B2) ) ) ).

% psubsetD
tff(fact_1496_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc7004362771917097900on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_nat,Uv2: option_nat] : X != produc6699827698519632094on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw2: fun_nat_fun_nat_nat,V2: nat] : X != produc6699827698519632094on_nat(Uw2,produc5098337634421038937on_nat(some_nat(V2),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_nat,A3: nat,B3: nat] : X != produc6699827698519632094on_nat(F2,produc5098337634421038937on_nat(some_nat(A3),some_nat(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1497_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc1260151622200656695at_nat] :
      ( ! [Uu2: fun_Pr2477239776880934807at_nat,Uv2: option4927543243414619207at_nat] : X != produc7814092408141220007at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw2: fun_Pr2477239776880934807at_nat,V2: product_prod_nat_nat] : X != produc7814092408141220007at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr2477239776880934807at_nat,A3: product_prod_nat_nat,B3: product_prod_nat_nat] : X != produc7814092408141220007at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A3),some_P7363390416028606310at_nat(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1498_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc2703417854250991842on_num] :
      ( ! [Uu2: fun_num_fun_num_num,Uv2: option_num] : X != produc6360833703471310868on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
     => ( ! [Uw2: fun_num_fun_num_num,V2: num] : X != produc6360833703471310868on_num(Uw2,produc8585076106096196333on_num(some_num(V2),none_num))
       => ~ ! [F2: fun_num_fun_num_num,A3: num,B3: num] : X != produc6360833703471310868on_num(F2,produc8585076106096196333on_num(some_num(A3),some_num(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1499_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc1615844007061411358on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_o,Uv2: option_nat] : X != produc7967433646104525454on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw2: fun_nat_fun_nat_o,V2: nat] : X != produc7967433646104525454on_nat(Uw2,produc5098337634421038937on_nat(some_nat(V2),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_o,X3: nat,Y3: nat] : X != produc7967433646104525454on_nat(F2,produc5098337634421038937on_nat(some_nat(X3),some_nat(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1500_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc3739950874144486240at_nat] :
      ( ! [Uu2: fun_Pr2028404973800837934_nat_o,Uv2: option4927543243414619207at_nat] : X != produc3653971535690468304at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw2: fun_Pr2028404973800837934_nat_o,V2: product_prod_nat_nat] : X != produc3653971535690468304at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr2028404973800837934_nat_o,X3: product_prod_nat_nat,Y3: product_prod_nat_nat] : X != produc3653971535690468304at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(X3),some_P7363390416028606310at_nat(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1501_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc50614890452615238on_num] :
      ( ! [Uu2: fun_num_fun_num_o,Uv2: option_num] : X != produc1140783415258738358on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
     => ( ! [Uw2: fun_num_fun_num_o,V2: num] : X != produc1140783415258738358on_num(Uw2,produc8585076106096196333on_num(some_num(V2),none_num))
       => ~ ! [F2: fun_num_fun_num_o,X3: num,Y3: num] : X != produc1140783415258738358on_num(F2,produc8585076106096196333on_num(some_num(X3),some_num(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1502_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A2)
     => ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A2 ) ).

% finite_list
tff(fact_1503_finite__list,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ? [Xs2: list_nat] : set_nat2(Xs2) = A2 ) ).

% finite_list
tff(fact_1504_finite__list,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ? [Xs2: list_int] : set_int2(Xs2) = A2 ) ).

% finite_list
tff(fact_1505_finite__list,axiom,
    ! [A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ? [Xs2: list_complex] : set_complex2(Xs2) = A2 ) ).

% finite_list
tff(fact_1506_finite__list,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ? [Xs2: list_P6011104703257516679at_nat] : set_Pr5648618587558075414at_nat(Xs2) = A2 ) ).

% finite_list
tff(fact_1507_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)),X) ).

% vebt_member.simps(2)
tff(fact_1508_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_1509_ex__min__if__finite,axiom,
    ! [S3: set_real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( ( S3 != bot_bot_set_real )
       => ? [X3: real] :
            ( aa_set_real_o(member_real(X3),S3)
            & ~ ? [Xa: real] :
                  ( aa_set_real_o(member_real(Xa),S3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1510_ex__min__if__finite,axiom,
    ! [S3: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,S3)
     => ( ( S3 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),S3)
            & ~ ? [Xa: rat] :
                  ( aa_set_rat_o(member_rat(Xa),S3)
                  & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1511_ex__min__if__finite,axiom,
    ! [S3: set_num] :
      ( aa_set_num_o(finite_finite_num,S3)
     => ( ( S3 != bot_bot_set_num )
       => ? [X3: num] :
            ( aa_set_num_o(member_num(X3),S3)
            & ~ ? [Xa: num] :
                  ( aa_set_num_o(member_num(Xa),S3)
                  & aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1512_ex__min__if__finite,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( ( S3 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),S3)
            & ~ ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),S3)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1513_ex__min__if__finite,axiom,
    ! [S3: set_int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ( S3 != bot_bot_set_int )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),S3)
            & ~ ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),S3)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1514_infinite__growing,axiom,
    ! [X5: set_real] :
      ( ( X5 != bot_bot_set_real )
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),X5)
           => ? [Xa: real] :
                ( aa_set_real_o(member_real(Xa),X5)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Xa) ) )
       => ~ aa_set_real_o(finite_finite_real,X5) ) ) ).

% infinite_growing
tff(fact_1515_infinite__growing,axiom,
    ! [X5: set_rat] :
      ( ( X5 != bot_bot_set_rat )
     => ( ! [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),X5)
           => ? [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),X5)
                & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Xa) ) )
       => ~ aa_set_rat_o(finite_finite_rat,X5) ) ) ).

% infinite_growing
tff(fact_1516_infinite__growing,axiom,
    ! [X5: set_num] :
      ( ( X5 != bot_bot_set_num )
     => ( ! [X3: num] :
            ( aa_set_num_o(member_num(X3),X5)
           => ? [Xa: num] :
                ( aa_set_num_o(member_num(Xa),X5)
                & aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Xa) ) )
       => ~ aa_set_num_o(finite_finite_num,X5) ) ) ).

% infinite_growing
tff(fact_1517_infinite__growing,axiom,
    ! [X5: set_nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),X5)
           => ? [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),X5)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Xa) ) )
       => ~ aa_set_nat_o(finite_finite_nat,X5) ) ) ).

% infinite_growing
tff(fact_1518_infinite__growing,axiom,
    ! [X5: set_int] :
      ( ( X5 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),X5)
           => ? [Xa: int] :
                ( aa_set_int_o(member_int(Xa),X5)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Xa) ) )
       => ~ aa_set_int_o(finite_finite_int,X5) ) ) ).

% infinite_growing
tff(fact_1519_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_1520_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_1521_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_1522_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_1523_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_Pr2477239776880934807at_nat,Uv: option4927543243414619207at_nat] : vEBT_V1502963449132264192at_nat(Uu,none_P5556105721700978146at_nat,Uv) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_1524_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_num_fun_num_num,Uv: option_num] : vEBT_V819420779217536731ft_num(Uu,none_num,Uv) = none_num ).

% VEBT_internal.option_shift.simps(1)
tff(fact_1525_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uv: option_nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uu),none_nat),Uv) = none_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_1526_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_1527_old_Oprod_Oexhaust,axiom,
    ! [Y: produc7509114972854644512nteger] :
      ~ ! [A3: fun_Co3460408610320954864e_term,B3: produc8923325533196201883nteger] : Y != aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A3),B3) ).

% old.prod.exhaust
tff(fact_1528_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2453742332425399446nteger] :
      ~ ! [A3: fun_Pr5098669352780829690e_term,B3: produc8923325533196201883nteger] : Y != aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A3),B3) ).

% old.prod.exhaust
tff(fact_1529_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3043004577601930755nt_int] :
      ~ ! [A3: fun_Pr7763680713840462443e_term,B3: product_prod_int_int] : Y != aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A3),B3) ).

% old.prod.exhaust
tff(fact_1530_old_Oprod_Oexhaust,axiom,
    ! [Y: produc4788131091073775919nt_int] :
      ~ ! [A3: fun_in5730379987311621695e_term,B3: product_prod_int_int] : Y != aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A3),B3) ).

% old.prod.exhaust
tff(fact_1531_surj__pair,axiom,
    ! [P2: product_prod_int_int] :
    ? [X3: int,Y3: int] : P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) ).

% surj_pair
tff(fact_1532_surj__pair,axiom,
    ! [P2: produc7509114972854644512nteger] :
    ? [X3: fun_Co3460408610320954864e_term,Y3: produc8923325533196201883nteger] : P2 = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X3),Y3) ).

% surj_pair
tff(fact_1533_surj__pair,axiom,
    ! [P2: produc2453742332425399446nteger] :
    ? [X3: fun_Pr5098669352780829690e_term,Y3: produc8923325533196201883nteger] : P2 = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X3),Y3) ).

% surj_pair
tff(fact_1534_surj__pair,axiom,
    ! [P2: produc3043004577601930755nt_int] :
    ? [X3: fun_Pr7763680713840462443e_term,Y3: product_prod_int_int] : P2 = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X3),Y3) ).

% surj_pair
tff(fact_1535_surj__pair,axiom,
    ! [P2: produc4788131091073775919nt_int] :
    ? [X3: fun_in5730379987311621695e_term,Y3: product_prod_int_int] : P2 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X3),Y3) ).

% surj_pair
tff(fact_1536_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_1537_prod__cases,axiom,
    ! [P: fun_Pr1347628893830317649eger_o,P2: produc7509114972854644512nteger] :
      ( ! [A3: fun_Co3460408610320954864e_term,B3: produc8923325533196201883nteger] : aa_Pro4652312591602923194eger_o(P,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A3),B3))
     => aa_Pro4652312591602923194eger_o(P,P2) ) ).

% prod_cases
tff(fact_1538_prod__cases,axiom,
    ! [P: fun_Pr7015759342434529115eger_o,P2: produc2453742332425399446nteger] :
      ( ! [A3: fun_Pr5098669352780829690e_term,B3: produc8923325533196201883nteger] : aa_Pro4148064003102580676eger_o(P,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A3),B3))
     => aa_Pro4148064003102580676eger_o(P,P2) ) ).

% prod_cases
tff(fact_1539_prod__cases,axiom,
    ! [P: fun_Pr3540825087398332590_int_o,P2: produc3043004577601930755nt_int] :
      ( ! [A3: fun_Pr7763680713840462443e_term,B3: product_prod_int_int] : aa_Pro2601676399929874711_int_o(P,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A3),B3))
     => aa_Pro2601676399929874711_int_o(P,P2) ) ).

% prod_cases
tff(fact_1540_prod__cases,axiom,
    ! [P: fun_Pr1196100989263358594_int_o,P2: produc4788131091073775919nt_int] :
      ( ! [A3: fun_in5730379987311621695e_term,B3: product_prod_int_int] : aa_Pro5451197003417746155_int_o(P,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A3),B3))
     => aa_Pro5451197003417746155_int_o(P,P2) ) ).

% prod_cases
tff(fact_1541_Pair__inject,axiom,
    ! [A: int,B: int,A4: int,B4: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A4),B4) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
tff(fact_1542_Pair__inject,axiom,
    ! [A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger,A4: fun_Co3460408610320954864e_term,B4: produc8923325533196201883nteger] :
      ( ( aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A),B) = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A4),B4) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
tff(fact_1543_Pair__inject,axiom,
    ! [A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,A4: fun_Pr5098669352780829690e_term,B4: produc8923325533196201883nteger] :
      ( ( aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A),B) = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A4),B4) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
tff(fact_1544_Pair__inject,axiom,
    ! [A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,A4: fun_Pr7763680713840462443e_term,B4: product_prod_int_int] :
      ( ( aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B) = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A4),B4) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
tff(fact_1545_Pair__inject,axiom,
    ! [A: fun_in5730379987311621695e_term,B: product_prod_int_int,A4: fun_in5730379987311621695e_term,B4: product_prod_int_int] :
      ( ( aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B) = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A4),B4) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
tff(fact_1546_prod__cases3,axiom,
    ! [Y: produc7509114972854644512nteger] :
      ~ ! [A3: fun_Co3460408610320954864e_term,B3: code_integer,C2: code_integer] : Y != aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A3),produc1086072967326762835nteger(B3,C2)) ).

% prod_cases3
tff(fact_1547_prod__cases3,axiom,
    ! [Y: produc2453742332425399446nteger] :
      ~ ! [A3: fun_Pr5098669352780829690e_term,B3: code_integer,C2: code_integer] : Y != aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A3),produc1086072967326762835nteger(B3,C2)) ).

% prod_cases3
tff(fact_1548_prod__cases3,axiom,
    ! [Y: produc3043004577601930755nt_int] :
      ~ ! [A3: fun_Pr7763680713840462443e_term,B3: int,C2: int] : Y != aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A3),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B3),C2)) ).

% prod_cases3
tff(fact_1549_prod__cases3,axiom,
    ! [Y: produc4788131091073775919nt_int] :
      ~ ! [A3: fun_in5730379987311621695e_term,B3: int,C2: int] : Y != aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A3),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B3),C2)) ).

% prod_cases3
tff(fact_1550_prod__induct3,axiom,
    ! [P: fun_Pr1347628893830317649eger_o,X: produc7509114972854644512nteger] :
      ( ! [A3: fun_Co3460408610320954864e_term,B3: code_integer,C2: code_integer] : aa_Pro4652312591602923194eger_o(P,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A3),produc1086072967326762835nteger(B3,C2)))
     => aa_Pro4652312591602923194eger_o(P,X) ) ).

% prod_induct3
tff(fact_1551_prod__induct3,axiom,
    ! [P: fun_Pr7015759342434529115eger_o,X: produc2453742332425399446nteger] :
      ( ! [A3: fun_Pr5098669352780829690e_term,B3: code_integer,C2: code_integer] : aa_Pro4148064003102580676eger_o(P,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A3),produc1086072967326762835nteger(B3,C2)))
     => aa_Pro4148064003102580676eger_o(P,X) ) ).

% prod_induct3
tff(fact_1552_prod__induct3,axiom,
    ! [P: fun_Pr3540825087398332590_int_o,X: produc3043004577601930755nt_int] :
      ( ! [A3: fun_Pr7763680713840462443e_term,B3: int,C2: int] : aa_Pro2601676399929874711_int_o(P,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A3),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B3),C2)))
     => aa_Pro2601676399929874711_int_o(P,X) ) ).

% prod_induct3
tff(fact_1553_prod__induct3,axiom,
    ! [P: fun_Pr1196100989263358594_int_o,X: produc4788131091073775919nt_int] :
      ( ! [A3: fun_in5730379987311621695e_term,B3: int,C2: int] : aa_Pro5451197003417746155_int_o(P,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A3),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B3),C2)))
     => aa_Pro5451197003417746155_int_o(P,X) ) ).

% prod_induct3
tff(fact_1554_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) = none_nat ).

% vebt_maxt.simps(2)
tff(fact_1555_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) = none_nat ).

% vebt_mint.simps(2)
tff(fact_1556_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_Pr2477239776880934807at_nat,V: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(Uw,some_P7363390416028606310at_nat(V),none_P5556105721700978146at_nat) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1557_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_num_fun_num_num,V: num] : vEBT_V819420779217536731ft_num(Uw,some_num(V),none_num) = none_num ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1558_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_nat_fun_nat_nat,V: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uw),some_nat(V)),none_nat) = none_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1559_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: fun_Pr2477239776880934807at_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(X,Xa2,Xb) = Y )
     => ( ( ( Xa2 = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] : Xa2 = some_P7363390416028606310at_nat(V2)
           => ( ( 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(X,A3),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1560_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: fun_num_fun_num_num,Xa2: option_num,Xb: option_num,Y: option_num] :
      ( ( vEBT_V819420779217536731ft_num(X,Xa2,Xb) = Y )
     => ( ( ( Xa2 = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V2: num] : Xa2 = some_num(V2)
           => ( ( 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(X,A3),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1561_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa2: option_nat,Xb: option_nat,Y: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(X),Xa2),Xb) = Y )
     => ( ( ( Xa2 = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V2: nat] : Xa2 = some_nat(V2)
           => ( ( 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(X,A3),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1562_vebt__buildup_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va2: nat] : X != aa_nat_nat(suc,aa_nat_nat(suc,Va2)) ) ) ).

% vebt_buildup.cases
tff(fact_1563_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ? [X3: real] :
            ( aa_set_real_o(member_real(X3),A2)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1564_finite__has__minimal,axiom,
    ! [A2: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X3: set_int] :
            ( member_set_int(X3,A2)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( aa_set_int_o(ord_less_eq_set_int(Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1565_finite__has__minimal,axiom,
    ! [A2: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),A2)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A2)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1566_finite__has__minimal,axiom,
    ! [A2: set_num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( ( A2 != bot_bot_set_num )
       => ? [X3: num] :
            ( aa_set_num_o(member_num(X3),A2)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A2)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1567_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A2)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1568_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1569_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ? [X3: real] :
            ( aa_set_real_o(member_real(X3),A2)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1570_finite__has__maximal,axiom,
    ! [A2: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X3: set_int] :
            ( member_set_int(X3,A2)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( aa_set_int_o(ord_less_eq_set_int(X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1571_finite__has__maximal,axiom,
    ! [A2: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),A2)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A2)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1572_finite__has__maximal,axiom,
    ! [A2: set_num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( ( A2 != bot_bot_set_num )
       => ? [X3: num] :
            ( aa_set_num_o(member_num(X3),A2)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A2)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1573_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A2)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1574_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1575_bot__empty__eq,axiom,
    ! [X2: set_nat] :
      ( aa_set_nat_o(bot_bo3967814874310145751_nat_o,X2)
    <=> aa_set_set_nat_o(member_set_nat(X2),bot_bot_set_set_nat) ) ).

% bot_empty_eq
tff(fact_1576_bot__empty__eq,axiom,
    ! [X2: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(bot_bo5074786513356115630_nat_o,X2)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq
tff(fact_1577_bot__empty__eq,axiom,
    ! [X2: real] :
      ( aa_real_o(bot_bot_fun_real_o,X2)
    <=> aa_set_real_o(member_real(X2),bot_bot_set_real) ) ).

% bot_empty_eq
tff(fact_1578_bot__empty__eq,axiom,
    ! [X2: nat] :
      ( aa_nat_o(bot_bot_fun_nat_o,X2)
    <=> aa_set_nat_o(member_nat(X2),bot_bot_set_nat) ) ).

% bot_empty_eq
tff(fact_1579_bot__empty__eq,axiom,
    ! [X2: int] :
      ( aa_int_o(bot_bot_fun_int_o,X2)
    <=> aa_set_int_o(member_int(X2),bot_bot_set_int) ) ).

% bot_empty_eq
tff(fact_1580_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_1581_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_1582_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_1583_Collect__empty__eq__bot,axiom,
    ! [P: fun_real_o] :
      ( ( collect_real(P) = bot_bot_set_real )
    <=> ( P = bot_bot_fun_real_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1584_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_1585_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_1586_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
    <=> ! [M5: nat] :
        ? [N3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N3)
          & aa_set_nat_o(member_nat(N3),S3) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_1587_finite__nat__set__iff__bounded__le,axiom,
    ! [N4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N4)
    <=> ? [M5: nat] :
        ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),M5) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_1588_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set_nat] :
      ( ! [M: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),M)
         => ? [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N5)
              & aa_set_nat_o(member_nat(N5),S3) ) )
     => ~ aa_set_nat_o(finite_finite_nat,S3) ) ).

% unbounded_k_infinite
tff(fact_1589_bounded__nat__set__is__finite,axiom,
    ! [N4: set_nat,Na: nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Na) )
     => aa_set_nat_o(finite_finite_nat,N4) ) ).

% bounded_nat_set_is_finite
tff(fact_1590_finite__maxlen,axiom,
    ! [M6: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M6)
     => ? [N: nat] :
        ! [X2: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X2,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,X2)),N) ) ) ).

% finite_maxlen
tff(fact_1591_finite__maxlen,axiom,
    ! [M6: set_list_o] :
      ( finite_finite_list_o(M6)
     => ? [N: nat] :
        ! [X2: list_o] :
          ( member_list_o(X2,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_list_o_nat(size_size_list_o,X2)),N) ) ) ).

% finite_maxlen
tff(fact_1592_finite__maxlen,axiom,
    ! [M6: set_list_nat] :
      ( finite8100373058378681591st_nat(M6)
     => ? [N: nat] :
        ! [X2: list_nat] :
          ( member_list_nat(X2,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_list_nat_nat(size_size_list_nat,X2)),N) ) ) ).

% finite_maxlen
tff(fact_1593_finite__maxlen,axiom,
    ! [M6: set_list_int] :
      ( finite3922522038869484883st_int(M6)
     => ? [N: nat] :
        ! [X2: list_int] :
          ( member_list_int(X2,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_list_int_nat(size_size_list_int,X2)),N) ) ) ).

% finite_maxlen
tff(fact_1594_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3)),R2)
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3)),S) )
     => aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(R2),S) ) ).

% subrelI
tff(fact_1595_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Y3: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3),R2)
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3),S) )
     => ord_le2843351958646193337nt_int(R2,S) ) ).

% subrelI
tff(fact_1596_subrelI,axiom,
    ! [R2: set_Pr1200682188532975830nteger,S: set_Pr1200682188532975830nteger] :
      ( ! [X3: fun_Co3460408610320954864e_term,Y3: produc8923325533196201883nteger] :
          ( member6901806533512420791nteger(aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X3),Y3),R2)
         => member6901806533512420791nteger(aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X3),Y3),S) )
     => ord_le2645283422117218614nteger(R2,S) ) ).

% subrelI
tff(fact_1597_subrelI,axiom,
    ! [R2: set_Pr3486312109254157132nteger,S: set_Pr3486312109254157132nteger] :
      ( ! [X3: fun_Pr5098669352780829690e_term,Y3: produc8923325533196201883nteger] :
          ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X3),Y3),R2)
         => member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X3),Y3),S) )
     => ord_le2932375828722672556nteger(R2,S) ) ).

% subrelI
tff(fact_1598_subrelI,axiom,
    ! [R2: set_Pr6700895561065273913nt_int,S: set_Pr6700895561065273913nt_int] :
      ( ! [X3: fun_Pr7763680713840462443e_term,Y3: product_prod_int_int] :
          ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X3),Y3),R2)
         => member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X3),Y3),S) )
     => ord_le3990897393281686169nt_int(R2,S) ) ).

% subrelI
tff(fact_1599_subrelI,axiom,
    ! [R2: set_Pr4166680992137330533nt_int,S: set_Pr4166680992137330533nt_int] :
      ( ! [X3: fun_in5730379987311621695e_term,Y3: product_prod_int_int] :
          ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X3),Y3),R2)
         => member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X3),Y3),S) )
     => ord_le3197784583289274309nt_int(R2,S) ) ).

% subrelI
tff(fact_1600_bounded__Max__nat,axiom,
    ! [P: fun_nat_o,X: nat,M6: nat] :
      ( aa_nat_o(P,X)
     => ( ! [X3: nat] :
            ( aa_nat_o(P,X3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),M6) )
       => ~ ! [M: nat] :
              ( aa_nat_o(P,M)
             => ~ ! [X2: nat] :
                    ( aa_nat_o(P,X2)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),M) ) ) ) ) ).

% bounded_Max_nat
tff(fact_1601_finite__has__minimal2,axiom,
    ! [A2: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(A),A2)
       => ? [X3: real] :
            ( aa_set_real_o(member_real(X3),A2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),A)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1602_finite__has__minimal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(member_set_nat(A),A2)
       => ? [X3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X3),A2)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),A)
            & ! [Xa: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(Xa),A2)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1603_finite__has__minimal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(A,A2)
       => ? [X3: set_int] :
            ( member_set_int(X3,A2)
            & aa_set_int_o(ord_less_eq_set_int(X3),A)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( aa_set_int_o(ord_less_eq_set_int(Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1604_finite__has__minimal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(member_rat(A),A2)
       => ? [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),A2)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),A)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A2)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1605_finite__has__minimal2,axiom,
    ! [A2: set_num,A: num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( aa_set_num_o(member_num(A),A2)
       => ? [X3: num] :
            ( aa_set_num_o(member_num(X3),A2)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),A)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A2)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1606_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(A),A2)
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),A)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1607_finite__has__minimal2,axiom,
    ! [A2: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(A),A2)
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),A)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1608_finite__has__maximal2,axiom,
    ! [A2: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(A),A2)
       => ? [X3: real] :
            ( aa_set_real_o(member_real(X3),A2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1609_finite__has__maximal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(member_set_nat(A),A2)
       => ? [X3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X3),A2)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X3)
            & ! [Xa: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(Xa),A2)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1610_finite__has__maximal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(A,A2)
       => ? [X3: set_int] :
            ( member_set_int(X3,A2)
            & aa_set_int_o(ord_less_eq_set_int(A),X3)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( aa_set_int_o(ord_less_eq_set_int(X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1611_finite__has__maximal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(member_rat(A),A2)
       => ? [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),A2)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),X3)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A2)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1612_finite__has__maximal2,axiom,
    ! [A2: set_num,A: num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( aa_set_num_o(member_num(A),A2)
       => ? [X3: num] :
            ( aa_set_num_o(member_num(X3),A2)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),X3)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A2)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1613_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(A),A2)
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X3)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1614_finite__has__maximal2,axiom,
    ! [A2: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(A),A2)
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X3)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1615_finite__transitivity__chain,axiom,
    ! [A2: set_set_nat,R: fun_se2170385732985038782_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ! [X3: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(R,X3),X3)
       => ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(R,X3),Y3)
             => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(R,Y3),Z2)
               => aa_set_nat_o(aa_set7861427066675100071_nat_o(R,X3),Z2) ) )
         => ( ! [X3: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(X3),A2)
               => ? [Y4: set_nat] :
                    ( aa_set_set_nat_o(member_set_nat(Y4),A2)
                    & aa_set_nat_o(aa_set7861427066675100071_nat_o(R,X3),Y4) ) )
           => ( A2 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1616_finite__transitivity__chain,axiom,
    ! [A2: set_complex,R: fun_co1148497455491383662plex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X3: complex] : ~ aa_complex_o(aa_com5808801942499186519plex_o(R,X3),X3)
       => ( ! [X3: complex,Y3: complex,Z2: complex] :
              ( aa_complex_o(aa_com5808801942499186519plex_o(R,X3),Y3)
             => ( aa_complex_o(aa_com5808801942499186519plex_o(R,Y3),Z2)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,X3),Z2) ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),A2)
               => ? [Y4: complex] :
                    ( aa_set_complex_o(member_complex(Y4),A2)
                    & aa_complex_o(aa_com5808801942499186519plex_o(R,X3),Y4) ) )
           => ( A2 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1617_finite__transitivity__chain,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,R: fun_Pr2028404973800837934_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( ! [X3: product_prod_nat_nat] : ~ aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,X3),X3)
       => ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,Z2: product_prod_nat_nat] :
              ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,X3),Y3)
             => ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,Y3),Z2)
               => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,X3),Z2) ) )
         => ( ! [X3: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A2)
               => ? [Y4: product_prod_nat_nat] :
                    ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y4),A2)
                    & aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,X3),Y4) ) )
           => ( A2 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1618_finite__transitivity__chain,axiom,
    ! [A2: set_real,R: fun_real_fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ! [X3: real] : ~ aa_real_o(aa_real_fun_real_o(R,X3),X3)
       => ( ! [X3: real,Y3: real,Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(R,X3),Y3)
             => ( aa_real_o(aa_real_fun_real_o(R,Y3),Z2)
               => aa_real_o(aa_real_fun_real_o(R,X3),Z2) ) )
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),A2)
               => ? [Y4: real] :
                    ( aa_set_real_o(member_real(Y4),A2)
                    & aa_real_o(aa_real_fun_real_o(R,X3),Y4) ) )
           => ( A2 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1619_finite__transitivity__chain,axiom,
    ! [A2: set_nat,R: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [X3: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(R,X3),X3)
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X3),Y3)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y3),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(R,X3),Z2) ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),A2)
               => ? [Y4: nat] :
                    ( aa_set_nat_o(member_nat(Y4),A2)
                    & aa_nat_o(aa_nat_fun_nat_o(R,X3),Y4) ) )
           => ( A2 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1620_finite__transitivity__chain,axiom,
    ! [A2: set_int,R: fun_int_fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X3: int] : ~ aa_int_o(aa_int_fun_int_o(R,X3),X3)
       => ( ! [X3: int,Y3: int,Z2: int] :
              ( aa_int_o(aa_int_fun_int_o(R,X3),Y3)
             => ( aa_int_o(aa_int_fun_int_o(R,Y3),Z2)
               => aa_int_o(aa_int_fun_int_o(R,X3),Z2) ) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),A2)
               => ? [Y4: int] :
                    ( aa_set_int_o(member_int(Y4),A2)
                    & aa_int_o(aa_int_fun_int_o(R,X3),Y4) ) )
           => ( A2 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1621_infinite__imp__nonempty,axiom,
    ! [S3: set_complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( S3 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
tff(fact_1622_infinite__imp__nonempty,axiom,
    ! [S3: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S3)
     => ( S3 != bot_bo2099793752762293965at_nat ) ) ).

% infinite_imp_nonempty
tff(fact_1623_infinite__imp__nonempty,axiom,
    ! [S3: set_real] :
      ( ~ aa_set_real_o(finite_finite_real,S3)
     => ( S3 != bot_bot_set_real ) ) ).

% infinite_imp_nonempty
tff(fact_1624_infinite__imp__nonempty,axiom,
    ! [S3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
     => ( S3 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
tff(fact_1625_infinite__imp__nonempty,axiom,
    ! [S3: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S3)
     => ( S3 != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
tff(fact_1626_finite_OemptyI,axiom,
    aa_set_complex_o(finite3207457112153483333omplex,bot_bot_set_complex) ).

% finite.emptyI
tff(fact_1627_finite_OemptyI,axiom,
    aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,bot_bo2099793752762293965at_nat) ).

% finite.emptyI
tff(fact_1628_finite_OemptyI,axiom,
    aa_set_real_o(finite_finite_real,bot_bot_set_real) ).

% finite.emptyI
tff(fact_1629_finite_OemptyI,axiom,
    aa_set_nat_o(finite_finite_nat,bot_bot_set_nat) ).

% finite.emptyI
tff(fact_1630_finite_OemptyI,axiom,
    aa_set_int_o(finite_finite_int,bot_bot_set_int) ).

% finite.emptyI
tff(fact_1631_rev__finite__subset,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
       => aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% rev_finite_subset
tff(fact_1632_rev__finite__subset,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
       => aa_set_complex_o(finite3207457112153483333omplex,A2) ) ) ).

% rev_finite_subset
tff(fact_1633_rev__finite__subset,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
     => ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),B2)
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2) ) ) ).

% rev_finite_subset
tff(fact_1634_rev__finite__subset,axiom,
    ! [B2: set_int,A2: set_int] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
       => aa_set_int_o(finite_finite_int,A2) ) ) ).

% rev_finite_subset
tff(fact_1635_infinite__super,axiom,
    ! [S3: set_nat,T3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),T3)
     => ( ~ aa_set_nat_o(finite_finite_nat,S3)
       => ~ aa_set_nat_o(finite_finite_nat,T3) ) ) ).

% infinite_super
tff(fact_1636_infinite__super,axiom,
    ! [S3: set_complex,T3: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T3)
     => ( ~ aa_set_complex_o(finite3207457112153483333omplex,S3)
       => ~ aa_set_complex_o(finite3207457112153483333omplex,T3) ) ) ).

% infinite_super
tff(fact_1637_infinite__super,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(S3),T3)
     => ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S3)
       => ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,T3) ) ) ).

% infinite_super
tff(fact_1638_infinite__super,axiom,
    ! [S3: set_int,T3: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(S3),T3)
     => ( ~ aa_set_int_o(finite_finite_int,S3)
       => ~ aa_set_int_o(finite_finite_int,T3) ) ) ).

% infinite_super
tff(fact_1639_finite__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% finite_subset
tff(fact_1640_finite__subset,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => aa_set_complex_o(finite3207457112153483333omplex,A2) ) ) ).

% finite_subset
tff(fact_1641_finite__subset,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),B2)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2) ) ) ).

% finite_subset
tff(fact_1642_finite__subset,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => aa_set_int_o(finite_finite_int,A2) ) ) ).

% finite_subset
tff(fact_1643_finite__psubset__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [A5: set_nat] :
            ( aa_set_nat_o(finite_finite_nat,A5)
           => ( ! [B5: set_nat] :
                  ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B5),A5)
                 => aa_set_nat_o(P,B5) )
             => aa_set_nat_o(P,A5) ) )
       => aa_set_nat_o(P,A2) ) ) ).

% finite_psubset_induct
tff(fact_1644_finite__psubset__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [A5: set_int] :
            ( aa_set_int_o(finite_finite_int,A5)
           => ( ! [B5: set_int] :
                  ( aa_set_int_o(ord_less_set_int(B5),A5)
                 => aa_set_int_o(P,B5) )
             => aa_set_int_o(P,A5) ) )
       => aa_set_int_o(P,A2) ) ) ).

% finite_psubset_induct
tff(fact_1645_finite__psubset__induct,axiom,
    ! [A2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [A5: set_complex] :
            ( aa_set_complex_o(finite3207457112153483333omplex,A5)
           => ( ! [B5: set_complex] :
                  ( aa_set_complex_o(ord_less_set_complex(B5),A5)
                 => aa_set_complex_o(P,B5) )
             => aa_set_complex_o(P,A5) ) )
       => aa_set_complex_o(P,A2) ) ) ).

% finite_psubset_induct
tff(fact_1646_finite__psubset__induct,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( ! [A5: set_Pr1261947904930325089at_nat] :
            ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A5)
           => ( ! [B5: set_Pr1261947904930325089at_nat] :
                  ( aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(B5),A5)
                 => aa_set3702741264012733747_nat_o(P,B5) )
             => aa_set3702741264012733747_nat_o(P,A5) ) )
       => aa_set3702741264012733747_nat_o(P,A2) ) ) ).

% finite_psubset_induct
tff(fact_1647_finite__nat__set__iff__bounded,axiom,
    ! [N4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N4)
    <=> ? [M5: nat] :
        ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),M5) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_1648_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
    <=> ! [M5: nat] :
        ? [N3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),N3)
          & aa_set_nat_o(member_nat(N3),S3) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_1649_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ( S3 != bot_bot_set_complex )
       => ~ ? [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),S3)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X2)),aa_complex_real(F,lattic8794016678065449205x_real(F,S3))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1650_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( ( S3 != bot_bot_set_real )
       => ~ ? [X2: real] :
              ( aa_set_real_o(member_real(X2),S3)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,lattic8440615504127631091l_real(F,S3))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1651_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( ( S3 != bot_bot_set_nat )
       => ~ ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),S3)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X2)),aa_nat_real(F,aa_set_nat_nat(lattic488527866317076247t_real(F),S3))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1652_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ( S3 != bot_bot_set_int )
       => ~ ? [X2: int] :
              ( aa_set_int_o(member_int(X2),S3)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X2)),aa_int_real(F,lattic2675449441010098035t_real(F,S3))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1653_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ( S3 != bot_bot_set_complex )
       => ~ ? [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),S3)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X2)),aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S3))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1654_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( ( S3 != bot_bot_set_real )
       => ~ ? [X2: real] :
              ( aa_set_real_o(member_real(X2),S3)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,lattic4420706379359479199al_rat(F,S3))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1655_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( ( S3 != bot_bot_set_nat )
       => ~ ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),S3)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,aa_set_nat_nat(lattic6811802900495863747at_rat(F),S3))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1656_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ( S3 != bot_bot_set_int )
       => ~ ? [X2: int] :
              ( aa_set_int_o(member_int(X2),S3)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(F,lattic7811156612396918303nt_rat(F,S3))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1657_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_complex,F: fun_complex_num] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ( S3 != bot_bot_set_complex )
       => ~ ? [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),S3)
              & aa_num_o(aa_num_fun_num_o(ord_less_num,aa_complex_num(F,X2)),aa_complex_num(F,lattic1922116423962787043ex_num(F,S3))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1658_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_real,F: fun_real_num] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( ( S3 != bot_bot_set_real )
       => ~ ? [X2: real] :
              ( aa_set_real_o(member_real(X2),S3)
              & aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X2)),aa_real_num(F,lattic1613168225601753569al_num(F,S3))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1659_arg__min__least,axiom,
    ! [S3: set_complex,Y: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ( S3 != bot_bot_set_complex )
       => ( aa_set_complex_o(member_complex(Y),S3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S3))),aa_complex_rat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1660_arg__min__least,axiom,
    ! [S3: set_real,Y: real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( ( S3 != bot_bot_set_real )
       => ( aa_set_real_o(member_real(Y),S3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,lattic4420706379359479199al_rat(F,S3))),aa_real_rat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1661_arg__min__least,axiom,
    ! [S3: set_nat,Y: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( ( S3 != bot_bot_set_nat )
       => ( aa_set_nat_o(member_nat(Y),S3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_set_nat_nat(lattic6811802900495863747at_rat(F),S3))),aa_nat_rat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1662_arg__min__least,axiom,
    ! [S3: set_int,Y: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ( S3 != bot_bot_set_int )
       => ( aa_set_int_o(member_int(Y),S3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,lattic7811156612396918303nt_rat(F,S3))),aa_int_rat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1663_arg__min__least,axiom,
    ! [S3: set_complex,Y: complex,F: fun_complex_num] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ( S3 != bot_bot_set_complex )
       => ( aa_set_complex_o(member_complex(Y),S3)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_complex_num(F,lattic1922116423962787043ex_num(F,S3))),aa_complex_num(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1664_arg__min__least,axiom,
    ! [S3: set_real,Y: real,F: fun_real_num] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( ( S3 != bot_bot_set_real )
       => ( aa_set_real_o(member_real(Y),S3)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_real_num(F,lattic1613168225601753569al_num(F,S3))),aa_real_num(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1665_arg__min__least,axiom,
    ! [S3: set_nat,Y: nat,F: fun_nat_num] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( ( S3 != bot_bot_set_nat )
       => ( aa_set_nat_o(member_nat(Y),S3)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,aa_set_nat_nat(lattic4004264746738138117at_num(F),S3))),aa_nat_num(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1666_arg__min__least,axiom,
    ! [S3: set_int,Y: int,F: fun_int_num] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ( S3 != bot_bot_set_int )
       => ( aa_set_int_o(member_int(Y),S3)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,lattic5003618458639192673nt_num(F,S3))),aa_int_num(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1667_arg__min__least,axiom,
    ! [S3: set_complex,Y: complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ( S3 != bot_bot_set_complex )
       => ( aa_set_complex_o(member_complex(Y),S3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,lattic5364784637807008409ex_nat(F,S3))),aa_complex_nat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1668_arg__min__least,axiom,
    ! [S3: set_real,Y: real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( ( S3 != bot_bot_set_real )
       => ( aa_set_real_o(member_real(Y),S3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,lattic5055836439445974935al_nat(F,S3))),aa_real_nat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1669_nat__descend__induct,axiom,
    ! [Na: nat,P: fun_nat_o,Mb: nat] :
      ( ! [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K2)
         => aa_nat_o(P,K2) )
     => ( ! [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Na)
           => ( ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),I4)
                 => aa_nat_o(P,I4) )
             => aa_nat_o(P,K2) ) )
       => aa_nat_o(P,Mb) ) ) ).

% nat_descend_induct
tff(fact_1670_subset__emptyI,axiom,
    ! [A2: set_set_nat] :
      ( ! [X3: set_nat] : ~ aa_set_set_nat_o(member_set_nat(X3),A2)
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),bot_bot_set_set_nat) ) ).

% subset_emptyI
tff(fact_1671_subset__emptyI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A2)
     => aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),bot_bo2099793752762293965at_nat) ) ).

% subset_emptyI
tff(fact_1672_subset__emptyI,axiom,
    ! [A2: set_real] :
      ( ! [X3: real] : ~ aa_set_real_o(member_real(X3),A2)
     => aa_set_real_o(ord_less_eq_set_real(A2),bot_bot_set_real) ) ).

% subset_emptyI
tff(fact_1673_subset__emptyI,axiom,
    ! [A2: set_nat] :
      ( ! [X3: nat] : ~ aa_set_nat_o(member_nat(X3),A2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),bot_bot_set_nat) ) ).

% subset_emptyI
tff(fact_1674_subset__emptyI,axiom,
    ! [A2: set_int] :
      ( ! [X3: int] : ~ aa_set_int_o(member_int(X3),A2)
     => aa_set_int_o(ord_less_eq_set_int(A2),bot_bot_set_int) ) ).

% subset_emptyI
tff(fact_1675_exists__least__lemma,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_nat_o(P,zero_zero_nat)
     => ( ? [X_12: nat] : aa_nat_o(P,X_12)
       => ? [N: nat] :
            ( ~ aa_nat_o(P,N)
            & aa_nat_o(P,aa_nat_nat(suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_1676_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N: nat] : X != aa_nat_nat(suc,N) ) ).

% list_decode.cases
tff(fact_1677_verit__sum__simplify,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% verit_sum_simplify
tff(fact_1678_verit__sum__simplify,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% verit_sum_simplify
tff(fact_1679_verit__sum__simplify,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_1680_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_1681_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_1682_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B = aa_complex_complex(plus_plus_complex(B),A) )
    <=> ( A = zero_zero_complex ) ) ).

% add_0_iff
tff(fact_1683_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
    <=> ( A = zero_zero_real ) ) ).

% add_0_iff
tff(fact_1684_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B = aa_rat_rat(plus_plus_rat(B),A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_1685_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_1686_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_1687_field__lbound__gt__zero,axiom,
    ! [D1: real,D2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
       => ? [E: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E),D1)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E),D2) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1688_field__lbound__gt__zero,axiom,
    ! [D1: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D1)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D2)
       => ? [E: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E),D1)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E),D2) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1689_less__by__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( A2 = bot_bo2099793752762293965at_nat )
     => aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),B2) ) ).

% less_by_empty
tff(fact_1690_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(A),A) ).

% verit_comp_simplify1(2)
tff(fact_1691_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_1692_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_1693_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_1694_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_1695_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_1696_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_1697_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_1698_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_1699_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1700_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1701_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1702_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1703_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1704_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S: nat,R: set_Pr1261947904930325089at_nat,S4: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(R2),S)),R)
     => ( ( S4 = S )
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(R2),S4)),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1705_ssubst__Pair__rhs,axiom,
    ! [R2: int,S: int,R: set_Pr958786334691620121nt_int,S4: int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R2),S),R)
     => ( ( S4 = S )
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R2),S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1706_ssubst__Pair__rhs,axiom,
    ! [R2: fun_Co3460408610320954864e_term,S: produc8923325533196201883nteger,R: set_Pr1200682188532975830nteger,S4: produc8923325533196201883nteger] :
      ( member6901806533512420791nteger(aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,R2),S),R)
     => ( ( S4 = S )
       => member6901806533512420791nteger(aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,R2),S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1707_ssubst__Pair__rhs,axiom,
    ! [R2: fun_Pr5098669352780829690e_term,S: produc8923325533196201883nteger,R: set_Pr3486312109254157132nteger,S4: produc8923325533196201883nteger] :
      ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,R2),S),R)
     => ( ( S4 = S )
       => member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,R2),S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1708_ssubst__Pair__rhs,axiom,
    ! [R2: fun_Pr7763680713840462443e_term,S: product_prod_int_int,R: set_Pr6700895561065273913nt_int,S4: product_prod_int_int] :
      ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,R2),S),R)
     => ( ( S4 = S )
       => member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,R2),S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1709_ssubst__Pair__rhs,axiom,
    ! [R2: fun_in5730379987311621695e_term,S: product_prod_int_int,R: set_Pr4166680992137330533nt_int,S4: product_prod_int_int] :
      ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,R2),S),R)
     => ( ( S4 = S )
       => member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,R2),S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1710_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K2: nat,M: nat] : X != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K2),M) ).

% prod_decode_aux.cases
tff(fact_1711_verit__comp__simplify1_I3_J,axiom,
    ! [B4: real,A4: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B4),A4)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_1712_verit__comp__simplify1_I3_J,axiom,
    ! [B4: rat,A4: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B4),A4)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_1713_verit__comp__simplify1_I3_J,axiom,
    ! [B4: num,A4: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B4),A4)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_1714_verit__comp__simplify1_I3_J,axiom,
    ! [B4: nat,A4: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B4),A4)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_1715_verit__comp__simplify1_I3_J,axiom,
    ! [B4: int,A4: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),A4)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_1716_triangle__Suc,axiom,
    ! [Na: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,Na)),aa_nat_nat(suc,Na)) ).

% triangle_Suc
tff(fact_1717_less__numeral__extra_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),zero_zero_real) ).

% less_numeral_extra(3)
tff(fact_1718_less__numeral__extra_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_zero_rat) ).

% less_numeral_extra(3)
tff(fact_1719_less__numeral__extra_I3_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),zero_zero_nat) ).

% less_numeral_extra(3)
tff(fact_1720_less__numeral__extra_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_numeral_extra(3)
tff(fact_1721_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),X2) ) ).

% minf(8)
tff(fact_1722_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,T),X2) ) ).

% minf(8)
tff(fact_1723_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,T),X2) ) ).

% minf(8)
tff(fact_1724_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,T),X2) ) ).

% minf(8)
tff(fact_1725_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X2) ) ).

% minf(8)
tff(fact_1726_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),T) ) ).

% minf(6)
tff(fact_1727_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),T) ) ).

% minf(6)
tff(fact_1728_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),T) ) ).

% minf(6)
tff(fact_1729_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),T) ) ).

% minf(6)
tff(fact_1730_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),T) ) ).

% minf(6)
tff(fact_1731_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),X2) ) ).

% pinf(8)
tff(fact_1732_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,T),X2) ) ).

% pinf(8)
tff(fact_1733_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,T),X2) ) ).

% pinf(8)
tff(fact_1734_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,T),X2) ) ).

% pinf(8)
tff(fact_1735_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X2) ) ).

% pinf(8)
tff(fact_1736_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),T) ) ).

% pinf(6)
tff(fact_1737_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),T) ) ).

% pinf(6)
tff(fact_1738_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),T) ) ).

% pinf(6)
tff(fact_1739_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),T) ) ).

% pinf(6)
tff(fact_1740_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),T) ) ).

% pinf(6)
tff(fact_1741_complete__interval,axiom,
    ! [A: real,B: real,P: fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(P,A)
       => ( ~ aa_real_o(P,B)
         => ? [C2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C2)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),B)
              & ! [X2: real] :
                  ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
                    & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),C2) )
                 => aa_real_o(P,X2) )
              & ! [D3: real] :
                  ( ! [X3: real] :
                      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
                        & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),D3) )
                     => aa_real_o(P,X3) )
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D3),C2) ) ) ) ) ) ).

% complete_interval
tff(fact_1742_complete__interval,axiom,
    ! [A: nat,B: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(P,A)
       => ( ~ aa_nat_o(P,B)
         => ? [C2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B)
              & ! [X2: nat] :
                  ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X2)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),C2) )
                 => aa_nat_o(P,X2) )
              & ! [D3: nat] :
                  ( ! [X3: 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_nat,X3),D3) )
                     => aa_nat_o(P,X3) )
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D3),C2) ) ) ) ) ) ).

% complete_interval
tff(fact_1743_complete__interval,axiom,
    ! [A: int,B: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(P,A)
       => ( ~ aa_int_o(P,B)
         => ? [C2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B)
              & ! [X2: int] :
                  ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X2)
                    & aa_int_o(aa_int_fun_int_o(ord_less_int,X2),C2) )
                 => aa_int_o(P,X2) )
              & ! [D3: int] :
                  ( ! [X3: 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_int,X3),D3) )
                     => aa_int_o(P,X3) )
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D3),C2) ) ) ) ) ) ).

% complete_interval
tff(fact_1744_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_1745_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_1746_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_1747_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_1748_find__Some__iff,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( ( find_P8199882355184865565at_nat(P,Xs) = some_P7363390416028606310at_nat(X) )
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s5460976970255530739at_nat(Xs))
          & aa_Pro6150637604445964947_nat_o(P,nth_Pr7617993195940197384at_nat(Xs,I2))
          & ( X = nth_Pr7617993195940197384at_nat(Xs,I2) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I2)
             => ~ aa_Pro6150637604445964947_nat_o(P,nth_Pr7617993195940197384at_nat(Xs,J3)) ) ) ) ).

% find_Some_iff
tff(fact_1749_find__Some__iff,axiom,
    ! [P: fun_num_o,Xs: list_num,X: num] :
      ( ( find_num(P,Xs) = some_num(X) )
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_num(Xs))
          & aa_num_o(P,aa_nat_num(nth_num(Xs),I2))
          & ( X = aa_nat_num(nth_num(Xs),I2) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I2)
             => ~ aa_num_o(P,aa_nat_num(nth_num(Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_1750_find__Some__iff,axiom,
    ! [P: fun_VEBT_VEBT_o,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( find_VEBT_VEBT(P,Xs) = some_VEBT_VEBT(X) )
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
          & aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I2))
          & ( X = nth_VEBT_VEBT(Xs,I2) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I2)
             => ~ aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,J3)) ) ) ) ).

% find_Some_iff
tff(fact_1751_find__Some__iff,axiom,
    ! [P: fun_o_o,Xs: list_o,X: $o] :
      ( ( find_o(P,Xs) = some_o((X)) )
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_o_nat(size_size_list_o,Xs))
          & aa_o_o(P,aa_nat_o(nth_o(Xs),I2))
          & ( (X)
          <=> aa_nat_o(nth_o(Xs),I2) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I2)
             => ~ aa_o_o(P,aa_nat_o(nth_o(Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_1752_find__Some__iff,axiom,
    ! [P: fun_nat_o,Xs: list_nat,X: nat] :
      ( ( find_nat(P,Xs) = some_nat(X) )
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs))
          & aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I2))
          & ( X = aa_nat_nat(nth_nat(Xs),I2) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I2)
             => ~ aa_nat_o(P,aa_nat_nat(nth_nat(Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_1753_find__Some__iff,axiom,
    ! [P: fun_int_o,Xs: list_int,X: int] :
      ( ( find_int(P,Xs) = some_int(X) )
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_int_nat(size_size_list_int,Xs))
          & aa_int_o(P,aa_nat_int(nth_int(Xs),I2))
          & ( X = aa_nat_int(nth_int(Xs),I2) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I2)
             => ~ aa_int_o(P,aa_nat_int(nth_int(Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_1754_find__Some__iff2,axiom,
    ! [X: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o,Xs: list_P6011104703257516679at_nat] :
      ( ( some_P7363390416028606310at_nat(X) = find_P8199882355184865565at_nat(P,Xs) )
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s5460976970255530739at_nat(Xs))
          & aa_Pro6150637604445964947_nat_o(P,nth_Pr7617993195940197384at_nat(Xs,I2))
          & ( X = nth_Pr7617993195940197384at_nat(Xs,I2) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I2)
             => ~ aa_Pro6150637604445964947_nat_o(P,nth_Pr7617993195940197384at_nat(Xs,J3)) ) ) ) ).

% find_Some_iff2
tff(fact_1755_find__Some__iff2,axiom,
    ! [X: num,P: fun_num_o,Xs: list_num] :
      ( ( some_num(X) = find_num(P,Xs) )
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_num(Xs))
          & aa_num_o(P,aa_nat_num(nth_num(Xs),I2))
          & ( X = aa_nat_num(nth_num(Xs),I2) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I2)
             => ~ aa_num_o(P,aa_nat_num(nth_num(Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_1756_find__Some__iff2,axiom,
    ! [X: vEBT_VEBT,P: fun_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ( some_VEBT_VEBT(X) = find_VEBT_VEBT(P,Xs) )
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
          & aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I2))
          & ( X = nth_VEBT_VEBT(Xs,I2) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I2)
             => ~ aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,J3)) ) ) ) ).

% find_Some_iff2
tff(fact_1757_find__Some__iff2,axiom,
    ! [X: $o,P: fun_o_o,Xs: list_o] :
      ( ( some_o((X)) = find_o(P,Xs) )
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_o_nat(size_size_list_o,Xs))
          & aa_o_o(P,aa_nat_o(nth_o(Xs),I2))
          & ( (X)
          <=> aa_nat_o(nth_o(Xs),I2) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I2)
             => ~ aa_o_o(P,aa_nat_o(nth_o(Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_1758_find__Some__iff2,axiom,
    ! [X: nat,P: fun_nat_o,Xs: list_nat] :
      ( ( some_nat(X) = find_nat(P,Xs) )
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs))
          & aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I2))
          & ( X = aa_nat_nat(nth_nat(Xs),I2) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I2)
             => ~ aa_nat_o(P,aa_nat_nat(nth_nat(Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_1759_find__Some__iff2,axiom,
    ! [X: int,P: fun_int_o,Xs: list_int] :
      ( ( some_int(X) = find_int(P,Xs) )
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_int_nat(size_size_list_int,Xs))
          & aa_int_o(P,aa_nat_int(nth_int(Xs),I2))
          & ( X = aa_nat_int(nth_int(Xs),I2) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I2)
             => ~ aa_int_o(P,aa_nat_int(nth_int(Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_1760_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_1761_find__cong,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,P: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] :
      ( ( Xs = Ys )
     => ( ! [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),set_Pr5648618587558075414at_nat(Ys))
           => ( aa_Pro6150637604445964947_nat_o(P,X3)
            <=> aa_Pro6150637604445964947_nat_o(Q,X3) ) )
       => ( find_P8199882355184865565at_nat(P,Xs) = find_P8199882355184865565at_nat(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1762_find__cong,axiom,
    ! [Xs: list_real,Ys: list_real,P: fun_real_o,Q: fun_real_o] :
      ( ( Xs = Ys )
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),set_real2(Ys))
           => ( aa_real_o(P,X3)
            <=> aa_real_o(Q,X3) ) )
       => ( find_real(P,Xs) = find_real(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1763_find__cong,axiom,
    ! [Xs: list_set_nat,Ys: list_set_nat,P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ( Xs = Ys )
     => ( ! [X3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X3),set_set_nat2(Ys))
           => ( aa_set_nat_o(P,X3)
            <=> aa_set_nat_o(Q,X3) ) )
       => ( find_set_nat(P,Xs) = find_set_nat(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1764_find__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Q: fun_VEBT_VEBT_o] :
      ( ( Xs = Ys )
     => ( ! [X3: vEBT_VEBT] :
            ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Ys))
           => ( aa_VEBT_VEBT_o(P,X3)
            <=> aa_VEBT_VEBT_o(Q,X3) ) )
       => ( find_VEBT_VEBT(P,Xs) = find_VEBT_VEBT(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1765_find__cong,axiom,
    ! [Xs: list_int,Ys: list_int,P: fun_int_o,Q: fun_int_o] :
      ( ( Xs = Ys )
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),set_int2(Ys))
           => ( aa_int_o(P,X3)
            <=> aa_int_o(Q,X3) ) )
       => ( find_int(P,Xs) = find_int(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1766_find__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,P: fun_nat_o,Q: fun_nat_o] :
      ( ( Xs = Ys )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),set_nat2(Ys))
           => ( aa_nat_o(P,X3)
            <=> aa_nat_o(Q,X3) ) )
       => ( find_nat(P,Xs) = find_nat(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1767_ex__gt__or__lt,axiom,
    ! [A: real] :
    ? [B3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B3)
      | aa_real_o(aa_real_fun_real_o(ord_less_real,B3),A) ) ).

% ex_gt_or_lt
tff(fact_1768_pinf_I1_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q3: fun_real_o] :
      ( ? [Z3: real] :
        ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
         => ( aa_real_o(P,X3)
          <=> aa_real_o(P3,X3) ) )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
           => ( aa_real_o(Q,X3)
            <=> aa_real_o(Q3,X3) ) )
       => ? [Z2: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X2)
           => ( ( aa_real_o(P,X2)
                & aa_real_o(Q,X2) )
            <=> ( aa_real_o(P3,X2)
                & aa_real_o(Q3,X2) ) ) ) ) ) ).

% pinf(1)
tff(fact_1769_pinf_I1_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q3: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
         => ( aa_rat_o(P,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z3: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
           => ( aa_rat_o(Q,X3)
            <=> aa_rat_o(Q3,X3) ) )
       => ? [Z2: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X2)
           => ( ( aa_rat_o(P,X2)
                & aa_rat_o(Q,X2) )
            <=> ( aa_rat_o(P3,X2)
                & aa_rat_o(Q3,X2) ) ) ) ) ) ).

% pinf(1)
tff(fact_1770_pinf_I1_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q3: fun_num_o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
         => ( aa_num_o(P,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
           => ( aa_num_o(Q,X3)
            <=> aa_num_o(Q3,X3) ) )
       => ? [Z2: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X2)
           => ( ( aa_num_o(P,X2)
                & aa_num_o(Q,X2) )
            <=> ( aa_num_o(P3,X2)
                & aa_num_o(Q3,X2) ) ) ) ) ) ).

% pinf(1)
tff(fact_1771_pinf_I1_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q3: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
         => ( aa_nat_o(P,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
           => ( aa_nat_o(Q,X3)
            <=> aa_nat_o(Q3,X3) ) )
       => ? [Z2: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X2)
           => ( ( aa_nat_o(P,X2)
                & aa_nat_o(Q,X2) )
            <=> ( aa_nat_o(P3,X2)
                & aa_nat_o(Q3,X2) ) ) ) ) ) ).

% pinf(1)
tff(fact_1772_pinf_I1_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q3: fun_int_o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
         => ( aa_int_o(P,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
           => ( aa_int_o(Q,X3)
            <=> aa_int_o(Q3,X3) ) )
       => ? [Z2: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X2)
           => ( ( aa_int_o(P,X2)
                & aa_int_o(Q,X2) )
            <=> ( aa_int_o(P3,X2)
                & aa_int_o(Q3,X2) ) ) ) ) ) ).

% pinf(1)
tff(fact_1773_pinf_I2_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q3: fun_real_o] :
      ( ? [Z3: real] :
        ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
         => ( aa_real_o(P,X3)
          <=> aa_real_o(P3,X3) ) )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
           => ( aa_real_o(Q,X3)
            <=> aa_real_o(Q3,X3) ) )
       => ? [Z2: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X2)
           => ( ( aa_real_o(P,X2)
                | aa_real_o(Q,X2) )
            <=> ( aa_real_o(P3,X2)
                | aa_real_o(Q3,X2) ) ) ) ) ) ).

% pinf(2)
tff(fact_1774_pinf_I2_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q3: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
         => ( aa_rat_o(P,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z3: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
           => ( aa_rat_o(Q,X3)
            <=> aa_rat_o(Q3,X3) ) )
       => ? [Z2: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X2)
           => ( ( aa_rat_o(P,X2)
                | aa_rat_o(Q,X2) )
            <=> ( aa_rat_o(P3,X2)
                | aa_rat_o(Q3,X2) ) ) ) ) ) ).

% pinf(2)
tff(fact_1775_pinf_I2_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q3: fun_num_o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
         => ( aa_num_o(P,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
           => ( aa_num_o(Q,X3)
            <=> aa_num_o(Q3,X3) ) )
       => ? [Z2: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X2)
           => ( ( aa_num_o(P,X2)
                | aa_num_o(Q,X2) )
            <=> ( aa_num_o(P3,X2)
                | aa_num_o(Q3,X2) ) ) ) ) ) ).

% pinf(2)
tff(fact_1776_pinf_I2_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q3: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
         => ( aa_nat_o(P,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
           => ( aa_nat_o(Q,X3)
            <=> aa_nat_o(Q3,X3) ) )
       => ? [Z2: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X2)
           => ( ( aa_nat_o(P,X2)
                | aa_nat_o(Q,X2) )
            <=> ( aa_nat_o(P3,X2)
                | aa_nat_o(Q3,X2) ) ) ) ) ) ).

% pinf(2)
tff(fact_1777_pinf_I2_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q3: fun_int_o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
         => ( aa_int_o(P,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
           => ( aa_int_o(Q,X3)
            <=> aa_int_o(Q3,X3) ) )
       => ? [Z2: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X2)
           => ( ( aa_int_o(P,X2)
                | aa_int_o(Q,X2) )
            <=> ( aa_int_o(P3,X2)
                | aa_int_o(Q3,X2) ) ) ) ) ) ).

% pinf(2)
tff(fact_1778_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X2)
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_1779_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X2)
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_1780_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X2)
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_1781_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X2)
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_1782_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X2)
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_1783_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X2)
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_1784_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X2)
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_1785_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X2)
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_1786_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X2)
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_1787_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X2)
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_1788_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X2),T) ) ).

% pinf(5)
tff(fact_1789_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),T) ) ).

% pinf(5)
tff(fact_1790_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X2),T) ) ).

% pinf(5)
tff(fact_1791_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),T) ) ).

% pinf(5)
tff(fact_1792_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X2),T) ) ).

% pinf(5)
tff(fact_1793_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,T),X2) ) ).

% pinf(7)
tff(fact_1794_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,T),X2) ) ).

% pinf(7)
tff(fact_1795_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X2)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,T),X2) ) ).

% pinf(7)
tff(fact_1796_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,T),X2) ) ).

% pinf(7)
tff(fact_1797_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,T),X2) ) ).

% pinf(7)
tff(fact_1798_minf_I1_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q3: fun_real_o] :
      ( ? [Z3: real] :
        ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
         => ( aa_real_o(P,X3)
          <=> aa_real_o(P3,X3) ) )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
           => ( aa_real_o(Q,X3)
            <=> aa_real_o(Q3,X3) ) )
       => ? [Z2: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z2)
           => ( ( aa_real_o(P,X2)
                & aa_real_o(Q,X2) )
            <=> ( aa_real_o(P3,X2)
                & aa_real_o(Q3,X2) ) ) ) ) ) ).

% minf(1)
tff(fact_1799_minf_I1_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q3: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
         => ( aa_rat_o(P,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z3: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
           => ( aa_rat_o(Q,X3)
            <=> aa_rat_o(Q3,X3) ) )
       => ? [Z2: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z2)
           => ( ( aa_rat_o(P,X2)
                & aa_rat_o(Q,X2) )
            <=> ( aa_rat_o(P3,X2)
                & aa_rat_o(Q3,X2) ) ) ) ) ) ).

% minf(1)
tff(fact_1800_minf_I1_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q3: fun_num_o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
         => ( aa_num_o(P,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
           => ( aa_num_o(Q,X3)
            <=> aa_num_o(Q3,X3) ) )
       => ? [Z2: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z2)
           => ( ( aa_num_o(P,X2)
                & aa_num_o(Q,X2) )
            <=> ( aa_num_o(P3,X2)
                & aa_num_o(Q3,X2) ) ) ) ) ) ).

% minf(1)
tff(fact_1801_minf_I1_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q3: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
         => ( aa_nat_o(P,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
           => ( aa_nat_o(Q,X3)
            <=> aa_nat_o(Q3,X3) ) )
       => ? [Z2: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z2)
           => ( ( aa_nat_o(P,X2)
                & aa_nat_o(Q,X2) )
            <=> ( aa_nat_o(P3,X2)
                & aa_nat_o(Q3,X2) ) ) ) ) ) ).

% minf(1)
tff(fact_1802_minf_I1_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q3: fun_int_o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
         => ( aa_int_o(P,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
           => ( aa_int_o(Q,X3)
            <=> aa_int_o(Q3,X3) ) )
       => ? [Z2: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z2)
           => ( ( aa_int_o(P,X2)
                & aa_int_o(Q,X2) )
            <=> ( aa_int_o(P3,X2)
                & aa_int_o(Q3,X2) ) ) ) ) ) ).

% minf(1)
tff(fact_1803_minf_I2_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q3: fun_real_o] :
      ( ? [Z3: real] :
        ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
         => ( aa_real_o(P,X3)
          <=> aa_real_o(P3,X3) ) )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
           => ( aa_real_o(Q,X3)
            <=> aa_real_o(Q3,X3) ) )
       => ? [Z2: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z2)
           => ( ( aa_real_o(P,X2)
                | aa_real_o(Q,X2) )
            <=> ( aa_real_o(P3,X2)
                | aa_real_o(Q3,X2) ) ) ) ) ) ).

% minf(2)
tff(fact_1804_minf_I2_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q3: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
         => ( aa_rat_o(P,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z3: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
           => ( aa_rat_o(Q,X3)
            <=> aa_rat_o(Q3,X3) ) )
       => ? [Z2: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z2)
           => ( ( aa_rat_o(P,X2)
                | aa_rat_o(Q,X2) )
            <=> ( aa_rat_o(P3,X2)
                | aa_rat_o(Q3,X2) ) ) ) ) ) ).

% minf(2)
tff(fact_1805_minf_I2_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q3: fun_num_o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
         => ( aa_num_o(P,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
           => ( aa_num_o(Q,X3)
            <=> aa_num_o(Q3,X3) ) )
       => ? [Z2: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z2)
           => ( ( aa_num_o(P,X2)
                | aa_num_o(Q,X2) )
            <=> ( aa_num_o(P3,X2)
                | aa_num_o(Q3,X2) ) ) ) ) ) ).

% minf(2)
tff(fact_1806_minf_I2_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q3: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
         => ( aa_nat_o(P,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
           => ( aa_nat_o(Q,X3)
            <=> aa_nat_o(Q3,X3) ) )
       => ? [Z2: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z2)
           => ( ( aa_nat_o(P,X2)
                | aa_nat_o(Q,X2) )
            <=> ( aa_nat_o(P3,X2)
                | aa_nat_o(Q3,X2) ) ) ) ) ) ).

% minf(2)
tff(fact_1807_minf_I2_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q3: fun_int_o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
         => ( aa_int_o(P,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
           => ( aa_int_o(Q,X3)
            <=> aa_int_o(Q3,X3) ) )
       => ? [Z2: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z2)
           => ( ( aa_int_o(P,X2)
                | aa_int_o(Q,X2) )
            <=> ( aa_int_o(P3,X2)
                | aa_int_o(Q3,X2) ) ) ) ) ) ).

% minf(2)
tff(fact_1808_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z2)
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_1809_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z2)
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_1810_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z2)
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_1811_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z2)
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_1812_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z2)
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_1813_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z2)
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_1814_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z2)
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_1815_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z2)
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_1816_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z2)
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_1817_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z2)
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_1818_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X2),T) ) ).

% minf(5)
tff(fact_1819_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),T) ) ).

% minf(5)
tff(fact_1820_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z2)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,X2),T) ) ).

% minf(5)
tff(fact_1821_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),T) ) ).

% minf(5)
tff(fact_1822_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X2),T) ) ).

% minf(5)
tff(fact_1823_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,T),X2) ) ).

% minf(7)
tff(fact_1824_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,T),X2) ) ).

% minf(7)
tff(fact_1825_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,T),X2) ) ).

% minf(7)
tff(fact_1826_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,T),X2) ) ).

% minf(7)
tff(fact_1827_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,T),X2) ) ).

% minf(7)
tff(fact_1828_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% is_num_normalize(1)
tff(fact_1829_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% is_num_normalize(1)
tff(fact_1830_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_1831_find__None__iff2,axiom,
    ! [P: fun_real_o,Xs: list_real] :
      ( ( none_real = find_real(P,Xs) )
    <=> ~ ? [X4: real] :
            ( aa_set_real_o(member_real(X4),set_real2(Xs))
            & aa_real_o(P,X4) ) ) ).

% find_None_iff2
tff(fact_1832_find__None__iff2,axiom,
    ! [P: fun_set_nat_o,Xs: list_set_nat] :
      ( ( none_set_nat = find_set_nat(P,Xs) )
    <=> ~ ? [X4: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X4),set_set_nat2(Xs))
            & aa_set_nat_o(P,X4) ) ) ).

% find_None_iff2
tff(fact_1833_find__None__iff2,axiom,
    ! [P: fun_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ( none_VEBT_VEBT = find_VEBT_VEBT(P,Xs) )
    <=> ~ ? [X4: vEBT_VEBT] :
            ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
            & aa_VEBT_VEBT_o(P,X4) ) ) ).

% find_None_iff2
tff(fact_1834_find__None__iff2,axiom,
    ! [P: fun_int_o,Xs: list_int] :
      ( ( none_int = find_int(P,Xs) )
    <=> ~ ? [X4: int] :
            ( aa_set_int_o(member_int(X4),set_int2(Xs))
            & aa_int_o(P,X4) ) ) ).

% find_None_iff2
tff(fact_1835_find__None__iff2,axiom,
    ! [P: fun_nat_o,Xs: list_nat] :
      ( ( none_nat = find_nat(P,Xs) )
    <=> ~ ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
            & aa_nat_o(P,X4) ) ) ).

% find_None_iff2
tff(fact_1836_find__None__iff2,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Xs: list_P6011104703257516679at_nat] :
      ( ( none_P5556105721700978146at_nat = find_P8199882355184865565at_nat(P,Xs) )
    <=> ~ ? [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),set_Pr5648618587558075414at_nat(Xs))
            & aa_Pro6150637604445964947_nat_o(P,X4) ) ) ).

% find_None_iff2
tff(fact_1837_find__None__iff2,axiom,
    ! [P: fun_num_o,Xs: list_num] :
      ( ( none_num = find_num(P,Xs) )
    <=> ~ ? [X4: num] :
            ( aa_set_num_o(member_num(X4),set_num2(Xs))
            & aa_num_o(P,X4) ) ) ).

% find_None_iff2
tff(fact_1838_find__None__iff,axiom,
    ! [P: fun_real_o,Xs: list_real] :
      ( ( find_real(P,Xs) = none_real )
    <=> ~ ? [X4: real] :
            ( aa_set_real_o(member_real(X4),set_real2(Xs))
            & aa_real_o(P,X4) ) ) ).

% find_None_iff
tff(fact_1839_find__None__iff,axiom,
    ! [P: fun_set_nat_o,Xs: list_set_nat] :
      ( ( find_set_nat(P,Xs) = none_set_nat )
    <=> ~ ? [X4: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X4),set_set_nat2(Xs))
            & aa_set_nat_o(P,X4) ) ) ).

% find_None_iff
tff(fact_1840_find__None__iff,axiom,
    ! [P: fun_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ( find_VEBT_VEBT(P,Xs) = none_VEBT_VEBT )
    <=> ~ ? [X4: vEBT_VEBT] :
            ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
            & aa_VEBT_VEBT_o(P,X4) ) ) ).

% find_None_iff
tff(fact_1841_find__None__iff,axiom,
    ! [P: fun_int_o,Xs: list_int] :
      ( ( find_int(P,Xs) = none_int )
    <=> ~ ? [X4: int] :
            ( aa_set_int_o(member_int(X4),set_int2(Xs))
            & aa_int_o(P,X4) ) ) ).

% find_None_iff
tff(fact_1842_find__None__iff,axiom,
    ! [P: fun_nat_o,Xs: list_nat] :
      ( ( find_nat(P,Xs) = none_nat )
    <=> ~ ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
            & aa_nat_o(P,X4) ) ) ).

% find_None_iff
tff(fact_1843_find__None__iff,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Xs: list_P6011104703257516679at_nat] :
      ( ( find_P8199882355184865565at_nat(P,Xs) = none_P5556105721700978146at_nat )
    <=> ~ ? [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),set_Pr5648618587558075414at_nat(Xs))
            & aa_Pro6150637604445964947_nat_o(P,X4) ) ) ).

% find_None_iff
tff(fact_1844_find__None__iff,axiom,
    ! [P: fun_num_o,Xs: list_num] :
      ( ( find_num(P,Xs) = none_num )
    <=> ~ ? [X4: num] :
            ( aa_set_num_o(member_num(X4),set_num2(Xs))
            & aa_num_o(P,X4) ) ) ).

% find_None_iff
tff(fact_1845_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys))
       => ( nth_Pr4953567300277697838T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys),I) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,I),nth_VEBT_VEBT(Ys,I)) ) ) ) ).

% nth_zip
tff(fact_1846_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_o_nat(size_size_list_o,Ys))
       => ( nth_Pr4606735188037164562VEBT_o(zip_VEBT_VEBT_o(Xs,Ys),I) = produc8721562602347293563VEBT_o(nth_VEBT_VEBT(Xs,I),aa_nat_o(nth_o(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_1847_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Ys))
       => ( nth_Pr1791586995822124652BT_nat(zip_VEBT_VEBT_nat(Xs,Ys),I) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,I),aa_nat_nat(nth_nat(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_1848_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Ys))
       => ( nth_Pr6837108013167703752BT_int(zip_VEBT_VEBT_int(Xs,Ys),I) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,I),aa_nat_int(nth_int(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_1849_nth__zip,axiom,
    ! [I: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_o_nat(size_size_list_o,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys))
       => ( nth_Pr6777367263587873994T_VEBT(zip_o_VEBT_VEBT(Xs,Ys),I) = produc2982872950893828659T_VEBT(aa_nat_o(nth_o(Xs),I),nth_VEBT_VEBT(Ys,I)) ) ) ) ).

% nth_zip
tff(fact_1850_nth__zip,axiom,
    ! [I: nat,Xs: list_o,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_o_nat(size_size_list_o,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_o_nat(size_size_list_o,Ys))
       => ( nth_Product_prod_o_o(zip_o_o(Xs,Ys),I) = product_Pair_o_o(aa_nat_o(nth_o(Xs),I),aa_nat_o(nth_o(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_1851_nth__zip,axiom,
    ! [I: nat,Xs: list_o,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_o_nat(size_size_list_o,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Ys))
       => ( nth_Pr5826913651314560976_o_nat(zip_o_nat(Xs,Ys),I) = product_Pair_o_nat(aa_nat_o(nth_o(Xs),I),aa_nat_nat(nth_nat(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_1852_nth__zip,axiom,
    ! [I: nat,Xs: list_o,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_o_nat(size_size_list_o,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Ys))
       => ( nth_Pr1649062631805364268_o_int(zip_o_int(Xs,Ys),I) = product_Pair_o_int(aa_nat_o(nth_o(Xs),I),aa_nat_int(nth_int(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_1853_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys))
       => ( nth_Pr744662078594809490T_VEBT(zip_nat_VEBT_VEBT(Xs,Ys),I) = produc599794634098209291T_VEBT(aa_nat_nat(nth_nat(Xs),I),nth_VEBT_VEBT(Ys,I)) ) ) ) ).

% nth_zip
tff(fact_1854_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_o_nat(size_size_list_o,Ys))
       => ( nth_Pr112076138515278198_nat_o(zip_nat_o(Xs,Ys),I) = product_Pair_nat_o(aa_nat_nat(nth_nat(Xs),I),aa_nat_o(nth_o(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_1855__C5_C_I10_J,axiom,
    ( ( mi != ma )
   => ! [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))),m))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),n) = I4 )
           => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),n)) )
          & ! [X2: nat] :
              ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),n) = I4 )
                & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),n)) )
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,mi),X2)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),ma) ) ) ) ) ) ).

% "5"(10)
tff(fact_1856_vebt__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),X3)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),X3)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),X3)
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X3)
           => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),X3) ) ) ) ) ).

% vebt_member.cases
tff(fact_1857_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Uw2)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Uz2)
       => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),X3)
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),X3)
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd),X3) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_1858_count__notin,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs))
     => ( aa_Pro4243173770044397973at_nat(count_4203492906077236349at_nat(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1859_count__notin,axiom,
    ! [X: real,Xs: list_real] :
      ( ~ aa_set_real_o(member_real(X),set_real2(Xs))
     => ( aa_real_nat(count_list_real(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1860_count__notin,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(X),set_set_nat2(Xs))
     => ( aa_set_nat_nat(count_list_set_nat(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1861_count__notin,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ~ member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
     => ( aa_VEBT_VEBT_nat(count_list_VEBT_VEBT(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1862_count__notin,axiom,
    ! [X: int,Xs: list_int] :
      ( ~ aa_set_int_o(member_int(X),set_int2(Xs))
     => ( aa_int_nat(count_list_int(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1863_count__notin,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ aa_set_nat_o(member_nat(X),set_nat2(Xs))
     => ( aa_nat_nat(count_list_nat(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1864_listrel1__iff__update,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
      ( member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(Xs,Ys),listrel1_VEBT_VEBT(R2))
    <=> ? [Y5: vEBT_VEBT,N3: nat] :
          ( member568628332442017744T_VEBT(produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,N3),Y5),R2)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs))
          & ( Ys = list_u1324408373059187874T_VEBT(Xs,N3,Y5) ) ) ) ).

% listrel1_iff_update
tff(fact_1865_listrel1__iff__update,axiom,
    ! [Xs: list_o,Ys: list_o,R2: set_Product_prod_o_o] :
      ( member4159035015898711888list_o(produc8435520187683070743list_o(Xs,Ys),listrel1_o(R2))
    <=> ? [Y5: $o,N3: nat] :
          ( member7466972457876170832od_o_o(product_Pair_o_o(aa_nat_o(nth_o(Xs),N3),(Y5)),R2)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),aa_list_o_nat(size_size_list_o,Xs))
          & ( Ys = list_update_o(Xs,N3,(Y5)) ) ) ) ).

% listrel1_iff_update
tff(fact_1866_listrel1__iff__update,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),listrel1_nat(R2))
    <=> ? [Y5: nat,N3: nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(nth_nat(Xs),N3)),Y5)),R2)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),aa_list_nat_nat(size_size_list_nat,Xs))
          & ( Ys = list_update_nat(Xs,N3,Y5) ) ) ) ).

% listrel1_iff_update
tff(fact_1867_listrel1__iff__update,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),listrel1_int(R2))
    <=> ? [Y5: int,N3: nat] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(nth_int(Xs),N3)),Y5),R2)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),aa_list_int_nat(size_size_list_int,Xs))
          & ( Ys = list_update_int(Xs,N3,Y5) ) ) ) ).

% listrel1_iff_update
tff(fact_1868_pair__lessI2,axiom,
    ! [A: nat,B: nat,S: nat,T: 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,S),T)
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_1869_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z)),fun_pair_less)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z) ) ).

% pair_less_iff1
tff(fact_1870_gen__length__def,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT] : aa_lis7044684121217598542BT_nat(gen_length_VEBT_VEBT(Na),Xs) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)) ).

% gen_length_def
tff(fact_1871_gen__length__def,axiom,
    ! [Na: nat,Xs: list_o] : aa_list_o_nat(gen_length_o(Na),Xs) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_list_o_nat(size_size_list_o,Xs)) ).

% gen_length_def
tff(fact_1872_gen__length__def,axiom,
    ! [Na: nat,Xs: list_nat] : aa_list_nat_nat(gen_length_nat(Na),Xs) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_list_nat_nat(size_size_list_nat,Xs)) ).

% gen_length_def
tff(fact_1873_gen__length__def,axiom,
    ! [Na: nat,Xs: list_int] : aa_list_int_nat(gen_length_int(Na),Xs) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_list_int_nat(size_size_list_int,Xs)) ).

% gen_length_def
tff(fact_1874_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_1875_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_1876_power__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,some_nat(X)),some_nat(Y)) = some_nat(Z) ) ) ).

% power_shift
tff(fact_1877__C5_C_I3_J,axiom,
    aa_lis7044684121217598542BT_nat(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) ).

% "5"(3)
tff(fact_1878__C5_C_I9_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)) ).

% "5"(9)
tff(fact_1879_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_1880_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_1881_high__bound__aux,axiom,
    ! [Maa: nat,Na: 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,Na),Mb)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Na)),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_1882_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,Na: nat] :
      ( aa_nat_o(vEBT_vebt_member(Tree),X)
     => ( aa_nat_o(vEBT_invar_vebt(Tree),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ).

% member_bound
tff(fact_1883__C5_C_I6_J,axiom,
    ! [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))),m))
     => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I4)),X_13)
      <=> aa_nat_o(vEBT_V8194947554948674370ptions(summary),I4) ) ) ).

% "5"(6)
tff(fact_1884_numeral__le__iff,axiom,
    ! [Mb: num,Na: num] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Mb)),numera1916890842035813515d_enat(Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Na) ) ).

% numeral_le_iff
tff(fact_1885_numeral__le__iff,axiom,
    ! [Mb: num,Na: 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,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Na) ) ).

% numeral_le_iff
tff(fact_1886_numeral__le__iff,axiom,
    ! [Mb: num,Na: 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,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Na) ) ).

% numeral_le_iff
tff(fact_1887_numeral__le__iff,axiom,
    ! [Mb: num,Na: 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,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Na) ) ).

% numeral_le_iff
tff(fact_1888_numeral__le__iff,axiom,
    ! [Mb: num,Na: 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,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Na) ) ).

% numeral_le_iff
tff(fact_1889_numeral__less__iff,axiom,
    ! [Mb: num,Na: num] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(Mb)),numera1916890842035813515d_enat(Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Na) ) ).

% numeral_less_iff
tff(fact_1890_numeral__less__iff,axiom,
    ! [Mb: num,Na: 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,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Na) ) ).

% numeral_less_iff
tff(fact_1891_numeral__less__iff,axiom,
    ! [Mb: num,Na: 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,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Na) ) ).

% numeral_less_iff
tff(fact_1892_numeral__less__iff,axiom,
    ! [Mb: num,Na: 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,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Na) ) ).

% numeral_less_iff
tff(fact_1893_numeral__less__iff,axiom,
    ! [Mb: num,Na: 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,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Na) ) ).

% numeral_less_iff
tff(fact_1894_numeral__plus__numeral,axiom,
    ! [Mb: num,Na: num] : plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Mb),numera1916890842035813515d_enat(Na)) = numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Na)) ).

% numeral_plus_numeral
tff(fact_1895_numeral__plus__numeral,axiom,
    ! [Mb: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Na)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Na)) ).

% numeral_plus_numeral
tff(fact_1896_numeral__plus__numeral,axiom,
    ! [Mb: num,Na: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Na)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Na)) ).

% numeral_plus_numeral
tff(fact_1897_numeral__plus__numeral,axiom,
    ! [Mb: num,Na: 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,Na)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Na)) ).

% numeral_plus_numeral
tff(fact_1898_numeral__plus__numeral,axiom,
    ! [Mb: num,Na: 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,Na)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Na)) ).

% numeral_plus_numeral
tff(fact_1899_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: extended_enat] : plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(V),plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(W2),Z)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2)),Z) ).

% add_numeral_left
tff(fact_1900_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,W2)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2))),Z) ).

% add_numeral_left
tff(fact_1901_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: rat] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,W2)),Z)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2))),Z) ).

% add_numeral_left
tff(fact_1902_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,W2)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2))),Z) ).

% add_numeral_left
tff(fact_1903_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,W2)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2))),Z) ).

% add_numeral_left
tff(fact_1904_mi__ma__2__deg,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Na)
     => ( 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_1905_Suc__numeral,axiom,
    ! [Na: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Na),one)) ).

% Suc_numeral
tff(fact_1906_add__2__eq__Suc_H,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,Na)) ).

% add_2_eq_Suc'
tff(fact_1907_add__2__eq__Suc,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) = aa_nat_nat(suc,aa_nat_nat(suc,Na)) ).

% add_2_eq_Suc
tff(fact_1908_numeral__Bit0,axiom,
    ! [Na: num] : numera1916890842035813515d_enat(aa_num_num(bit0,Na)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Na),numera1916890842035813515d_enat(Na)) ).

% numeral_Bit0
tff(fact_1909_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Na)),aa_num_real(numeral_numeral_real,Na)) ).

% numeral_Bit0
tff(fact_1910_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Na)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Na)),aa_num_rat(numeral_numeral_rat,Na)) ).

% numeral_Bit0
tff(fact_1911_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,Na)) ).

% numeral_Bit0
tff(fact_1912_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,Na)) ).

% numeral_Bit0
tff(fact_1913_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_1914_less__2__cases,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
     => ( ( Na = zero_zero_nat )
        | ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_1915_less__2__cases__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
    <=> ( ( Na = zero_zero_nat )
        | ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_1916_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Mb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),Na)),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_1917_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Mb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_1918_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_1919_Suc__nat__number__of__add,axiom,
    ! [V: num,Na: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),one))),Na) ).

% Suc_nat_number_of_add
tff(fact_1920_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_1921_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D4: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),D4)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Deg2) ) ).

% VEBT_internal.valid'.cases
tff(fact_1922_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_1923_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_1924_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_complex != numera6690914467698888265omplex(Na) ).

% zero_neq_numeral
tff(fact_1925_zero__neq__numeral,axiom,
    ! [Na: num] : zero_z5237406670263579293d_enat != numera1916890842035813515d_enat(Na) ).

% zero_neq_numeral
tff(fact_1926_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_real != aa_num_real(numeral_numeral_real,Na) ).

% zero_neq_numeral
tff(fact_1927_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,Na) ).

% zero_neq_numeral
tff(fact_1928_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,Na) ).

% zero_neq_numeral
tff(fact_1929_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_int != aa_num_int(numeral_numeral_int,Na) ).

% zero_neq_numeral
tff(fact_1930_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,Uw: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_Leaf((Uu),(Uv))),Uw) ).

% VEBT_internal.membermima.simps(1)
tff(fact_1931_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_1932_listrel1__eq__len,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
      ( member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(Xs,Ys),listrel1_VEBT_VEBT(R2))
     => ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys) ) ) ).

% listrel1_eq_len
tff(fact_1933_listrel1__eq__len,axiom,
    ! [Xs: list_o,Ys: list_o,R2: set_Product_prod_o_o] :
      ( member4159035015898711888list_o(produc8435520187683070743list_o(Xs,Ys),listrel1_o(R2))
     => ( aa_list_o_nat(size_size_list_o,Xs) = aa_list_o_nat(size_size_list_o,Ys) ) ) ).

% listrel1_eq_len
tff(fact_1934_listrel1__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),listrel1_nat(R2))
     => ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) ) ) ).

% listrel1_eq_len
tff(fact_1935_listrel1__eq__len,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),listrel1_int(R2))
     => ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_int_nat(size_size_list_int,Ys) ) ) ).

% listrel1_eq_len
tff(fact_1936_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf((Uu),$true)) ).

% VEBT_internal.minNull.simps(3)
tff(fact_1937_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf($true,(Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_1938_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_1939_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Na: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X3),Na) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( aa_lis7044684121217598542BT_nat(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 = Na )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Mb) )
             => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_1)
               => ( ! [X3: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_1) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_1940_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Na: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X3),Na) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( aa_lis7044684121217598542BT_nat(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,Na) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Mb) )
             => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_1)
               => ( ! [X3: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_1) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_1941_zip__same,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(A,B),set_Pr5518436109238095868at_nat(zip_Pr4664179122662387191at_nat(Xs,Xs)))
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),set_Pr5648618587558075414at_nat(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1942_zip__same,axiom,
    ! [A: real,B: real,Xs: list_real] :
      ( member7849222048561428706l_real(produc4511245868158468465l_real(A,B),set_Pr5999470521830281550l_real(zip_real_real(Xs,Xs)))
    <=> ( aa_set_real_o(member_real(A),set_real2(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1943_zip__same,axiom,
    ! [A: set_nat,B: set_nat,Xs: list_set_nat] :
      ( member8277197624267554838et_nat(produc4532415448927165861et_nat(A,B),set_Pr9040384385603167362et_nat(zip_set_nat_set_nat(Xs,Xs)))
    <=> ( aa_set_set_nat_o(member_set_nat(A),set_set_nat2(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1944_zip__same,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member568628332442017744T_VEBT(produc537772716801021591T_VEBT(A,B),set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Xs)))
    <=> ( member_VEBT_VEBT(A,set_VEBT_VEBT2(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1945_zip__same,axiom,
    ! [A: nat,B: nat,Xs: list_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Xs)))
    <=> ( aa_set_nat_o(member_nat(A),set_nat2(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1946_zip__same,axiom,
    ! [A: int,B: int,Xs: list_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Xs)))
    <=> ( aa_set_int_o(member_int(A),set_int2(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1947_in__set__zipE,axiom,
    ! [X: real,Y: real,Xs: list_real,Ys: list_real] :
      ( member7849222048561428706l_real(produc4511245868158468465l_real(X,Y),set_Pr5999470521830281550l_real(zip_real_real(Xs,Ys)))
     => ~ ( aa_set_real_o(member_real(X),set_real2(Xs))
         => ~ aa_set_real_o(member_real(Y),set_real2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1948_in__set__zipE,axiom,
    ! [X: real,Y: vEBT_VEBT,Xs: list_real,Ys: list_VEBT_VEBT] :
      ( member7262085504369356948T_VEBT(produc6931449550656315951T_VEBT(X,Y),set_Pr8897343066327330088T_VEBT(zip_real_VEBT_VEBT(Xs,Ys)))
     => ~ ( aa_set_real_o(member_real(X),set_real2(Xs))
         => ~ member_VEBT_VEBT(Y,set_VEBT_VEBT2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1949_in__set__zipE,axiom,
    ! [X: real,Y: int,Xs: list_real,Ys: list_int] :
      ( member1627681773268152802al_int(produc3179012173361985393al_int(X,Y),set_Pr8219819362198175822al_int(zip_real_int(Xs,Ys)))
     => ~ ( aa_set_real_o(member_real(X),set_real2(Xs))
         => ~ aa_set_int_o(member_int(Y),set_int2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1950_in__set__zipE,axiom,
    ! [X: real,Y: nat,Xs: list_real,Ys: list_nat] :
      ( member5805532792777349510al_nat(produc3181502643871035669al_nat(X,Y),set_Pr3174298344852596722al_nat(zip_real_nat(Xs,Ys)))
     => ~ ( aa_set_real_o(member_real(X),set_real2(Xs))
         => ~ aa_set_nat_o(member_nat(Y),set_nat2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1951_in__set__zipE,axiom,
    ! [X: vEBT_VEBT,Y: real,Xs: list_VEBT_VEBT,Ys: list_real] :
      ( member8675245146396747942T_real(produc8117437818029410057T_real(X,Y),set_Pr1087130671499945274T_real(zip_VEBT_VEBT_real(Xs,Ys)))
     => ~ ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
         => ~ aa_set_real_o(member_real(Y),set_real2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1952_in__set__zipE,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( member568628332442017744T_VEBT(produc537772716801021591T_VEBT(X,Y),set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys)))
     => ~ ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
         => ~ member_VEBT_VEBT(Y,set_VEBT_VEBT2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1953_in__set__zipE,axiom,
    ! [X: vEBT_VEBT,Y: int,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( member5419026705395827622BT_int(produc736041933913180425BT_int(X,Y),set_Pr2853735649769556538BT_int(zip_VEBT_VEBT_int(Xs,Ys)))
     => ~ ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
         => ~ aa_set_int_o(member_int(Y),set_int2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1954_in__set__zipE,axiom,
    ! [X: vEBT_VEBT,Y: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( member373505688050248522BT_nat(produc738532404422230701BT_nat(X,Y),set_Pr7031586669278753246BT_nat(zip_VEBT_VEBT_nat(Xs,Ys)))
     => ~ ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
         => ~ aa_set_nat_o(member_nat(Y),set_nat2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1955_in__set__zipE,axiom,
    ! [X: int,Y: real,Xs: list_int,Ys: list_real] :
      ( member2744130022092475746t_real(produc801115645435158769t_real(X,Y),set_Pr112895574167722958t_real(zip_int_real(Xs,Ys)))
     => ~ ( aa_set_int_o(member_int(X),set_int2(Xs))
         => ~ aa_set_real_o(member_real(Y),set_real2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1956_in__set__zipE,axiom,
    ! [X: int,Y: vEBT_VEBT,Xs: list_int,Ys: list_VEBT_VEBT] :
      ( member2056185340421749780T_VEBT(produc3329399203697025711T_VEBT(X,Y),set_Pr8714266321650254504T_VEBT(zip_int_VEBT_VEBT(Xs,Ys)))
     => ~ ( aa_set_int_o(member_int(X),set_int2(Xs))
         => ~ member_VEBT_VEBT(Y,set_VEBT_VEBT2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1957_set__zip__leftD,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Ys)))
     => aa_set_nat_o(member_nat(X),set_nat2(Xs)) ) ).

% set_zip_leftD
tff(fact_1958_set__zip__leftD,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Ys)))
     => aa_set_int_o(member_int(X),set_int2(Xs)) ) ).

% set_zip_leftD
tff(fact_1959_set__zip__leftD,axiom,
    ! [X: fun_Co3460408610320954864e_term,Y: produc8923325533196201883nteger,Xs: list_f8851413139013959296e_term,Ys: list_P5578671422887162913nteger] :
      ( member6901806533512420791nteger(aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X),Y),set_Pr6295439789841872971nteger(zip_fu7084642285130276018nteger(Xs,Ys)))
     => member7516538326899768071e_term(X,set_fu2578164854786925467e_term(Xs)) ) ).

% set_zip_leftD
tff(fact_1960_set__zip__leftD,axiom,
    ! [X: fun_Pr5098669352780829690e_term,Y: produc8923325533196201883nteger,Xs: list_f4306079564415238026e_term,Ys: list_P5578671422887162913nteger] :
      ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X),Y),set_Pr5524914931742404033nteger(zip_fu4451626411769283112nteger(Xs,Ys)))
     => member8464590939221061905e_term(X,set_fu3771136004723407269e_term(Xs)) ) ).

% set_zip_leftD
tff(fact_1961_set__zip__leftD,axiom,
    ! [X: fun_Pr7763680713840462443e_term,Y: product_prod_int_int,Xs: list_f280343417242313083e_term,Ys: list_P5707943133018811711nt_int] :
      ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X),Y),set_Pr8602918816529533742nt_int(zip_fu5048005673116055189nt_int(Xs,Ys)))
     => member1495369047980753794e_term(X,set_fu3262800575041149974e_term(Xs)) ) ).

% set_zip_leftD
tff(fact_1962_set__zip__leftD,axiom,
    ! [X: fun_in5730379987311621695e_term,Y: product_prod_int_int,Xs: list_f628289730585369423e_term,Ys: list_P5707943133018811711nt_int] :
      ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X),Y),set_Pr4881580080894854746nt_int(zip_fu2910578966527623617nt_int(Xs,Ys)))
     => member4871225538671630166e_term(X,set_fu1748814359475392490e_term(Xs)) ) ).

% set_zip_leftD
tff(fact_1963_set__zip__rightD,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Ys)))
     => aa_set_nat_o(member_nat(Y),set_nat2(Ys)) ) ).

% set_zip_rightD
tff(fact_1964_set__zip__rightD,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Ys)))
     => aa_set_int_o(member_int(Y),set_int2(Ys)) ) ).

% set_zip_rightD
tff(fact_1965_set__zip__rightD,axiom,
    ! [X: fun_Co3460408610320954864e_term,Y: produc8923325533196201883nteger,Xs: list_f8851413139013959296e_term,Ys: list_P5578671422887162913nteger] :
      ( member6901806533512420791nteger(aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X),Y),set_Pr6295439789841872971nteger(zip_fu7084642285130276018nteger(Xs,Ys)))
     => member157494554546826820nteger(Y,set_Pr920681315882439344nteger(Ys)) ) ).

% set_zip_rightD
tff(fact_1966_set__zip__rightD,axiom,
    ! [X: fun_Pr5098669352780829690e_term,Y: produc8923325533196201883nteger,Xs: list_f4306079564415238026e_term,Ys: list_P5578671422887162913nteger] :
      ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X),Y),set_Pr5524914931742404033nteger(zip_fu4451626411769283112nteger(Xs,Ys)))
     => member157494554546826820nteger(Y,set_Pr920681315882439344nteger(Ys)) ) ).

% set_zip_rightD
tff(fact_1967_set__zip__rightD,axiom,
    ! [X: fun_Pr7763680713840462443e_term,Y: product_prod_int_int,Xs: list_f280343417242313083e_term,Ys: list_P5707943133018811711nt_int] :
      ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X),Y),set_Pr8602918816529533742nt_int(zip_fu5048005673116055189nt_int(Xs,Ys)))
     => member5262025264175285858nt_int(Y,set_Pr2470121279949933262nt_int(Ys)) ) ).

% set_zip_rightD
tff(fact_1968_set__zip__rightD,axiom,
    ! [X: fun_in5730379987311621695e_term,Y: product_prod_int_int,Xs: list_f628289730585369423e_term,Ys: list_P5707943133018811711nt_int] :
      ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X),Y),set_Pr4881580080894854746nt_int(zip_fu2910578966527623617nt_int(Xs,Ys)))
     => member5262025264175285858nt_int(Y,set_Pr2470121279949933262nt_int(Ys)) ) ).

% set_zip_rightD
tff(fact_1969_zip__update,axiom,
    ! [Xs: list_int,I: nat,X: int,Ys: list_int,Y: int] : zip_int_int(list_update_int(Xs,I,X),list_update_int(Ys,I,Y)) = list_u3002344382305578791nt_int(zip_int_int(Xs,Ys),I,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)) ).

% zip_update
tff(fact_1970_zip__update,axiom,
    ! [Xs: list_f8851413139013959296e_term,I: nat,X: fun_Co3460408610320954864e_term,Ys: list_P5578671422887162913nteger,Y: produc8923325533196201883nteger] : zip_fu7084642285130276018nteger(list_u2080314064485400322e_term(Xs,I,X),list_u2254550707601501961nteger(Ys,I,Y)) = list_u5986614341521392946nteger(zip_fu7084642285130276018nteger(Xs,Ys),I,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X),Y)) ).

% zip_update
tff(fact_1971_zip__update,axiom,
    ! [Xs: list_f4306079564415238026e_term,I: nat,X: fun_Pr5098669352780829690e_term,Ys: list_P5578671422887162913nteger,Y: produc8923325533196201883nteger] : zip_fu4451626411769283112nteger(list_u3429793794183195916e_term(Xs,I,X),list_u2254550707601501961nteger(Ys,I,Y)) = list_u5815421275070861992nteger(zip_fu4451626411769283112nteger(Xs,Ys),I,aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X),Y)) ).

% zip_update
tff(fact_1972_zip__update,axiom,
    ! [Xs: list_f280343417242313083e_term,I: nat,X: fun_Pr7763680713840462443e_term,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] : zip_fu5048005673116055189nt_int(list_u7283322402012212093e_term(Xs,I,X),list_u3002344382305578791nt_int(Ys,I,Y)) = list_u7912136156437333013nt_int(zip_fu5048005673116055189nt_int(Xs,Ys),I,aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X),Y)) ).

% zip_update
tff(fact_1973_zip__update,axiom,
    ! [Xs: list_f628289730585369423e_term,I: nat,X: fun_in5730379987311621695e_term,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] : zip_fu2910578966527623617nt_int(list_u528859344425011025e_term(Xs,I,X),list_u3002344382305578791nt_int(Ys,I,Y)) = list_u4453756226229724993nt_int(zip_fu2910578966527623617nt_int(Xs,Ys),I,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X),Y)) ).

% zip_update
tff(fact_1974_zero__le__numeral,axiom,
    ! [Na: num] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(Na)) ).

% zero_le_numeral
tff(fact_1975_zero__le__numeral,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,Na)) ).

% zero_le_numeral
tff(fact_1976_zero__le__numeral,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Na)) ).

% zero_le_numeral
tff(fact_1977_zero__le__numeral,axiom,
    ! [Na: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Na)) ).

% zero_le_numeral
tff(fact_1978_zero__le__numeral,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,Na)) ).

% zero_le_numeral
tff(fact_1979_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Na)),zero_z5237406670263579293d_enat) ).

% not_numeral_le_zero
tff(fact_1980_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Na)),zero_zero_real) ).

% not_numeral_le_zero
tff(fact_1981_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Na)),zero_zero_rat) ).

% not_numeral_le_zero
tff(fact_1982_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Na)),zero_zero_nat) ).

% not_numeral_le_zero
tff(fact_1983_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),zero_zero_int) ).

% not_numeral_le_zero
tff(fact_1984_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(Na)),zero_z5237406670263579293d_enat) ).

% not_numeral_less_zero
tff(fact_1985_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Na)),zero_zero_real) ).

% not_numeral_less_zero
tff(fact_1986_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Na)),zero_zero_rat) ).

% not_numeral_less_zero
tff(fact_1987_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Na)),zero_zero_nat) ).

% not_numeral_less_zero
tff(fact_1988_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),zero_zero_int) ).

% not_numeral_less_zero
tff(fact_1989_zero__less__numeral,axiom,
    ! [Na: num] : aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(Na)) ).

% zero_less_numeral
tff(fact_1990_zero__less__numeral,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,Na)) ).

% zero_less_numeral
tff(fact_1991_zero__less__numeral,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Na)) ).

% zero_less_numeral
tff(fact_1992_zero__less__numeral,axiom,
    ! [Na: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Na)) ).

% zero_less_numeral
tff(fact_1993_zero__less__numeral,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,Na)) ).

% zero_less_numeral
tff(fact_1994_set__encode__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ( aa_set_nat_nat(nat_set_encode,A2) = aa_set_nat_nat(nat_set_encode,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% set_encode_eq
tff(fact_1995_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_1996_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_real,Y: real] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = size_size_list_real(Ys) )
     => ( aa_set_real_o(member_real(Y),set_real2(Ys))
       => ~ ! [X3: vEBT_VEBT] : ~ member8675245146396747942T_real(produc8117437818029410057T_real(X3,Y),set_Pr1087130671499945274T_real(zip_VEBT_VEBT_real(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1997_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys) )
     => ( member_VEBT_VEBT(Y,set_VEBT_VEBT2(Ys))
       => ~ ! [X3: vEBT_VEBT] : ~ member568628332442017744T_VEBT(produc537772716801021591T_VEBT(X3,Y),set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1998_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_o,Y: $o] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_list_o_nat(size_size_list_o,Ys) )
     => ( member_o((Y),set_o2(Ys))
       => ~ ! [X3: vEBT_VEBT] : ~ member3307348790968139188VEBT_o(produc8721562602347293563VEBT_o(X3,(Y)),set_Pr7708085864119495200VEBT_o(zip_VEBT_VEBT_o(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1999_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,Y: nat] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( aa_set_nat_o(member_nat(Y),set_nat2(Ys))
       => ~ ! [X3: vEBT_VEBT] : ~ member373505688050248522BT_nat(produc738532404422230701BT_nat(X3,Y),set_Pr7031586669278753246BT_nat(zip_VEBT_VEBT_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_2000_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int,Y: int] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
     => ( aa_set_int_o(member_int(Y),set_int2(Ys))
       => ~ ! [X3: vEBT_VEBT] : ~ member5419026705395827622BT_int(produc736041933913180425BT_int(X3,Y),set_Pr2853735649769556538BT_int(zip_VEBT_VEBT_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_2001_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_o,Ys: list_real,Y: real] :
      ( ( aa_list_o_nat(size_size_list_o,Xs) = size_size_list_real(Ys) )
     => ( aa_set_real_o(member_real(Y),set_real2(Ys))
       => ~ ! [X3: $o] : ~ member7400031367953476362o_real(product_Pair_o_real((X3),Y),set_Pr2600826154070092190o_real(zip_o_real(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_2002_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_o,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( aa_list_o_nat(size_size_list_o,Xs) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys) )
     => ( member_VEBT_VEBT(Y,set_VEBT_VEBT2(Ys))
       => ~ ! [X3: $o] : ~ member5477980866518848620T_VEBT(produc2982872950893828659T_VEBT((X3),Y),set_Pr655345902815428824T_VEBT(zip_o_VEBT_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_2003_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_o,Ys: list_o,Y: $o] :
      ( ( aa_list_o_nat(size_size_list_o,Xs) = aa_list_o_nat(size_size_list_o,Ys) )
     => ( member_o((Y),set_o2(Ys))
       => ~ ! [X3: $o] : ~ member7466972457876170832od_o_o(product_Pair_o_o((X3),(Y)),set_Product_prod_o_o2(zip_o_o(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_2004_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_o,Ys: list_nat,Y: nat] :
      ( ( aa_list_o_nat(size_size_list_o,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( aa_set_nat_o(member_nat(Y),set_nat2(Ys))
       => ~ ! [X3: $o] : ~ member2802428098988154798_o_nat(product_Pair_o_nat((X3),Y),set_Pr7006799604034136130_o_nat(zip_o_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_2005_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_o,Ys: list_int,Y: int] :
      ( ( aa_list_o_nat(size_size_list_o,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
     => ( aa_set_int_o(member_int(Y),set_int2(Ys))
       => ~ ! [X3: $o] : ~ member7847949116333733898_o_int(product_Pair_o_int((X3),Y),set_Pr2828948584524939422_o_int(zip_o_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_2006_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_VEBT_VEBT,X: real] :
      ( ( size_size_list_real(Xs) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys) )
     => ( aa_set_real_o(member_real(X),set_real2(Xs))
       => ~ ! [Y3: vEBT_VEBT] : ~ member7262085504369356948T_VEBT(produc6931449550656315951T_VEBT(X,Y3),set_Pr8897343066327330088T_VEBT(zip_real_VEBT_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2007_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_o,X: real] :
      ( ( size_size_list_real(Xs) = aa_list_o_nat(size_size_list_o,Ys) )
     => ( aa_set_real_o(member_real(X),set_real2(Xs))
       => ~ ! [Y3: $o] : ~ member772602641336174712real_o(product_Pair_real_o(X,(Y3)),set_Pr5196769464307566348real_o(zip_real_o(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2008_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_nat,X: real] :
      ( ( size_size_list_real(Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( aa_set_real_o(member_real(X),set_real2(Xs))
       => ~ ! [Y3: nat] : ~ member5805532792777349510al_nat(produc3181502643871035669al_nat(X,Y3),set_Pr3174298344852596722al_nat(zip_real_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2009_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_int,X: real] :
      ( ( size_size_list_real(Xs) = aa_list_int_nat(size_size_list_int,Ys) )
     => ( aa_set_real_o(member_real(X),set_real2(Xs))
       => ~ ! [Y3: int] : ~ member1627681773268152802al_int(produc3179012173361985393al_int(X,Y3),set_Pr8219819362198175822al_int(zip_real_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2010_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys) )
     => ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
       => ~ ! [Y3: vEBT_VEBT] : ~ member568628332442017744T_VEBT(produc537772716801021591T_VEBT(X,Y3),set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2011_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_o,X: vEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_list_o_nat(size_size_list_o,Ys) )
     => ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
       => ~ ! [Y3: $o] : ~ member3307348790968139188VEBT_o(produc8721562602347293563VEBT_o(X,(Y3)),set_Pr7708085864119495200VEBT_o(zip_VEBT_VEBT_o(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2012_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,X: vEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
       => ~ ! [Y3: nat] : ~ member373505688050248522BT_nat(produc738532404422230701BT_nat(X,Y3),set_Pr7031586669278753246BT_nat(zip_VEBT_VEBT_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2013_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int,X: vEBT_VEBT] :
      ( ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
     => ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
       => ~ ! [Y3: int] : ~ member5419026705395827622BT_int(produc736041933913180425BT_int(X,Y3),set_Pr2853735649769556538BT_int(zip_VEBT_VEBT_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2014_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_o,Ys: list_VEBT_VEBT,X: $o] :
      ( ( aa_list_o_nat(size_size_list_o,Xs) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys) )
     => ( member_o((X),set_o2(Xs))
       => ~ ! [Y3: vEBT_VEBT] : ~ member5477980866518848620T_VEBT(produc2982872950893828659T_VEBT((X),Y3),set_Pr655345902815428824T_VEBT(zip_o_VEBT_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2015_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_o,Ys: list_o,X: $o] :
      ( ( aa_list_o_nat(size_size_list_o,Xs) = aa_list_o_nat(size_size_list_o,Ys) )
     => ( member_o((X),set_o2(Xs))
       => ~ ! [Y3: $o] : ~ member7466972457876170832od_o_o(product_Pair_o_o((X),(Y3)),set_Product_prod_o_o2(zip_o_o(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2016_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),X3)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2),X3) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_2017_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_2018_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_2019_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
    <=> ( ( ? [A6: $o,B6: $o] : A1 = vEBT_Leaf((A6),(B6))
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X4),N3) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),N3)
            & ( aa_lis7044684121217598542BT_nat(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))),N3) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3) )
            & ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_13)
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) )
        | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X4),N3) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N3))
            & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N3)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) )
            & ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_13)
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) )
        | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X4),N3) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),N3)
            & ( aa_lis7044684121217598542BT_nat(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))),N3) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3) )
            & ! [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))),N3))
               => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I2)),X_13)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I2) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N3) = I2 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I2)),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) = I2 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I2)),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) ) ) ) ) ) )
        | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X4),N3) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N3))
            & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N3)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) )
            & ! [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N3)))
               => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I2)),X_13)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I2) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N3)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N3) = I2 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I2)),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) = I2 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I2)),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_2020_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,Summary: vEBT_VEBT,M: nat,Deg: nat] :
              ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
             => ( ( A22 = Deg )
               => ( ! [X2: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                     => aa_nat_o(vEBT_invar_vebt(X2),N) )
                 => ( aa_nat_o(vEBT_invar_vebt(Summary),M)
                   => ( ( aa_lis7044684121217598542BT_nat(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_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_12)
                           => ~ ! [X2: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_12) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
                ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
               => ( ( A22 = Deg )
                 => ( ! [X2: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                       => aa_nat_o(vEBT_invar_vebt(X2),N) )
                   => ( aa_nat_o(vEBT_invar_vebt(Summary),M)
                     => ( ( aa_lis7044684121217598542BT_nat(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_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_12)
                             => ~ ! [X2: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_12) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: 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,Summary) )
                 => ( ( A22 = Deg )
                   => ( ! [X2: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(vEBT_invar_vebt(X2),N) )
                     => ( aa_nat_o(vEBT_invar_vebt(Summary),M)
                       => ( ( aa_lis7044684121217598542BT_nat(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) )
                             => ( ! [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))),M))
                                   => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),X_13)
                                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I4) ) )
                               => ( ( ( Mi = Ma )
                                   => ! [X2: vEBT_VEBT] :
                                        ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                                       => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_12) ) )
                                 => ( 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 )
                                         => ! [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))),M))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I4 )
                                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N)) )
                                                & ! [X2: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N) = I4 )
                                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N)) )
                                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X2)
                                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list_VEBT_VEBT,N: nat,Summary: 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,Summary) )
                   => ( ( A22 = Deg )
                     => ( ! [X2: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(vEBT_invar_vebt(X2),N) )
                       => ( aa_nat_o(vEBT_invar_vebt(Summary),M)
                         => ( ( aa_lis7044684121217598542BT_nat(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) )
                               => ( ! [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))),M))
                                     => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),X_13)
                                      <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I4) ) )
                                 => ( ( ( Mi = Ma )
                                     => ! [X2: vEBT_VEBT] :
                                          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                                         => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_12) ) )
                                   => ( 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 )
                                           => ! [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))),M))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I4 )
                                                   => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N)) )
                                                  & ! [X2: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N) = I4 )
                                                        & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N)) )
                                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X2)
                                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_2021_VEBT__internal_OminNull_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X != vEBT_Leaf($false,$false) )
     => ( ! [Uv2: $o] : X != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : X != vEBT_Leaf((Uu2),$true)
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_2022_num_Osize_I5_J,axiom,
    ! [X22: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_2023_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Na: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X3),Na) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( aa_lis7044684121217598542BT_nat(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 = Na )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),X_13)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I3) ) )
               => ( ( ( Mia = Maa )
                   => ! [X3: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_1) ) )
                 => ( 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),Na) = I3 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Na)) )
                                & ! [X3: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),Na) = I3 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),Na)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X3)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Maa) ) ) ) ) )
                       => aa_nat_o(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_2024_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,X)
     => ( ! [Uv2: $o] : X != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : X != vEBT_Leaf((Uu2),$true)
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_2025_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,X)
     => ( ( X != vEBT_Leaf($false,$false) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_2026_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Na: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X3),Na) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( aa_lis7044684121217598542BT_nat(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,Na) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),X_13)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I3) ) )
               => ( ( ( Mia = Maa )
                   => ! [X3: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_1) ) )
                 => ( 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),Na) = I3 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Na)) )
                                & ! [X3: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),Na) = I3 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),Na)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X3)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Maa) ) ) ) ) )
                       => aa_nat_o(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_2027_set__encode__inf,axiom,
    ! [A2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(nat_set_encode,A2) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_2028_pair__lessI1,axiom,
    ! [A: nat,B: nat,S: nat,T: 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),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_2029_count__le__length,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(count_list_VEBT_VEBT(Xs),X)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)) ).

% count_le_length
tff(fact_2030_count__le__length,axiom,
    ! [Xs: list_o,X: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(count_list_o(Xs),(X))),aa_list_o_nat(size_size_list_o,Xs)) ).

% count_le_length
tff(fact_2031_count__le__length,axiom,
    ! [Xs: list_nat,X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(count_list_nat(Xs),X)),aa_list_nat_nat(size_size_list_nat,Xs)) ).

% count_le_length
tff(fact_2032_count__le__length,axiom,
    ! [Xs: list_int,X: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(count_list_int(Xs),X)),aa_list_int_nat(size_size_list_int,Xs)) ).

% count_le_length
tff(fact_2033_vebt__mint_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A3: $o,B3: $o] : X != vEBT_Leaf((A3),(B3))
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
       => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2) ) ) ).

% vebt_mint.cases
tff(fact_2034_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,X)
      <=> (Y) )
     => ( ( ( X = vEBT_Leaf($false,$false) )
         => ~ (Y) )
       => ( ( ? [Uv2: $o] : X = vEBT_Leaf($true,(Uv2))
           => (Y) )
         => ( ( ? [Uu2: $o] : X = vEBT_Leaf((Uu2),$true)
             => (Y) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
               => ~ (Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)
                 => (Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_2035_length__code,axiom,
    size_s6755466524823107622T_VEBT = gen_length_VEBT_VEBT(zero_zero_nat) ).

% length_code
tff(fact_2036_length__code,axiom,
    size_size_list_o = gen_length_o(zero_zero_nat) ).

% length_code
tff(fact_2037_length__code,axiom,
    size_size_list_nat = gen_length_nat(zero_zero_nat) ).

% length_code
tff(fact_2038_length__code,axiom,
    size_size_list_int = gen_length_int(zero_zero_nat) ).

% length_code
tff(fact_2039_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_2040_sum__power2__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_2041_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_2042_zero__less__power2,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_2043_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_2044_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_2045_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_2046_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_2047_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_2048_power2__eq__iff__nonneg,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2049_power2__eq__iff__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2050_power2__eq__iff__nonneg,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X)
     => ( 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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2051_power2__eq__iff__nonneg,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2052_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_2053_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_2054_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_2055_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_2056_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_2057_power__mono__iff,axiom,
    ! [A: real,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2058_power__mono__iff,axiom,
    ! [A: rat,B: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2059_power__mono__iff,axiom,
    ! [A: nat,B: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2060_power__mono__iff,axiom,
    ! [A: int,B: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2061_power__eq__0__iff,axiom,
    ! [A: rat,Na: nat] :
      ( ( aa_nat_rat(power_power_rat(A),Na) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_2062_power__eq__0__iff,axiom,
    ! [A: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_2063_power__eq__0__iff,axiom,
    ! [A: real,Na: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_2064_power__eq__0__iff,axiom,
    ! [A: int,Na: nat] :
      ( ( aa_nat_int(power_power_int(A),Na) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_2065_power__eq__0__iff,axiom,
    ! [A: complex,Na: nat] :
      ( ( aa_nat_complex(power_power_complex(A),Na) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_2066_member__inv,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,X: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
        & ( ( X = Mia )
          | ( X = Maa )
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Maa)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_lis7044684121217598542BT_nat(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,X),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,X),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_2067_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Na: nat,X: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),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(Info,Dega,TreeLista,Summarya)),X) ) ) ) ).

% both_member_options_ding
tff(fact_2068_nat__zero__less__power__iff,axiom,
    ! [X: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( Na = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_2069_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_2070_high__def,axiom,
    ! [X: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),Na) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% high_def
tff(fact_2071_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_2072_division__ring__divide__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_2073_division__ring__divide__zero,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_2074_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_2075_divide__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_2076_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_2077_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(C),A) = aa_complex_complex(divide1717551699836669952omplex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_2078_divide__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_2079_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(C),A) = aa_rat_rat(divide_divide_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_2080_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_2081_div__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).

% div_by_0
tff(fact_2082_div__by__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_2083_div__by__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_2084_div__by__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_2085_div__by__0,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% div_by_0
tff(fact_2086_divide__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
tff(fact_2087_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_2088_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_2089_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_2090_div__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).

% div_0
tff(fact_2091_div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_2092_div__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,zero_zero_real),A) = zero_zero_real ).

% div_0
tff(fact_2093_div__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(zero_zero_rat),A) = zero_zero_rat ).

% div_0
tff(fact_2094_div__0,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% div_0
tff(fact_2095_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,Na)) = zero_zero_rat ).

% power_0_Suc
tff(fact_2096_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,Na)) = zero_zero_nat ).

% power_0_Suc
tff(fact_2097_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,Na)) = zero_zero_real ).

% power_0_Suc
tff(fact_2098_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,Na)) = zero_zero_int ).

% power_0_Suc
tff(fact_2099_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,Na)) = zero_zero_complex ).

% power_0_Suc
tff(fact_2100_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_rat ).

% power_zero_numeral
tff(fact_2101_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_nat ).

% power_zero_numeral
tff(fact_2102_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_real ).

% power_zero_numeral
tff(fact_2103_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_int ).

% power_zero_numeral
tff(fact_2104_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_2105_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_2106_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_2107_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_2108_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_2109_power__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_2110_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Mb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Mb = zero_zero_nat )
        | ( X = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_2111_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ).

% add_divide_distrib
tff(fact_2112_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ).

% add_divide_distrib
tff(fact_2113_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(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)) ) ) ).

% divide_right_mono_neg
tff(fact_2114_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(divide_divide_rat(B),C)),aa_rat_rat(divide_divide_rat(A),C)) ) ) ).

% divide_right_mono_neg
tff(fact_2115_divide__nonpos__nonpos,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),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(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_2116_divide__nonpos__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),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(divide_divide_rat(X),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_2117_divide__nonpos__nonneg,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),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(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real) ) ) ).

% divide_nonpos_nonneg
tff(fact_2118_divide__nonpos__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),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(divide_divide_rat(X),Y)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_2119_divide__nonneg__nonpos,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( 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(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real) ) ) ).

% divide_nonneg_nonpos
tff(fact_2120_divide__nonneg__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( 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(divide_divide_rat(X),Y)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_2121_divide__nonneg__nonneg,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( 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(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_2122_divide__nonneg__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( 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(divide_divide_rat(X),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_2123_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(aa_rea7340605660300375871l_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_2124_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(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_2125_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(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ) ) ).

% divide_right_mono
tff(fact_2126_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(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).

% divide_right_mono
tff(fact_2127_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_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_2128_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_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_2129_divide__neg__neg,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),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_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% divide_neg_neg
tff(fact_2130_divide__neg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),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_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y)) ) ) ).

% divide_neg_neg
tff(fact_2131_divide__neg__pos,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),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_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real) ) ) ).

% divide_neg_pos
tff(fact_2132_divide__neg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),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_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat) ) ) ).

% divide_neg_pos
tff(fact_2133_divide__pos__neg,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( 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_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real) ) ) ).

% divide_pos_neg
tff(fact_2134_divide__pos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ( 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_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat) ) ) ).

% divide_pos_neg
tff(fact_2135_divide__pos__pos,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( 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,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% divide_pos_pos
tff(fact_2136_divide__pos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ( 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_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y)) ) ) ).

% divide_pos_pos
tff(fact_2137_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% divide_less_0_iff
tff(fact_2138_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% divide_less_0_iff
tff(fact_2139_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_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_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_real,B),A) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_2140_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_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_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_rat,B),A) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_2141_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B))
    <=> ( ( 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,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_divide_iff
tff(fact_2142_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B))
    <=> ( ( 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,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_divide_iff
tff(fact_2143_divide__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,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ) ) ).

% divide_strict_right_mono
tff(fact_2144_divide__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,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).

% divide_strict_right_mono
tff(fact_2145_divide__strict__right__mono__neg,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),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2146_divide__strict__right__mono__neg,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),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2147_frac__le,axiom,
    ! [Y: real,X: real,W2: 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,X),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W2),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2)) ) ) ) ) ).

% frac_le
tff(fact_2148_frac__le,axiom,
    ! [Y: rat,X: rat,W2: 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,X),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W2),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Z)),aa_rat_rat(divide_divide_rat(Y),W2)) ) ) ) ) ).

% frac_le
tff(fact_2149_frac__less,axiom,
    ! [X: real,Y: real,W2: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W2),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2)) ) ) ) ) ).

% frac_less
tff(fact_2150_frac__less,axiom,
    ! [X: rat,Y: rat,W2: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W2),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Z)),aa_rat_rat(divide_divide_rat(Y),W2)) ) ) ) ) ).

% frac_less
tff(fact_2151_frac__less2,axiom,
    ! [X: real,Y: real,W2: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2)) ) ) ) ) ).

% frac_less2
tff(fact_2152_frac__less2,axiom,
    ! [X: rat,Y: rat,W2: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Z)),aa_rat_rat(divide_divide_rat(Y),W2)) ) ) ) ) ).

% frac_less2
tff(fact_2153_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(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_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_2154_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(divide_divide_rat(A),C)),aa_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_2155_divide__nonneg__neg,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( 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(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real) ) ) ).

% divide_nonneg_neg
tff(fact_2156_divide__nonneg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( 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(divide_divide_rat(X),Y)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_2157_divide__nonneg__pos,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( 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(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_2158_divide__nonneg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( 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(divide_divide_rat(X),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_2159_divide__nonpos__neg,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),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(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_2160_divide__nonpos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),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(divide_divide_rat(X),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_2161_divide__nonpos__pos,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),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(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real) ) ) ).

% divide_nonpos_pos
tff(fact_2162_divide__nonpos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),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(divide_divide_rat(X),Y)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_2163_field__sum__of__halves,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = X ).

% field_sum_of_halves
tff(fact_2164_field__sum__of__halves,axiom,
    ! [X: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(divide_divide_rat(X),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) = X ).

% field_sum_of_halves
tff(fact_2165_half__gt__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_2166_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(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_2167_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% half_gt_zero_iff
tff(fact_2168_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_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_2169_field__less__half__sum,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_2170_field__less__half__sum,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_2171_power__not__zero,axiom,
    ! [A: rat,Na: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),Na) != zero_zero_rat ) ) ).

% power_not_zero
tff(fact_2172_power__not__zero,axiom,
    ! [A: nat,Na: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) != zero_zero_nat ) ) ).

% power_not_zero
tff(fact_2173_power__not__zero,axiom,
    ! [A: real,Na: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_2174_power__not__zero,axiom,
    ! [A: int,Na: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),Na) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_2175_power__not__zero,axiom,
    ! [A: complex,Na: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),Na) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_2176_zero__le__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ).

% zero_le_power
tff(fact_2177_zero__le__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na)) ) ).

% zero_le_power
tff(fact_2178_zero__le__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ).

% zero_le_power
tff(fact_2179_zero__le__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Na)) ) ).

% zero_le_power
tff(fact_2180_power__mono,axiom,
    ! [A: real,B: real,Na: 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(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na)) ) ) ).

% power_mono
tff(fact_2181_power__mono,axiom,
    ! [A: rat,B: rat,Na: 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),Na)),aa_nat_rat(power_power_rat(B),Na)) ) ) ).

% power_mono
tff(fact_2182_power__mono,axiom,
    ! [A: nat,B: nat,Na: 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),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)) ) ) ).

% power_mono
tff(fact_2183_power__mono,axiom,
    ! [A: int,B: int,Na: 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),Na)),aa_nat_int(power_power_int(B),Na)) ) ) ).

% power_mono
tff(fact_2184_zero__less__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ).

% zero_less_power
tff(fact_2185_zero__less__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na)) ) ).

% zero_less_power
tff(fact_2186_zero__less__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ).

% zero_less_power
tff(fact_2187_zero__less__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Na)) ) ).

% zero_less_power
tff(fact_2188_nat__power__less__imp__less,axiom,
    ! [I: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ) ).

% nat_power_less_imp_less
tff(fact_2189_power__less__imp__less__base,axiom,
    ! [A: real,Na: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na))
     => ( 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_2190_power__less__imp__less__base,axiom,
    ! [A: rat,Na: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na))
     => ( 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_2191_power__less__imp__less__base,axiom,
    ! [A: nat,Na: 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),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na))
     => ( 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_2192_power__less__imp__less__base,axiom,
    ! [A: int,Na: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na))
     => ( 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_2193_power__le__imp__le__base,axiom,
    ! [A: real,Na: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Na)))
     => ( 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_2194_power__le__imp__le__base,axiom,
    ! [A: rat,Na: 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,Na))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Na)))
     => ( 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_2195_power__le__imp__le__base,axiom,
    ! [A: nat,Na: 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,Na))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Na)))
     => ( 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_2196_power__le__imp__le__base,axiom,
    ! [A: int,Na: 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,Na))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,Na)))
     => ( 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_2197_power__inject__base,axiom,
    ! [A: real,Na: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Na)) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2198_power__inject__base,axiom,
    ! [A: rat,Na: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Na)) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2199_power__inject__base,axiom,
    ! [A: nat,Na: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Na)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2200_power__inject__base,axiom,
    ! [A: int,Na: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,Na)) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2201_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),Na) = zero_zero_rat ) ) ).

% zero_power
tff(fact_2202_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Na) = zero_zero_nat ) ) ).

% zero_power
tff(fact_2203_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Na) = zero_zero_real ) ) ).

% zero_power
tff(fact_2204_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_int(power_power_int(zero_zero_int),Na) = zero_zero_int ) ) ).

% zero_power
tff(fact_2205_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),Na) = zero_zero_complex ) ) ).

% zero_power
tff(fact_2206_power__gt__expt,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),K)) ) ).

% power_gt_expt
tff(fact_2207_nat__one__le__power,axiom,
    ! [I: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Na)) ) ).

% nat_one_le_power
tff(fact_2208_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2209_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( ( aa_nat_rat(power_power_rat(A),Na) = aa_nat_rat(power_power_rat(B),Na) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2210_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2211_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( ( aa_nat_int(power_power_int(A),Na) = aa_nat_int(power_power_int(B),Na) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2212_power__eq__imp__eq__base,axiom,
    ! [A: real,Na: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2213_power__eq__imp__eq__base,axiom,
    ! [A: rat,Na: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),Na) = aa_nat_rat(power_power_rat(B),Na) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2214_power__eq__imp__eq__base,axiom,
    ! [A: nat,Na: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2215_power__eq__imp__eq__base,axiom,
    ! [A: int,Na: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),Na) = aa_nat_int(power_power_int(B),Na) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2216_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_2217_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_2218_zero__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real ).

% zero_power2
tff(fact_2219_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_2220_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_2221_less__exp,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% less_exp
tff(fact_2222_power2__nat__le__imp__le,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% power2_nat_le_imp_le
tff(fact_2223_power2__nat__le__eq__le,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% power2_nat_le_eq_le
tff(fact_2224_self__le__ge2__pow,axiom,
    ! [K: 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))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Mb)) ) ).

% self_le_ge2_pow
tff(fact_2225_power2__le__imp__le,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_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,X),Y) ) ) ).

% power2_le_imp_le
tff(fact_2226_power2__le__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( 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,X),Y) ) ) ).

% power2_le_imp_le
tff(fact_2227_power2__le__imp__le,axiom,
    ! [X: 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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_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,X),Y) ) ) ).

% power2_le_imp_le
tff(fact_2228_power2__le__imp__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( 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,X),Y) ) ) ).

% power2_le_imp_le
tff(fact_2229_power2__eq__imp__eq,axiom,
    ! [X: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_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),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2230_power2__eq__imp__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2231_power2__eq__imp__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2232_power2__eq__imp__eq,axiom,
    ! [X: int,Y: int] :
      ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2233_zero__le__power2,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_2234_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_2235_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_2236_power__strict__mono,axiom,
    ! [A: real,B: real,Na: 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),Na)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na)) ) ) ) ).

% power_strict_mono
tff(fact_2237_power__strict__mono,axiom,
    ! [A: rat,B: rat,Na: 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),Na)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na)) ) ) ) ).

% power_strict_mono
tff(fact_2238_power__strict__mono,axiom,
    ! [A: nat,B: nat,Na: 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),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)) ) ) ) ).

% power_strict_mono
tff(fact_2239_power__strict__mono,axiom,
    ! [A: int,B: int,Na: 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),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na)) ) ) ) ).

% power_strict_mono
tff(fact_2240_power2__less__0,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real) ).

% power2_less_0
tff(fact_2241_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_2242_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_2243_power2__less__imp__less,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_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,X),Y) ) ) ).

% power2_less_imp_less
tff(fact_2244_power2__less__imp__less,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( 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,X),Y) ) ) ).

% power2_less_imp_less
tff(fact_2245_power2__less__imp__less,axiom,
    ! [X: 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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_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,X),Y) ) ) ).

% power2_less_imp_less
tff(fact_2246_power2__less__imp__less,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( 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,X),Y) ) ) ).

% power2_less_imp_less
tff(fact_2247_sum__power2__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2248_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2249_sum__power2__le__zero__iff,axiom,
    ! [X: 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(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2250_sum__power2__ge__zero,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_2251_sum__power2__ge__zero,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_2252_sum__power2__ge__zero,axiom,
    ! [X: 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(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_2253_not__sum__power2__lt__zero,axiom,
    ! [X: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_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_2254_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat) ).

% not_sum_power2_lt_zero
tff(fact_2255_not__sum__power2__lt__zero,axiom,
    ! [X: 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(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int) ).

% not_sum_power2_lt_zero
tff(fact_2256_sum__power2__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2257_sum__power2__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2258_sum__power2__gt__zero__iff,axiom,
    ! [X: 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(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2259_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: 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,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_lis7044684121217598542BT_nat(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,X),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,X),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)),X) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_2260_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_2261_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_2262_both__member__options__from__complete__tree__to__child,axiom,
    ! [Dega: nat,Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,X: 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)),X)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),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,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
          | ( X = Mia )
          | ( X = Maa ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_2263_semiring__norm_I76_J,axiom,
    ! [Na: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit0,Na)) ).

% semiring_norm(76)
tff(fact_2264_div__less,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na) = zero_zero_nat ) ) ).

% div_less
tff(fact_2265_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_2266_set__n__deg__not__0,axiom,
    ! [TreeLista: list_VEBT_VEBT,Na: nat,Mb: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X3),Na) )
     => ( ( aa_lis7044684121217598542BT_nat(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),Na) ) ) ).

% set_n_deg_not_0
tff(fact_2267_Suc__n__div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_2268_div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% div_2_gt_zero
tff(fact_2269_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(T),one_one_nat)
    <=> ? [A6: $o,B6: $o] : T = vEBT_Leaf((A6),(B6)) ) ).

% deg1Leaf
tff(fact_2270_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(T),one_one_nat)
     => ? [A3: $o,B3: $o] : T = vEBT_Leaf((A3),(B3)) ) ).

% deg_1_Leaf
tff(fact_2271_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => ( ( Na = one_one_nat )
       => ? [A3: $o,B3: $o] : T = vEBT_Leaf((A3),(B3)) ) ) ).

% deg_1_Leafy
tff(fact_2272_div__by__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),one_one_complex) = A ).

% div_by_1
tff(fact_2273_div__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).

% div_by_1
tff(fact_2274_div__by__1,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),one_one_int) = A ).

% div_by_1
tff(fact_2275_div__by__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),one_one_real) = A ).

% div_by_1
tff(fact_2276_div__by__1,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),one_one_rat) = A ).

% div_by_1
tff(fact_2277_div__by__1,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,one_one_Code_integer) = A ).

% div_by_1
tff(fact_2278_semiring__norm_I78_J,axiom,
    ! [Mb: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Na) ) ).

% semiring_norm(78)
tff(fact_2279_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_2280_divide__eq__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = one_one_complex )
    <=> ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2281_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = one_one_real )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2282_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2283_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_2284_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),A) = one_one_nat ) ) ).

% div_self
tff(fact_2285_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(A),A) = one_one_int ) ) ).

% div_self
tff(fact_2286_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ).

% div_self
tff(fact_2287_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% div_self
tff(fact_2288_div__self,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(A,A) = one_one_Code_integer ) ) ).

% div_self
tff(fact_2289_one__eq__divide__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( one_one_complex = aa_complex_complex(divide1717551699836669952omplex(A),B) )
    <=> ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2290_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2291_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(A),B) )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2292_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_2293_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ).

% divide_self
tff(fact_2294_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% divide_self
tff(fact_2295_divide__self__if,axiom,
    ! [A: complex] :
      aa_complex_complex(divide1717551699836669952omplex(A),A) = $ite(A = zero_zero_complex,zero_zero_complex,one_one_complex) ).

% divide_self_if
tff(fact_2296_divide__self__if,axiom,
    ! [A: real] :
      aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = $ite(A = zero_zero_real,zero_zero_real,one_one_real) ).

% divide_self_if
tff(fact_2297_divide__self__if,axiom,
    ! [A: rat] :
      aa_rat_rat(divide_divide_rat(A),A) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% divide_self_if
tff(fact_2298_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) = one_one_real )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_2299_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_2300_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_2301_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(B),A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_2302_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_2303_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(one_one_rat),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_2304_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_2305_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(divide_divide_rat(one_one_rat),A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_2306_power__inject__exp,axiom,
    ! [A: real,Mb: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) )
      <=> ( Mb = Na ) ) ) ).

% power_inject_exp
tff(fact_2307_power__inject__exp,axiom,
    ! [A: rat,Mb: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( ( aa_nat_rat(power_power_rat(A),Mb) = aa_nat_rat(power_power_rat(A),Na) )
      <=> ( Mb = Na ) ) ) ).

% power_inject_exp
tff(fact_2308_power__inject__exp,axiom,
    ! [A: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) )
      <=> ( Mb = Na ) ) ) ).

% power_inject_exp
tff(fact_2309_power__inject__exp,axiom,
    ! [A: int,Mb: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( ( aa_nat_int(power_power_int(A),Mb) = aa_nat_int(power_power_int(A),Na) )
      <=> ( Mb = Na ) ) ) ).

% power_inject_exp
tff(fact_2310_less__one,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),one_one_nat)
    <=> ( Na = zero_zero_nat ) ) ).

% less_one
tff(fact_2311_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% divide_le_0_1_iff
tff(fact_2312_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_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_2313_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_divide_1_iff
tff(fact_2314_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(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_2315_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_divide_1_iff
tff(fact_2316_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_divide_1_iff
tff(fact_2317_less__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_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_2318_less__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_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_2319_less__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_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_2320_less__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_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_2321_divide__less__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_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_2322_divide__less__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_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_2323_divide__less__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_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_2324_divide__less__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_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_2325_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% divide_less_0_1_iff
tff(fact_2326_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% divide_less_0_1_iff
tff(fact_2327_power__strict__increasing__iff,axiom,
    ! [B: real,X: 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(aa_real_fun_nat_real(power_power_real,B),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_2328_power__strict__increasing__iff,axiom,
    ! [B: rat,X: 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),X)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_2329_power__strict__increasing__iff,axiom,
    ! [B: nat,X: 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),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_2330_power__strict__increasing__iff,axiom,
    ! [B: int,X: 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),X)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_2331_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(aa_rea7340605660300375871l_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_2332_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(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_2333_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(aa_rea7340605660300375871l_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_2334_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(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_2335_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(aa_rea7340605660300375871l_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_2336_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(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_2337_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(aa_rea7340605660300375871l_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_2338_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(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_2339_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_2340_one__add__one,axiom,
    plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2341_one__add__one,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2342_one__add__one,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2343_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_2344_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_2345_power__strict__decreasing__iff,axiom,
    ! [B: real,Mb: nat,Na: 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(aa_real_fun_nat_real(power_power_real,B),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2346_power__strict__decreasing__iff,axiom,
    ! [B: rat,Mb: nat,Na: 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),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2347_power__strict__decreasing__iff,axiom,
    ! [B: nat,Mb: nat,Na: 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),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2348_power__strict__decreasing__iff,axiom,
    ! [B: int,Mb: nat,Na: 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),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2349_power__increasing__iff,axiom,
    ! [B: real,X: 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(aa_real_fun_nat_real(power_power_real,B),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_2350_power__increasing__iff,axiom,
    ! [B: rat,X: 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),X)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_2351_power__increasing__iff,axiom,
    ! [B: nat,X: 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),X)),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,X),Y) ) ) ).

% power_increasing_iff
tff(fact_2352_power__increasing__iff,axiom,
    ! [B: int,X: 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),X)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_2353_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_2354_one__plus__numeral,axiom,
    ! [Na: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Na)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Na)) ).

% one_plus_numeral
tff(fact_2355_one__plus__numeral,axiom,
    ! [Na: num] : plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(Na)) = numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Na)) ).

% one_plus_numeral
tff(fact_2356_one__plus__numeral,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,Na)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Na)) ).

% one_plus_numeral
tff(fact_2357_one__plus__numeral,axiom,
    ! [Na: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Na)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Na)) ).

% one_plus_numeral
tff(fact_2358_one__plus__numeral,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Na)) ).

% one_plus_numeral
tff(fact_2359_one__plus__numeral,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Na)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Na)) ).

% one_plus_numeral
tff(fact_2360_numeral__plus__one,axiom,
    ! [Na: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Na)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Na),one)) ).

% numeral_plus_one
tff(fact_2361_numeral__plus__one,axiom,
    ! [Na: num] : plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Na),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(plus_plus_num,Na),one)) ).

% numeral_plus_one
tff(fact_2362_numeral__plus__one,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Na)),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Na),one)) ).

% numeral_plus_one
tff(fact_2363_numeral__plus__one,axiom,
    ! [Na: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Na)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Na),one)) ).

% numeral_plus_one
tff(fact_2364_numeral__plus__one,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Na)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Na),one)) ).

% numeral_plus_one
tff(fact_2365_numeral__plus__one,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Na)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Na),one)) ).

% numeral_plus_one
tff(fact_2366_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Na)),one_on7984719198319812577d_enat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_2367_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Na)),one_one_real)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_2368_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Na)),one_one_rat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_2369_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Na)),one_one_nat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_2370_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),one_one_int)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_2371_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_2372_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_num_real(numeral_numeral_real,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_2373_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_2374_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_2375_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_2376_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_2377_one__div__two__eq__zero,axiom,
    aa_int_int(divide_divide_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_2378_one__div__two__eq__zero,axiom,
    divide6298287555418463151nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% one_div_two_eq_zero
tff(fact_2379_power__decreasing__iff,axiom,
    ! [B: real,Mb: nat,Na: 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(aa_real_fun_nat_real(power_power_real,B),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2380_power__decreasing__iff,axiom,
    ! [B: rat,Mb: nat,Na: 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),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2381_power__decreasing__iff,axiom,
    ! [B: nat,Mb: nat,Na: 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),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2382_power__decreasing__iff,axiom,
    ! [B: int,Mb: nat,Na: 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),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2383_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
    <=> ( X = one_one_complex ) ) ).

% one_reorient
tff(fact_2384_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
    <=> ( X = one_one_real ) ) ).

% one_reorient
tff(fact_2385_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
    <=> ( X = one_one_rat ) ) ).

% one_reorient
tff(fact_2386_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
    <=> ( X = one_one_nat ) ) ).

% one_reorient
tff(fact_2387_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
    <=> ( X = one_one_int ) ) ).

% one_reorient
tff(fact_2388_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_2389_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_2390_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_2391_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_2392_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_2393_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_2394_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_2395_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_2396_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_2397_less__numeral__extra_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),one_one_real) ).

% less_numeral_extra(4)
tff(fact_2398_less__numeral__extra_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),one_one_rat) ).

% less_numeral_extra(4)
tff(fact_2399_less__numeral__extra_I4_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),one_one_nat) ).

% less_numeral_extra(4)
tff(fact_2400_less__numeral__extra_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),one_one_int) ).

% less_numeral_extra(4)
tff(fact_2401_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_2402_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),B)),one_one_int) ) ) ).

% div_add_self2
tff(fact_2403_div__add__self2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),B) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,B),one_one_Code_integer) ) ) ).

% div_add_self2
tff(fact_2404_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_2405_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),B)),one_one_int) ) ) ).

% div_add_self1
tff(fact_2406_div__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(B,A),B) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,B),one_one_Code_integer) ) ) ).

% div_add_self1
tff(fact_2407_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_2408_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_2409_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_2410_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_2411_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_2412_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_2413_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_2414_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_2415_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_2416_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_2417_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_2418_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_2419_not__one__less__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),zero_zero_real) ).

% not_one_less_zero
tff(fact_2420_not__one__less__zero,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),zero_zero_rat) ).

% not_one_less_zero
tff(fact_2421_not__one__less__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),zero_zero_nat) ).

% not_one_less_zero
tff(fact_2422_not__one__less__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),zero_zero_int) ).

% not_one_less_zero
tff(fact_2423_zero__less__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).

% zero_less_one
tff(fact_2424_zero__less__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).

% zero_less_one
tff(fact_2425_zero__less__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one
tff(fact_2426_zero__less__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% zero_less_one
tff(fact_2427_less__numeral__extra_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).

% less_numeral_extra(1)
tff(fact_2428_less__numeral__extra_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).

% less_numeral_extra(1)
tff(fact_2429_less__numeral__extra_I1_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% less_numeral_extra(1)
tff(fact_2430_less__numeral__extra_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% less_numeral_extra(1)
tff(fact_2431_one__le__numeral,axiom,
    ! [Na: num] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(Na)) ).

% one_le_numeral
tff(fact_2432_one__le__numeral,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_num_real(numeral_numeral_real,Na)) ).

% one_le_numeral
tff(fact_2433_one__le__numeral,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Na)) ).

% one_le_numeral
tff(fact_2434_one__le__numeral,axiom,
    ! [Na: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Na)) ).

% one_le_numeral
tff(fact_2435_one__le__numeral,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,Na)) ).

% one_le_numeral
tff(fact_2436_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(Na)),one_on7984719198319812577d_enat) ).

% not_numeral_less_one
tff(fact_2437_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Na)),one_one_real) ).

% not_numeral_less_one
tff(fact_2438_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Na)),one_one_rat) ).

% not_numeral_less_one
tff(fact_2439_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Na)),one_one_nat) ).

% not_numeral_less_one
tff(fact_2440_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),one_one_int) ).

% not_numeral_less_one
tff(fact_2441_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(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),one_one_real)) ) ).

% add_mono1
tff(fact_2442_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(plus_plus_rat(A),one_one_rat)),aa_rat_rat(plus_plus_rat(B),one_one_rat)) ) ).

% add_mono1
tff(fact_2443_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_2444_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_2445_less__add__one,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)) ).

% less_add_one
tff(fact_2446_less__add__one,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),one_one_rat)) ).

% less_add_one
tff(fact_2447_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_2448_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_2449_right__inverse__eq,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = one_one_complex )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2450_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = one_one_real )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2451_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2452_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(X)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(X)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_2453_one__plus__numeral__commute,axiom,
    ! [X: num] : plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(X)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(X),one_on7984719198319812577d_enat) ).

% one_plus_numeral_commute
tff(fact_2454_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,X)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_2455_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,X)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,X)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_2456_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_2457_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_2458_one__le__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ).

% one_le_power
tff(fact_2459_one__le__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Na)) ) ).

% one_le_power
tff(fact_2460_one__le__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ).

% one_le_power
tff(fact_2461_one__le__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),Na)) ) ).

% one_le_power
tff(fact_2462_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_2463_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_2464_power__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_2465_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_2466_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_2467_div__eq__dividend__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na) = Mb )
      <=> ( Na = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_2468_div__less__dividend,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na)),Mb) ) ) ).

% div_less_dividend
tff(fact_2469_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_2470_Suc__eq__plus1,axiom,
    ! [Na: nat] : aa_nat_nat(suc,Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat) ).

% Suc_eq_plus1
tff(fact_2471_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_2472_Suc__eq__plus1__left,axiom,
    ! [Na: nat] : aa_nat_nat(suc,Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),Na) ).

% Suc_eq_plus1_left
tff(fact_2473_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(vEBT_Leaf((Uu),(Uv))),D)
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_2474_zero__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real)) ).

% zero_less_two
tff(fact_2475_zero__less__two,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat)) ).

% zero_less_two
tff(fact_2476_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_2477_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_2478_power__le__one,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),one_one_real) ) ) ).

% power_le_one
tff(fact_2479_power__le__one,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),one_one_rat) ) ) ).

% power_le_one
tff(fact_2480_power__le__one,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),one_one_nat) ) ) ).

% power_le_one
tff(fact_2481_power__le__one,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),one_one_int) ) ) ).

% power_le_one
tff(fact_2482_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,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_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_real,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_2483_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_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_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_rat,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_2484_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_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_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_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_2485_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_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_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_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_2486_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(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))) ) ).

% less_half_sum
tff(fact_2487_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(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))) ) ).

% less_half_sum
tff(fact_2488_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(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))),B) ) ).

% gt_half_sum
tff(fact_2489_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(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))),B) ) ).

% gt_half_sum
tff(fact_2490_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_rat(power_power_rat(zero_zero_rat),Na) = $ite(Na = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% power_0_left
tff(fact_2491_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Na) = $ite(Na = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% power_0_left
tff(fact_2492_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Na) = $ite(Na = zero_zero_nat,one_one_real,zero_zero_real) ).

% power_0_left
tff(fact_2493_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_int(power_power_int(zero_zero_int),Na) = $ite(Na = zero_zero_nat,one_one_int,zero_zero_int) ).

% power_0_left
tff(fact_2494_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_complex(power_power_complex(zero_zero_complex),Na) = $ite(Na = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% power_0_left
tff(fact_2495_power__gt1,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_2496_power__gt1,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_2497_power__gt1,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_2498_power__gt1,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_2499_power__less__imp__less__exp,axiom,
    ! [A: real,Mb: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_2500_power__less__imp__less__exp,axiom,
    ! [A: rat,Mb: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_2501_power__less__imp__less__exp,axiom,
    ! [A: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_2502_power__less__imp__less__exp,axiom,
    ! [A: int,Mb: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_2503_power__strict__increasing,axiom,
    ! [Na: nat,N4: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4)) ) ) ).

% power_strict_increasing
tff(fact_2504_power__strict__increasing,axiom,
    ! [Na: nat,N4: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N4)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(A),N4)) ) ) ).

% power_strict_increasing
tff(fact_2505_power__strict__increasing,axiom,
    ! [Na: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)) ) ) ).

% power_strict_increasing
tff(fact_2506_power__strict__increasing,axiom,
    ! [Na: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(A),N4)) ) ) ).

% power_strict_increasing
tff(fact_2507_power__increasing,axiom,
    ! [Na: nat,N4: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4)) ) ) ).

% power_increasing
tff(fact_2508_power__increasing,axiom,
    ! [Na: nat,N4: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(A),N4)) ) ) ).

% power_increasing
tff(fact_2509_power__increasing,axiom,
    ! [Na: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)) ) ) ).

% power_increasing
tff(fact_2510_power__increasing,axiom,
    ! [Na: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(A),N4)) ) ) ).

% power_increasing
tff(fact_2511_nat__induct__non__zero,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [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,Na) ) ) ) ).

% nat_induct_non_zero
tff(fact_2512_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Leaf((A),(B))),X)
    <=> $ite(
          X = zero_zero_nat,
          (A),
          $ite(X = one_one_nat,(B),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_2513_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Leaf((A),(B))),X)
    <=> $ite(
          X = zero_zero_nat,
          (A),
          $ite(X = one_one_nat,(B),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_2514_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(aa_rea7340605660300375871l_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_2515_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(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_2516_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_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_2517_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_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_2518_power__Suc__le__self,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_2519_power__Suc__le__self,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_2520_power__Suc__le__self,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_2521_power__Suc__le__self,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_2522_power__Suc__less__one,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na))),one_one_real) ) ) ).

% power_Suc_less_one
tff(fact_2523_power__Suc__less__one,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_2524_power__Suc__less__one,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_2525_power__Suc__less__one,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_2526_power__strict__decreasing,axiom,
    ! [Na: nat,N4: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_2527_power__strict__decreasing,axiom,
    ! [Na: nat,N4: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N4)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N4)),aa_nat_rat(power_power_rat(A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_2528_power__strict__decreasing,axiom,
    ! [Na: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_2529_power__strict__decreasing,axiom,
    ! [Na: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),N4)),aa_nat_int(power_power_int(A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_2530_power__decreasing,axiom,
    ! [Na: nat,N4: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ) ).

% power_decreasing
tff(fact_2531_power__decreasing,axiom,
    ! [Na: nat,N4: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N4)),aa_nat_rat(power_power_rat(A),Na)) ) ) ) ).

% power_decreasing
tff(fact_2532_power__decreasing,axiom,
    ! [Na: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ) ).

% power_decreasing
tff(fact_2533_power__decreasing,axiom,
    ! [Na: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),N4)),aa_nat_int(power_power_int(A),Na)) ) ) ) ).

% power_decreasing
tff(fact_2534_power__le__imp__le__exp,axiom,
    ! [A: real,Mb: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_2535_power__le__imp__le__exp,axiom,
    ! [A: rat,Mb: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_2536_power__le__imp__le__exp,axiom,
    ! [A: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_2537_power__le__imp__le__exp,axiom,
    ! [A: int,Mb: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_2538_self__le__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ).

% self_le_power
tff(fact_2539_self__le__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),Na)) ) ) ).

% self_le_power
tff(fact_2540_self__le__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ).

% self_le_power
tff(fact_2541_self__le__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(A),Na)) ) ) ).

% self_le_power
tff(fact_2542_one__less__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ).

% one_less_power
tff(fact_2543_one__less__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Na)) ) ) ).

% one_less_power
tff(fact_2544_one__less__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ).

% one_less_power
tff(fact_2545_one__less__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),Na)) ) ) ).

% one_less_power
tff(fact_2546_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_2547_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_2548_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_2549_div__le__mono,axiom,
    ! [Mb: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),K)) ) ).

% div_le_mono
tff(fact_2550_div__le__dividend,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na)),Mb) ).

% div_le_dividend
tff(fact_2551_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_2552_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),K)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_2553_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
        | ( Na = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_2554_Suc__div__le__mono,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Mb)),Na)) ).

% Suc_div_le_mono
tff(fact_2555_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(X) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X = vEBT_Leaf((A3),(B3)) )
           => ( Y != $ite(
                  (B3),
                  some_nat(one_one_nat),
                  $ite((A3),some_nat(zero_zero_nat),none_nat) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi: nat,Ma: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = 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_2556_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(X) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X = vEBT_Leaf((A3),(B3)) )
           => ( Y != $ite(
                  (A3),
                  some_nat(zero_zero_nat),
                  $ite((B3),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi: nat] :
                ( ? [Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = 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_2557_div__greater__zero__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% div_greater_zero_iff
tff(fact_2558_div__le__mono2,axiom,
    ! [Mb: nat,Na: nat,K: 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),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Mb)) ) ) ).

% div_le_mono2
tff(fact_2559_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_2560_bits__1__div__2,axiom,
    aa_int_int(divide_divide_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_2561_bits__1__div__2,axiom,
    divide6298287555418463151nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% bits_1_div_2
tff(fact_2562_nat__induct2,axiom,
    ! [P: fun_nat_o,Na: 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,Na) ) ) ) ).

% nat_induct2
tff(fact_2563_div__exp__eq,axiom,
    ! [A: nat,Mb: nat,Na: 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))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na))) ).

% div_exp_eq
tff(fact_2564_div__exp__eq,axiom,
    ! [A: int,Mb: nat,Na: nat] : aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na))) ).

% div_exp_eq
tff(fact_2565_div__exp__eq,axiom,
    ! [A: code_integer,Mb: nat,Na: nat] : divide6298287555418463151nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na))) ).

% div_exp_eq
tff(fact_2566_exp__add__not__zero__imp__right,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2567_exp__add__not__zero__imp__right,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2568_exp__add__not__zero__imp__left,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2569_exp__add__not__zero__imp__left,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)) != 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_2570_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_2571_low__inv,axiom,
    ! [X: nat,Na: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),X)),Na) = X ) ) ).

% low_inv
tff(fact_2572_high__inv,axiom,
    ! [X: nat,Na: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),X)),Na) = Y ) ) ).

% high_inv
tff(fact_2573_enat__ord__number_I1_J,axiom,
    ! [Mb: num,Na: num] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Mb)),numera1916890842035813515d_enat(Na))
    <=> 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,Na)) ) ).

% enat_ord_number(1)
tff(fact_2574_enat__ord__number_I2_J,axiom,
    ! [Mb: num,Na: num] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(Mb)),numera1916890842035813515d_enat(Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Na)) ) ).

% enat_ord_number(2)
tff(fact_2575_mul__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(X)),some_nat(Y)) = some_nat(Z) ) ) ).

% mul_shift
tff(fact_2576_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_2577_i0__less,axiom,
    ! [Na: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Na)
    <=> ( Na != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_2578_mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2579_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2580_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2581_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2582_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2583_mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2584_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2585_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2586_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2587_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2588_mult__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
tff(fact_2589_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_2590_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_2591_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_2592_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_2593_mult__zero__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_2594_mult__zero__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),zero_zero_real) = zero_zero_real ).

% mult_zero_right
tff(fact_2595_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_2596_mult__zero__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),zero_zero_nat) = zero_zero_nat ).

% mult_zero_right
tff(fact_2597_mult__zero__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),zero_zero_int) = zero_zero_int ).

% mult_zero_right
tff(fact_2598_mult__zero__left,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A) = zero_zero_complex ).

% mult_zero_left
tff(fact_2599_mult__zero__left,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_zero_real),A) = zero_zero_real ).

% mult_zero_left
tff(fact_2600_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_2601_mult__zero__left,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A) = zero_zero_nat ).

% mult_zero_left
tff(fact_2602_mult__zero__left,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),A) = zero_zero_int ).

% mult_zero_left
tff(fact_2603_bits__div__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).

% bits_div_by_0
tff(fact_2604_bits__div__by__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_2605_bits__div__by__0,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% bits_div_by_0
tff(fact_2606_bits__div__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).

% bits_div_0
tff(fact_2607_bits__div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_2608_bits__div__0,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% bits_div_0
tff(fact_2609_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_2610_mult_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),one_one_real) = A ).

% mult.right_neutral
tff(fact_2611_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_2612_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_2613_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_2614_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_2615_mult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% mult_1
tff(fact_2616_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% mult_1
tff(fact_2617_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_2618_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_2619_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% half_negative_int_iff
tff(fact_2620_times__divide__eq__left,axiom,
    ! [B: complex,C: complex,A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(B),C)),A) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(B),A)),C) ).

% times_divide_eq_left
tff(fact_2621_times__divide__eq__left,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),C) ).

% times_divide_eq_left
tff(fact_2622_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(B),C)),A) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(B),A)),C) ).

% times_divide_eq_left
tff(fact_2623_divide__divide__eq__left,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(B),C)) ).

% divide_divide_eq_left
tff(fact_2624_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% divide_divide_eq_left
tff(fact_2625_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A),B)),C) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% divide_divide_eq_left
tff(fact_2626_divide__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(divide1717551699836669952omplex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),B) ).

% divide_divide_eq_right
tff(fact_2627_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ).

% divide_divide_eq_right
tff(fact_2628_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(A),aa_rat_rat(divide_divide_rat(B),C)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),B) ).

% divide_divide_eq_right
tff(fact_2629_times__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),C) ).

% times_divide_eq_right
tff(fact_2630_times__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) ).

% times_divide_eq_right
tff(fact_2631_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(divide_divide_rat(B),C)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),C) ).

% times_divide_eq_right
tff(fact_2632_mult__cancel2,axiom,
    ! [Mb: nat,K: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K) )
    <=> ( ( Mb = Na )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_2633_mult__cancel1,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na) )
    <=> ( ( Mb = Na )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_2634_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_2635_mult__is__0,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na) = zero_zero_nat )
    <=> ( ( Mb = zero_zero_nat )
        | ( Na = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_2636_nat__1__eq__mult__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na) )
    <=> ( ( Mb = one_one_nat )
        & ( Na = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_2637_nat__mult__eq__1__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na) = one_one_nat )
    <=> ( ( Mb = one_one_nat )
        & ( Na = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_2638_mult__cancel__right2,axiom,
    ! [A: complex,C: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
tff(fact_2639_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_2640_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_2641_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_2642_mult__cancel__right1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(times_times_complex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_2643_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_2644_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_2645_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_2646_mult__cancel__left2,axiom,
    ! [C: complex,A: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
tff(fact_2647_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_2648_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_2649_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_2650_mult__cancel__left1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(times_times_complex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_2651_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_2652_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_2653_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_2654_sum__squares__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2655_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y)) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2656_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2657_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2658_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2659_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2660_nonzero__mult__div__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2661_nonzero__mult__div__cancel__right,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2662_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2663_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2664_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2665_nonzero__mult__div__cancel__right,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2666_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2667_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2668_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2669_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2670_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2671_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2672_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2673_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2674_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2675_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2676_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2677_nonzero__mult__div__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2678_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2679_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2680_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2681_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B: complex] :
      aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = $ite(C = zero_zero_complex,zero_zero_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_2682_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = $ite(C = zero_zero_real,zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_2683_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = $ite(C = zero_zero_rat,zero_zero_rat,aa_rat_rat(divide_divide_rat(A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_2684_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = $ite(C = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% div_mult_mult1_if
tff(fact_2685_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = $ite(C = zero_zero_int,zero_zero_int,aa_int_int(divide_divide_int(A),B)) ).

% div_mult_mult1_if
tff(fact_2686_div__mult__mult1__if,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) = $ite(C = zero_z3403309356797280102nteger,zero_z3403309356797280102nteger,divide6298287555418463151nteger(A,B)) ).

% div_mult_mult1_if
tff(fact_2687_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult2
tff(fact_2688_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(A),B) ) ) ).

% div_mult_mult2
tff(fact_2689_div__mult__mult2,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(A,B) ) ) ).

% div_mult_mult2
tff(fact_2690_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult1
tff(fact_2691_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(divide_divide_int(A),B) ) ) ).

% div_mult_mult1
tff(fact_2692_div__mult__mult1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) = divide6298287555418463151nteger(A,B) ) ) ).

% div_mult_mult1
tff(fact_2693_distrib__right__numeral,axiom,
    ! [A: complex,B: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V))) ).

% distrib_right_numeral
tff(fact_2694_distrib__right__numeral,axiom,
    ! [A: extended_enat,B: extended_enat,V: num] : times_7803423173614009249d_enat(plus_p3455044024723400733d_enat(A,B),numera1916890842035813515d_enat(V)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(A,numera1916890842035813515d_enat(V)),times_7803423173614009249d_enat(B,numera1916890842035813515d_enat(V))) ).

% distrib_right_numeral
tff(fact_2695_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_num_real(numeral_numeral_real,V))) ).

% distrib_right_numeral
tff(fact_2696_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B),aa_num_rat(numeral_numeral_rat,V))) ).

% distrib_right_numeral
tff(fact_2697_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,V))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_num_nat(numeral_numeral_nat,V))) ).

% distrib_right_numeral
tff(fact_2698_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_num_int(numeral_numeral_int,V))) ).

% distrib_right_numeral
tff(fact_2699_distrib__left__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(B),C)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C)) ).

% distrib_left_numeral
tff(fact_2700_distrib__left__numeral,axiom,
    ! [V: num,B: extended_enat,C: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),plus_p3455044024723400733d_enat(B,C)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),B),times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),C)) ).

% distrib_left_numeral
tff(fact_2701_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C)) ).

% distrib_left_numeral
tff(fact_2702_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).

% distrib_left_numeral
tff(fact_2703_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),C)) ).

% distrib_left_numeral
tff(fact_2704_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C)) ).

% distrib_left_numeral
tff(fact_2705_mult__eq__1__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
        & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_2706_one__eq__mult__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na) )
    <=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
        & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_2707_mult__less__cancel2,axiom,
    ! [Mb: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ) ).

% mult_less_cancel2
tff(fact_2708_nat__0__less__mult__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na))
    <=> ( 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),Na) ) ) ).

% nat_0_less_mult_iff
tff(fact_2709_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_2710_mult__Suc__right,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na)) ).

% mult_Suc_right
tff(fact_2711_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) = $ite(K = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na)) ).

% nat_mult_div_cancel_disj
tff(fact_2712_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2713_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2714_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2715_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2716_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W2: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W2)) = A )
    <=> $ite(numera6690914467698888265omplex(W2) != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W2)),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2717_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2)) = A )
    <=> $ite(aa_num_real(numeral_numeral_real,W2) != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2)),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2718_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2)) = A )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W2) != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2)),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2719_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W2: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W2)) )
    <=> $ite(numera6690914467698888265omplex(W2) != zero_zero_complex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W2)) = B,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2720_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2)) )
    <=> $ite(aa_num_real(numeral_numeral_real,W2) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2)) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2721_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2)) )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W2) != zero_zero_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2)) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2722_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_2723_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_2724_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_2725_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_2726_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2727_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2728_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2729_nonzero__divide__mult__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2730_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2731_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2732_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_2733_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self1
tff(fact_2734_div__mult__self1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),B) = plus_p5714425477246183910nteger(C,divide6298287555418463151nteger(A,B)) ) ) ).

% div_mult_self1
tff(fact_2735_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_2736_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self2
tff(fact_2737_div__mult__self2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),B) = plus_p5714425477246183910nteger(C,divide6298287555418463151nteger(A,B)) ) ) ).

% div_mult_self2
tff(fact_2738_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_2739_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self3
tff(fact_2740_div__mult__self3,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B),A),B) = plus_p5714425477246183910nteger(C,divide6298287555418463151nteger(A,B)) ) ) ).

% div_mult_self3
tff(fact_2741_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_2742_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self4
tff(fact_2743_div__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C),A),B) = plus_p5714425477246183910nteger(C,divide6298287555418463151nteger(A,B)) ) ) ).

% div_mult_self4
tff(fact_2744_one__le__mult__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na))
    <=> ( 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)),Na) ) ) ).

% one_le_mult_iff
tff(fact_2745_mult__le__cancel2,axiom,
    ! [Mb: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ) ).

% mult_le_cancel2
tff(fact_2746_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_2747_div__mult__self__is__m,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na)),Na) = Mb ) ) ).

% div_mult_self_is_m
tff(fact_2748_div__mult__self1__is__m,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Mb)),Na) = Mb ) ) ).

% div_mult_self1_is_m
tff(fact_2749_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(A),B)),C) = aa_complex_complex(times_times_complex(A),aa_complex_complex(times_times_complex(B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2750_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2751_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2752_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_2753_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_2754_mult_Oassoc,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(A),B)),C) = aa_complex_complex(times_times_complex(A),aa_complex_complex(times_times_complex(B),C)) ).

% mult.assoc
tff(fact_2755_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% mult.assoc
tff(fact_2756_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% mult.assoc
tff(fact_2757_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_2758_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_2759_mult_Ocommute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(A),B) = aa_complex_complex(times_times_complex(B),A) ).

% mult.commute
tff(fact_2760_mult_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A) ).

% mult.commute
tff(fact_2761_mult_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),B) = aa_rat_rat(times_times_rat(B),A) ).

% mult.commute
tff(fact_2762_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_2763_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_2764_mult_Oleft__commute,axiom,
    ! [B: complex,A: complex,C: complex] : aa_complex_complex(times_times_complex(B),aa_complex_complex(times_times_complex(A),C)) = aa_complex_complex(times_times_complex(A),aa_complex_complex(times_times_complex(B),C)) ).

% mult.left_commute
tff(fact_2765_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% mult.left_commute
tff(fact_2766_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(times_times_rat(A),C)) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% mult.left_commute
tff(fact_2767_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_2768_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_2769_not__iless0,axiom,
    ! [Na: extended_enat] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(Na),zero_z5237406670263579293d_enat) ).

% not_iless0
tff(fact_2770_enat__less__induct,axiom,
    ! [P: fun_Extended_enat_o,Na: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M4: extended_enat] :
              ( aa_Extended_enat_o(ord_le72135733267957522d_enat(M4),N)
             => aa_Extended_enat_o(P,M4) )
         => aa_Extended_enat_o(P,N) )
     => aa_Extended_enat_o(P,Na) ) ).

% enat_less_induct
tff(fact_2771_mult__right__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2772_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2773_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2774_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2775_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2776_mult__left__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2777_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2778_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2779_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2780_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2781_no__zero__divisors,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex ) ) ) ).

% no_zero_divisors
tff(fact_2782_no__zero__divisors,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_2783_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_2784_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_2785_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_2786_divisors__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
     => ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divisors_zero
tff(fact_2787_divisors__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_2788_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_2789_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_2790_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_2791_mult__not__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex )
     => ( ( A != zero_zero_complex )
        & ( B != zero_zero_complex ) ) ) ).

% mult_not_zero
tff(fact_2792_mult__not__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_2793_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_2794_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_2795_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_2796_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_2797_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2798_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2799_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_2800_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_2801_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_2802_mult_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),one_one_real) = A ).

% mult.comm_neutral
tff(fact_2803_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_2804_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_2805_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_2806_combine__common__factor,axiom,
    ! [A: complex,E2: complex,B: complex,C: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),E2)),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(B),E2)),C)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2807_combine__common__factor,axiom,
    ! [A: real,E2: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2808_combine__common__factor,axiom,
    ! [A: rat,E2: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2809_combine__common__factor,axiom,
    ! [A: nat,E2: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E2)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2810_combine__common__factor,axiom,
    ! [A: int,E2: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2811_distrib__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)) ).

% distrib_right
tff(fact_2812_distrib__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% distrib_right
tff(fact_2813_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% distrib_right
tff(fact_2814_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_2815_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_2816_distrib__left,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(plus_plus_complex(B),C)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),B)),aa_complex_complex(times_times_complex(A),C)) ).

% distrib_left
tff(fact_2817_distrib__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% distrib_left
tff(fact_2818_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% distrib_left
tff(fact_2819_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_2820_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_2821_comm__semiring__class_Odistrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)) ).

% comm_semiring_class.distrib
tff(fact_2822_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2823_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% comm_semiring_class.distrib
tff(fact_2824_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_2825_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_2826_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(plus_plus_complex(B),C)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),B)),aa_complex_complex(times_times_complex(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2827_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2828_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2829_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_2830_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2831_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2832_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2833_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_2834_crossproduct__noteq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),D)) != aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),D)),aa_complex_complex(times_times_complex(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2835_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2836_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) != aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),D)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2837_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_2838_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_2839_crossproduct__eq,axiom,
    ! [W2: complex,Y: complex,X: complex,Z: complex] :
      ( ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(W2),Y)),aa_complex_complex(times_times_complex(X),Z)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(W2),Z)),aa_complex_complex(times_times_complex(X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2840_crossproduct__eq,axiom,
    ! [W2: real,Y: real,X: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2841_crossproduct__eq,axiom,
    ! [W2: rat,Y: rat,X: rat,Z: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W2),Y)),aa_rat_rat(times_times_rat(X),Z)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W2),Z)),aa_rat_rat(times_times_rat(X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2842_crossproduct__eq,axiom,
    ! [W2: nat,Y: nat,X: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W2),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W2),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2843_crossproduct__eq,axiom,
    ! [W2: int,Y: int,X: int,Z: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2844_times__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(X),Z)),aa_complex_complex(times_times_complex(Y),W2)) ).

% times_divide_times_eq
tff(fact_2845_times__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),W2)) ).

% times_divide_times_eq
tff(fact_2846_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(Z),W2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(Y),W2)) ).

% times_divide_times_eq
tff(fact_2847_divide__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W2: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(X),W2)),aa_complex_complex(times_times_complex(Y),Z)) ).

% divide_divide_times_eq
tff(fact_2848_divide__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),W2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ).

% divide_divide_times_eq
tff(fact_2849_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W2: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(Z),W2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(X),W2)),aa_rat_rat(times_times_rat(Y),Z)) ).

% divide_divide_times_eq
tff(fact_2850_divide__divide__eq__left_H,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(C),B)) ).

% divide_divide_eq_left'
tff(fact_2851_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ).

% divide_divide_eq_left'
tff(fact_2852_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A),B)),C) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(C),B)) ).

% divide_divide_eq_left'
tff(fact_2853_Suc__mult__cancel1,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Na) )
    <=> ( Mb = Na ) ) ).

% Suc_mult_cancel1
tff(fact_2854_mult__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),Na) = zero_zero_nat ).

% mult_0
tff(fact_2855_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na) )
    <=> ( ( K = zero_zero_nat )
        | ( Mb = Na ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_2856_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ).

% mult_le_mono2
tff(fact_2857_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ).

% mult_le_mono1
tff(fact_2858_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L)) ) ) ).

% mult_le_mono
tff(fact_2859_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_2860_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_2861_add__mult__distrib2,axiom,
    ! [K: nat,Mb: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) ).

% add_mult_distrib2
tff(fact_2862_add__mult__distrib,axiom,
    ! [Mb: nat,Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)) ).

% add_mult_distrib
tff(fact_2863_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_2864_nat__mult__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Na) = Na ).

% nat_mult_1
tff(fact_2865_nat__mult__1__right,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),one_one_nat) = Na ).

% nat_mult_1_right
tff(fact_2866_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(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2867_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(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2868_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_2869_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_2870_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(aa_rea7340605660300375871l_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_2871_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(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_2872_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_2873_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(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2874_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(times_times_rat(B),A)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2875_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_2876_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_2877_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(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_nonpos_nonneg
tff(fact_2878_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(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_2879_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_2880_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_2881_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(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos
tff(fact_2882_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(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_2883_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_2884_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_2885_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(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2886_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(times_times_rat(A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2887_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_2888_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_2889_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(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ).

% split_mult_neg_le
tff(fact_2890_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(times_times_rat(A),B)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_2891_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_2892_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_2893_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_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_2894_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_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_2895_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_2896_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(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_right_mono
tff(fact_2897_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_right_mono
tff(fact_2898_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_2899_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_2900_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(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2901_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2902_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_2903_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(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_left_mono
tff(fact_2904_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(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_left_mono
tff(fact_2905_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_2906_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_2907_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(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2908_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(times_times_rat(A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2909_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_2910_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(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2911_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(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2912_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_2913_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(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ).

% split_mult_pos_le
tff(fact_2914_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(times_times_rat(A),B)) ) ).

% split_mult_pos_le
tff(fact_2915_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_2916_zero__le__square,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)) ).

% zero_le_square
tff(fact_2917_zero__le__square,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),A)) ).

% zero_le_square
tff(fact_2918_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_2919_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(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2920_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2921_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_2922_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_2923_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(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2924_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2925_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_2926_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_2927_mult__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,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_neg_neg
tff(fact_2928_mult__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,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_neg_neg
tff(fact_2929_mult__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,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_neg_neg
tff(fact_2930_not__square__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),zero_zero_real) ).

% not_square_less_zero
tff(fact_2931_not__square__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),A)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_2932_not__square__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),zero_zero_int) ).

% not_square_less_zero
tff(fact_2933_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% mult_less_0_iff
tff(fact_2934_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% mult_less_0_iff
tff(fact_2935_mult__less__0__iff,axiom,
    ! [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,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% mult_less_0_iff
tff(fact_2936_mult__neg__pos,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,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_neg_pos
tff(fact_2937_mult__neg__pos,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,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_2938_mult__neg__pos,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,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_neg_pos
tff(fact_2939_mult__neg__pos,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,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_neg_pos
tff(fact_2940_mult__pos__neg,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,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_pos_neg
tff(fact_2941_mult__pos__neg,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,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_2942_mult__pos__neg,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,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_pos_neg
tff(fact_2943_mult__pos__neg,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,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_pos_neg
tff(fact_2944_mult__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(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_pos_pos
tff(fact_2945_mult__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(times_times_rat(A),B)) ) ) ).

% mult_pos_pos
tff(fact_2946_mult__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(times_times_nat,A),B)) ) ) ).

% mult_pos_pos
tff(fact_2947_mult__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(times_times_int,A),B)) ) ) ).

% mult_pos_pos
tff(fact_2948_mult__pos__neg2,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,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real) ) ) ).

% mult_pos_neg2
tff(fact_2949_mult__pos__neg2,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,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_2950_mult__pos__neg2,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,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_pos_neg2
tff(fact_2951_mult__pos__neg2,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,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).

% mult_pos_neg2
tff(fact_2952_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
    <=> ( ( 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,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_mult_iff
tff(fact_2953_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
    <=> ( ( 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,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_2954_zero__less__mult__iff,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(times_times_int,A),B))
    <=> ( ( 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,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) ) ) ) ).

% zero_less_mult_iff
tff(fact_2955_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
     => ( 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) ) ) ).

% zero_less_mult_pos
tff(fact_2956_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
     => ( 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) ) ) ).

% zero_less_mult_pos
tff(fact_2957_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ( 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) ) ) ).

% zero_less_mult_pos
tff(fact_2958_zero__less__mult__pos,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(times_times_int,A),B))
     => ( 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) ) ) ).

% zero_less_mult_pos
tff(fact_2959_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,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_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2960_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_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_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2961_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2962_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B),A))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2963_mult__less__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_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2964_mult__less__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_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2965_mult__less__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_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,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2966_mult__less__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_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2967_mult__less__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_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2968_mult__less__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_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,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2969_mult__strict__left__mono__neg,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),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2970_mult__strict__left__mono__neg,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),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2971_mult__strict__left__mono__neg,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),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2972_mult__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,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2973_mult__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,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2974_mult__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,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,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2975_mult__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,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,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2976_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( 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,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_real,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2977_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_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_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_rat,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2978_mult__less__cancel__left__disj,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_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_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2979_mult__strict__right__mono__neg,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),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2980_mult__strict__right__mono__neg,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),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2981_mult__strict__right__mono__neg,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),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2982_mult__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,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2983_mult__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,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2984_mult__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,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),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2985_mult__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,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),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2986_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( 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,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_real,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2987_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_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_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_rat,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2988_mult__less__cancel__right__disj,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_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_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2989_linordered__comm__semiring__strict__class_Ocomm__mult__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,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2990_linordered__comm__semiring__strict__class_Ocomm__mult__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,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2991_linordered__comm__semiring__strict__class_Ocomm__mult__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,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,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2992_linordered__comm__semiring__strict__class_Ocomm__mult__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,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,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2993_add__scale__eq__noteq,axiom,
    ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(R2),C)) != aa_complex_complex(plus_plus_complex(B),aa_complex_complex(times_times_complex(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2994_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),C)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2995_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(plus_plus_rat(A),aa_rat_rat(times_times_rat(R2),C)) != aa_rat_rat(plus_plus_rat(B),aa_rat_rat(times_times_rat(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2996_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_2997_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_2998_less__1__mult,axiom,
    ! [Mb: real,Na: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Mb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Mb),Na)) ) ) ).

% less_1_mult
tff(fact_2999_less__1__mult,axiom,
    ! [Mb: rat,Na: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Mb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(Mb),Na)) ) ) ).

% less_1_mult
tff(fact_3000_less__1__mult,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na)) ) ) ).

% less_1_mult
tff(fact_3001_less__1__mult,axiom,
    ! [Mb: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Na)) ) ) ).

% less_1_mult
tff(fact_3002_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( aa_complex_complex(divide1717551699836669952omplex(X),Y) = aa_complex_complex(divide1717551699836669952omplex(W2),Z) )
        <=> ( aa_complex_complex(times_times_complex(X),Z) = aa_complex_complex(times_times_complex(W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_3003_frac__eq__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z) )
        <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_3004_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( aa_rat_rat(divide_divide_rat(X),Y) = aa_rat_rat(divide_divide_rat(W2),Z) )
        <=> ( aa_rat_rat(times_times_rat(X),Z) = aa_rat_rat(times_times_rat(W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_3005_divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),C),A = zero_zero_complex) ) ).

% divide_eq_eq
tff(fact_3006_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
    <=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C),A = zero_zero_real) ) ).

% divide_eq_eq
tff(fact_3007_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = A )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),C),A = zero_zero_rat) ) ).

% divide_eq_eq
tff(fact_3008_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(A),C) = B,A = zero_zero_complex) ) ).

% eq_divide_eq
tff(fact_3009_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B,A = zero_zero_real) ) ).

% eq_divide_eq
tff(fact_3010_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(A),C) = B,A = zero_zero_rat) ) ).

% eq_divide_eq
tff(fact_3011_divide__eq__imp,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B = aa_complex_complex(times_times_complex(A),C) )
       => ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_3012_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_3013_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B = aa_rat_rat(times_times_rat(A),C) )
       => ( aa_rat_rat(divide_divide_rat(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_3014_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = B )
       => ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_3015_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B )
       => ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) ) ) ) ).

% eq_divide_imp
tff(fact_3016_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = B )
       => ( A = aa_rat_rat(divide_divide_rat(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_3017_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
      <=> ( B = aa_complex_complex(times_times_complex(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_3018_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
      <=> ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_3019_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(B),C) = A )
      <=> ( B = aa_rat_rat(times_times_rat(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_3020_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
      <=> ( aa_complex_complex(times_times_complex(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_3021_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_3022_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A = aa_rat_rat(divide_divide_rat(B),C) )
      <=> ( aa_rat_rat(times_times_rat(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_3023_power__Suc,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Na)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),Na)) ).

% power_Suc
tff(fact_3024_power__Suc,axiom,
    ! [A: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ).

% power_Suc
tff(fact_3025_power__Suc,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Na)) ).

% power_Suc
tff(fact_3026_power__Suc,axiom,
    ! [A: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ).

% power_Suc
tff(fact_3027_power__Suc,axiom,
    ! [A: int,Na: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Na)) ).

% power_Suc
tff(fact_3028_power__Suc2,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Na)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Na)),A) ).

% power_Suc2
tff(fact_3029_power__Suc2,axiom,
    ! [A: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),A) ).

% power_Suc2
tff(fact_3030_power__Suc2,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Na)),A) ).

% power_Suc2
tff(fact_3031_power__Suc2,axiom,
    ! [A: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),A) ).

% power_Suc2
tff(fact_3032_power__Suc2,axiom,
    ! [A: int,Na: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Na)),A) ).

% power_Suc2
tff(fact_3033_Suc__mult__less__cancel1,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ).

% Suc_mult_less_cancel1
tff(fact_3034_power__add,axiom,
    ! [A: complex,Mb: nat,Na: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Na)) ).

% power_add
tff(fact_3035_power__add,axiom,
    ! [A: real,Mb: nat,Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ).

% power_add
tff(fact_3036_power__add,axiom,
    ! [A: rat,Mb: nat,Na: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Na)) ).

% power_add
tff(fact_3037_power__add,axiom,
    ! [A: nat,Mb: nat,Na: 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),Na)) = 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),Na)) ).

% power_add
tff(fact_3038_power__add,axiom,
    ! [A: int,Mb: nat,Na: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)) = 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),Na)) ).

% power_add
tff(fact_3039_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ) ).

% mult_less_mono1
tff(fact_3040_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ) ).

% mult_less_mono2
tff(fact_3041_nat__mult__eq__cancel1,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na) )
      <=> ( Mb = Na ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_3042_nat__mult__less__cancel1,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ) ).

% nat_mult_less_cancel1
tff(fact_3043_Suc__mult__le__cancel1,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% Suc_mult_le_cancel1
tff(fact_3044_mult__Suc,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Mb)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na)) ).

% mult_Suc
tff(fact_3045_mult__eq__self__implies__10,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na) )
     => ( ( Na = one_one_nat )
        | ( Mb = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_3046_less__mult__imp__div__less,axiom,
    ! [Mb: nat,I: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na)),I) ) ).

% less_mult_imp_div_less
tff(fact_3047_times__div__less__eq__dividend,axiom,
    ! [Na: 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,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na))),Mb) ).

% times_div_less_eq_dividend
tff(fact_3048_div__times__less__eq__dividend,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na)),Na)),Mb) ).

% div_times_less_eq_dividend
tff(fact_3049_power__odd__eq,axiom,
    ! [A: complex,Na: 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))),Na))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3050_power__odd__eq,axiom,
    ! [A: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3051_power__odd__eq,axiom,
    ! [A: rat,Na: 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))),Na))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3052_power__odd__eq,axiom,
    ! [A: nat,Na: 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))),Na))) = 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),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3053_power__odd__eq,axiom,
    ! [A: int,Na: 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))),Na))) = 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),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3054_Suc__double__not__eq__double,axiom,
    ! [Mb: nat,Na: 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))),Na) ).

% Suc_double_not_eq_double
tff(fact_3055_double__not__eq__Suc__double,axiom,
    ! [Mb: nat,Na: 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))),Na)) ).

% double_not_eq_Suc_double
tff(fact_3056_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(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_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_3057_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(times_times_rat(C),A)),aa_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_3058_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_3059_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(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_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_3060_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(times_times_rat(A),C)),aa_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_3061_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_3062_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(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_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_3063_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(times_times_rat(C),A)),aa_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_3064_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_3065_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_3066_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(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3067_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3068_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_3069_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_3070_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(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_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_3071_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(times_times_rat(C),A)),aa_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_3072_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_3073_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(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_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_3074_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(times_times_rat(A),C)),aa_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_3075_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_3076_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_3077_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(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3078_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3079_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_3080_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_3081_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(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_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_3082_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(times_times_rat(A),C)),aa_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_3083_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_3084_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(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_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_3085_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(times_times_rat(C),A)),aa_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_3086_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_3087_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(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_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_3088_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(times_times_rat(C),A)),aa_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_3089_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_3090_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(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_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_3091_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(times_times_rat(C),A)),aa_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_3092_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_3093_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_3094_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(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_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_3095_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(times_times_rat(A),C)),aa_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_3096_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_3097_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_3098_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(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3099_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3100_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_3101_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_3102_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(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3103_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(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3104_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_3105_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_3106_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(aa_rea7340605660300375871l_real(times_times_real,A),C)),A) ) ) ).

% mult_left_le
tff(fact_3107_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(times_times_rat(A),C)),A) ) ) ).

% mult_left_le
tff(fact_3108_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_3109_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_3110_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(aa_rea7340605660300375871l_real(times_times_real,A),B)),one_one_real) ) ) ) ).

% mult_le_one
tff(fact_3111_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(times_times_rat(A),B)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_3112_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_3113_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_3114_mult__right__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( 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(aa_rea7340605660300375871l_real(times_times_real,X),Y)),X) ) ) ) ).

% mult_right_le_one_le
tff(fact_3115_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( 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(times_times_rat(X),Y)),X) ) ) ) ).

% mult_right_le_one_le
tff(fact_3116_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( 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,X),Y)),X) ) ) ) ).

% mult_right_le_one_le
tff(fact_3117_mult__left__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( 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(aa_rea7340605660300375871l_real(times_times_real,Y),X)),X) ) ) ) ).

% mult_left_le_one_le
tff(fact_3118_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( 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(times_times_rat(Y),X)),X) ) ) ) ).

% mult_left_le_one_le
tff(fact_3119_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( 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),X)),X) ) ) ) ).

% mult_left_le_one_le
tff(fact_3120_sum__squares__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real)
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3121_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat)
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3122_sum__squares__le__zero__iff,axiom,
    ! [X: 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,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3123_sum__squares__ge__zero,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3124_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3125_sum__squares__ge__zero,axiom,
    ! [X: 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,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3126_sum__squares__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))
    <=> ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3127_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y)))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3128_sum__squares__gt__zero__iff,axiom,
    ! [X: 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,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3129_not__sum__squares__lt__zero,axiom,
    ! [X: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real) ).

% not_sum_squares_lt_zero
tff(fact_3130_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_3131_not__sum__squares__lt__zero,axiom,
    ! [X: 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,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_3132_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_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_real,B),aa_real_real(aa_rea7340605660300375871l_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_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% divide_less_eq
tff(fact_3133_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_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_rat,B),aa_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_rat,aa_rat_rat(times_times_rat(A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% divide_less_eq
tff(fact_3134_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_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_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_divide_eq
tff(fact_3135_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_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(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_rat,B),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_3136_neg__divide__less__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_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_3137_neg__divide__less__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_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_3138_neg__less__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_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_3139_neg__less__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_rat,A),aa_rat_rat(divide_divide_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_3140_pos__divide__less__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_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_3141_pos__divide__less__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_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_3142_pos__less__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_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_3143_pos__less__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_rat,A),aa_rat_rat(divide_divide_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_3144_mult__imp__div__pos__less,axiom,
    ! [Y: real,X: 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_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_3145_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: 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_rat,X),aa_rat_rat(times_times_rat(Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_3146_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,X: 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,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_3147_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,X: 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_rat,aa_rat_rat(times_times_rat(Z),Y)),X)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(divide_divide_rat(X),Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_3148_divide__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,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_3149_divide__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,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_3150_divide__strict__left__mono__neg,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,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_3151_divide__strict__left__mono__neg,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,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_3152_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W2: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = numera6690914467698888265omplex(W2) )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),C),numera6690914467698888265omplex(W2) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3153_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = aa_num_real(numeral_numeral_real,W2) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C),aa_num_real(numeral_numeral_real,W2) = zero_zero_real) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3154_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_num_rat(numeral_numeral_rat,W2) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C),aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3155_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: complex,C: complex] :
      ( ( numera6690914467698888265omplex(W2) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),C) = B,numera6690914467698888265omplex(W2) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3156_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( aa_num_real(numeral_numeral_real,W2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C) = B,aa_num_real(numeral_numeral_real,W2) = zero_zero_real) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3157_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W2) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C) = B,aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3158_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_3159_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = $ite(Z = zero_zero_real,B,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_3160_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_3161_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_3162_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_3163_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      aa_rat_rat(plus_plus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_3164_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(W2),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),Z)),aa_complex_complex(times_times_complex(W2),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_3165_add__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_3166_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W2),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(W2),Y))),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_3167_add__frac__num,axiom,
    ! [Y: complex,X: complex,Z: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_3168_add__frac__num,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_3169_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X),Y)),Z) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),aa_rat_rat(times_times_rat(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_3170_add__num__frac,axiom,
    ! [Y: complex,Z: complex,X: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_3171_add__num__frac,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_3172_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(X),Y)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),aa_rat_rat(times_times_rat(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_3173_add__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(X),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_3174_add__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_3175_add__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(X),aa_rat_rat(divide_divide_rat(Y),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_3176_divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_3177_divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_3178_divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_3179_power__gt1__lemma,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))) ) ).

% power_gt1_lemma
tff(fact_3180_power__gt1__lemma,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Na))) ) ).

% power_gt1_lemma
tff(fact_3181_power__gt1__lemma,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))) ) ).

% power_gt1_lemma
tff(fact_3182_power__gt1__lemma,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Na))) ) ).

% power_gt1_lemma
tff(fact_3183_power__less__power__Suc,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))) ) ).

% power_less_power_Suc
tff(fact_3184_power__less__power__Suc,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Na))) ) ).

% power_less_power_Suc
tff(fact_3185_power__less__power__Suc,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))) ) ).

% power_less_power_Suc
tff(fact_3186_power__less__power__Suc,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Na))) ) ).

% power_less_power_Suc
tff(fact_3187_one__less__mult,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),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),Na)) ) ) ).

% one_less_mult
tff(fact_3188_n__less__m__mult__n,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na)) ) ) ).

% n_less_m_mult_n
tff(fact_3189_n__less__n__mult__m,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Mb)) ) ) ).

% n_less_n_mult_m
tff(fact_3190_nat__mult__le__cancel1,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ) ).

% nat_mult_le_cancel1
tff(fact_3191_nat__mult__div__cancel1,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na) ) ) ).

% nat_mult_div_cancel1
tff(fact_3192_div__less__iff__less__mult,axiom,
    ! [Q4: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Q4)),Na)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q4)) ) ) ).

% div_less_iff_less_mult
tff(fact_3193_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(aa_rea7340605660300375871l_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_3194_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(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_3195_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_3196_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_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_3197_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_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_3198_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_3199_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(aa_rea7340605660300375871l_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_3200_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(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_3201_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_3202_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_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_3203_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_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_3204_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_3205_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_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_3206_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_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_3207_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_3208_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_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_3209_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_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_3210_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_3211_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_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_3212_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_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_3213_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_3214_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_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_3215_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_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_3216_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_3217_field__le__mult__one__interval,axiom,
    ! [X: 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(aa_rea7340605660300375871l_real(times_times_real,Z2),X)),Y) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ).

% field_le_mult_one_interval
tff(fact_3218_field__le__mult__one__interval,axiom,
    ! [X: 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(times_times_rat(Z2),X)),Y) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% field_le_mult_one_interval
tff(fact_3219_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_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_3220_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(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(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(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_3221_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_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_3222_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(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(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(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_3223_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(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_left_mono
tff(fact_3224_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(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_left_mono
tff(fact_3225_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(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_3226_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(divide_divide_rat(B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_3227_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(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_3228_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(divide_divide_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_3229_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(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_3230_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(divide_divide_rat(B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_3231_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(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_3232_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(divide_divide_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_3233_mult__imp__div__pos__le,axiom,
    ! [Y: real,X: 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,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_3234_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: 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,X),aa_rat_rat(times_times_rat(Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_3235_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,X: 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(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_3236_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X: 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(times_times_rat(Z),Y)),X)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(divide_divide_rat(X),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_3237_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(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_3238_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(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_3239_convex__bound__le,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),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),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3240_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),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),V)
           => ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),X)),aa_rat_rat(times_times_rat(V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3241_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),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),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3242_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_num_real(numeral_numeral_real,W2))
    <=> $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(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),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(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,W2))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_3243_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W2))
    <=> $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(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),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(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W2))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_3244_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,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(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),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(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W2)),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_3245_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_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(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),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(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_3246_power__Suc__less,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ).

% power_Suc_less
tff(fact_3247_power__Suc__less,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Na))),aa_nat_rat(power_power_rat(A),Na)) ) ) ).

% power_Suc_less
tff(fact_3248_power__Suc__less,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ).

% power_Suc_less
tff(fact_3249_power__Suc__less,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Na))),aa_nat_int(power_power_int(A),Na)) ) ) ).

% power_Suc_less
tff(fact_3250_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_3251_mult__2,axiom,
    ! [Z: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),Z) = plus_p3455044024723400733d_enat(Z,Z) ).

% mult_2
tff(fact_3252_mult__2,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2
tff(fact_3253_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z) = aa_rat_rat(plus_plus_rat(Z),Z) ).

% mult_2
tff(fact_3254_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_3255_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_3256_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_3257_mult__2__right,axiom,
    ! [Z: extended_enat] : times_7803423173614009249d_enat(Z,numera1916890842035813515d_enat(aa_num_num(bit0,one))) = plus_p3455044024723400733d_enat(Z,Z) ).

% mult_2_right
tff(fact_3258_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2_right
tff(fact_3259_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(Z),Z) ).

% mult_2_right
tff(fact_3260_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_3261_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_3262_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_3263_left__add__twice,axiom,
    ! [A: extended_enat,B: extended_enat] : plus_p3455044024723400733d_enat(A,plus_p3455044024723400733d_enat(A,B)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),A),B) ).

% left_add_twice
tff(fact_3264_left__add__twice,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_3265_left__add__twice,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_3266_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_3267_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_3268_div__nat__eqI,axiom,
    ! [Na: nat,Q4: 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,Na),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,Na),aa_nat_nat(suc,Q4)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na) = Q4 ) ) ) ).

% div_nat_eqI
tff(fact_3269_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q4: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),Q4))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q4)),Na) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_3270_split__div,axiom,
    ! [P: fun_nat_o,Mb: nat,Na: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na))
    <=> ( ( ( Na = zero_zero_nat )
         => aa_nat_o(P,zero_zero_nat) )
        & ( ( Na != zero_zero_nat )
         => ! [I2: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Na)
             => ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),I2)),J3) )
               => aa_nat_o(P,I2) ) ) ) ) ) ).

% split_div
tff(fact_3271_dividend__less__div__times,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na)),Na))) ) ).

% dividend_less_div_times
tff(fact_3272_dividend__less__times__div,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na)))) ) ).

% dividend_less_times_div
tff(fact_3273_nat__bit__induct,axiom,
    ! [P: fun_nat_o,Na: 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,Na) ) ) ) ).

% nat_bit_induct
tff(fact_3274_convex__bound__lt,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),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),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3275_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),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),V)
           => ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),X)),aa_rat_rat(times_times_rat(V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3276_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),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),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3277_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_num_real(numeral_numeral_real,W2))
    <=> $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(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),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(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,W2))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3278_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W2))
    <=> $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(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),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(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W2))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3279_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,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(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),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(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W2)),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3280_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_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(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),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(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3281_split__div_H,axiom,
    ! [P: fun_nat_o,Mb: nat,Na: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na))
    <=> ( ( ( Na = zero_zero_nat )
          & aa_nat_o(P,zero_zero_nat) )
        | ? [Q5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q5)),Mb)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Q5)))
            & aa_nat_o(P,Q5) ) ) ) ).

% split_div'
tff(fact_3282_power2__sum,axiom,
    ! [X: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_3283_power2__sum,axiom,
    ! [X: extended_enat,Y: extended_enat] : power_8040749407984259932d_enat(plus_p3455044024723400733d_enat(X,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = plus_p3455044024723400733d_enat(plus_p3455044024723400733d_enat(power_8040749407984259932d_enat(X,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),power_8040749407984259932d_enat(Y,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),times_7803423173614009249d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),X),Y)) ).

% power2_sum
tff(fact_3284_power2__sum,axiom,
    ! [X: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_3285_power2__sum,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_3286_power2__sum,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_3287_power2__sum,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_3288_zero__le__even__power_H,axiom,
    ! [A: real,Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% zero_le_even_power'
tff(fact_3289_zero__le__even__power_H,axiom,
    ! [A: rat,Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% zero_le_even_power'
tff(fact_3290_zero__le__even__power_H,axiom,
    ! [A: int,Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% zero_le_even_power'
tff(fact_3291_triangle__def,axiom,
    ! [Na: nat] : aa_nat_nat(nat_triangle,Na) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_3292_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3293_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3294_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3295_odd__power__less__zero,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),zero_zero_real) ) ).

% odd_power_less_zero
tff(fact_3296_odd__power__less__zero,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),zero_zero_rat) ) ).

% odd_power_less_zero
tff(fact_3297_odd__power__less__zero,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),zero_zero_int) ) ).

% odd_power_less_zero
tff(fact_3298_arith__geo__mean,axiom,
    ! [U: real,X: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( 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(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_3299_arith__geo__mean,axiom,
    ! [U: rat,X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(X),Y) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
       => ( 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(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_3300_sum__squares__bound,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_3301_sum__squares__bound,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_3302_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_3303_set__bit__0,axiom,
    ! [A: code_integer] : bit_se2793503036327961859nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_3304_set__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se7879613467334960850it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_3305_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_3306_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_3307_unset__bit__0,axiom,
    ! [A: code_integer] : bit_se8260200283734997820nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_3308_unset__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se4203085406695923979it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_3309_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3310_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_3311_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(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3312_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_3313_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_3314_unique__euclidean__semiring__numeral__class_Odiv__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)
       => ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3315_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_3316_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(divide_divide_int(A),B) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3317_div__positive,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,B),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),divide6298287555418463151nteger(A,B)) ) ) ).

% div_positive
tff(fact_3318_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_3319_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(divide_divide_int(A),B)) ) ) ).

% div_positive
tff(fact_3320_zle__add1__eq__le,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ).

% zle_add1_eq_le
tff(fact_3321_set__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se7879613467334960850it_int(Na),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% set_bit_negative_int_iff
tff(fact_3322_unset__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se4203085406695923979it_int(Na),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% unset_bit_negative_int_iff
tff(fact_3323_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(divide_divide_int(K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_3324_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(divide_divide_int(K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_3325_nonneg1__imp__zdiv__pos__iff,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),aa_int_int(divide_divide_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_int,zero_zero_int),B) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_3326_pos__imp__zdiv__nonneg__iff,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(divide_divide_int(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_3327_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: 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_eq_int,zero_zero_int),aa_int_int(divide_divide_int(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_3328_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_3329_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(I),K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_3330_div__nonpos__pos__le0,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,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int) ) ) ).

% div_nonpos_pos_le0
tff(fact_3331_div__nonneg__neg__le0,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,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int) ) ) ).

% div_nonneg_neg_le0
tff(fact_3332_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(K),L)) ) ) ).

% div_positive_int
tff(fact_3333_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(K),L))
    <=> ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ) ).

% div_int_pos_iff
tff(fact_3334_zless__imp__add1__zle,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z) ) ).

% zless_imp_add1_zle
tff(fact_3335_zdiv__mono2__neg,axiom,
    ! [A: int,B4: 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,zero_zero_int),B4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A),B4)),aa_int_int(divide_divide_int(A),B)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_3336_zdiv__mono1__neg,axiom,
    ! [A: int,A4: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A4)
     => ( 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_eq_int,aa_int_int(divide_divide_int(A4),B)),aa_int_int(divide_divide_int(A),B)) ) ) ).

% zdiv_mono1_neg
tff(fact_3337_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2) )
     => ( 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),B)
         => ( aa_int_int(divide_divide_int(A),B) = Q4 ) ) ) ) ).

% int_div_pos_eq
tff(fact_3338_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_int(divide_divide_int(A),B) = Q4 ) ) ) ) ).

% int_div_neg_eq
tff(fact_3339_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(divide_divide_int(I),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_3340_zdiv__mono2,axiom,
    ! [A: int,B4: 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),B4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(A),B4)) ) ) ) ).

% zdiv_mono2
tff(fact_3341_zdiv__mono1,axiom,
    ! [A: int,A4: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A4)
     => ( 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,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(A4),B)) ) ) ).

% zdiv_mono1
tff(fact_3342_split__zdiv,axiom,
    ! [P: fun_int_o,Na: int,K: int] :
      ( aa_int_o(P,aa_int_int(divide_divide_int(Na),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I2: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J3) ) )
             => aa_int_o(P,I2) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I2: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J3) ) )
             => aa_int_o(P,I2) ) ) ) ) ).

% split_zdiv
tff(fact_3343_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ).

% le_imp_0_less
tff(fact_3344_add1__zle__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% add1_zle_eq
tff(fact_3345_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% odd_less_0_iff
tff(fact_3346_pos__zmult__eq__1__iff,axiom,
    ! [Mb: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Mb)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Na) = one_one_int )
      <=> ( ( Mb = one_one_int )
          & ( Na = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_3347_zless__add1__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z)
        | ( W2 = Z ) ) ) ).

% zless_add1_eq
tff(fact_3348_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I3)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_gr_induct
tff(fact_3349_pos__imp__zdiv__neg__iff,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(divide_divide_int(A),B)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_3350_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: 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(divide_divide_int(A),B)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_3351_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(X),K)),X) ) ) ).

% int_div_less_self
tff(fact_3352_div__neg__pos__less0,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,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int) ) ) ).

% div_neg_pos_less0
tff(fact_3353_odd__nonzero,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z) != zero_zero_int ).

% odd_nonzero
tff(fact_3354_times__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),L) = zero_zero_int ).

% times_int_code(2)
tff(fact_3355_times__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(times_times_int,K),zero_zero_int) = zero_zero_int ).

% times_int_code(1)
tff(fact_3356_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_3357_incr__mult__lemma,axiom,
    ! [D: int,P: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X3: int] :
            ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X2: int] :
              ( aa_int_o(P,X2)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_3358_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_3359_less__int__code_I1_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_int_code(1)
tff(fact_3360_q__pos__lemma,axiom,
    ! [B4: int,Q6: int,R3: 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,B4),Q6)),R3))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R3),B4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q6) ) ) ) ).

% q_pos_lemma
tff(fact_3361_zdiv__mono2__lemma,axiom,
    ! [B: int,Q4: int,R2: int,B4: int,Q6: int,R3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q6)),R3) )
     => ( 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,B4),Q6)),R3))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R3),B4)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q4),Q6) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_3362_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q4: int,R2: int,B4: int,Q6: int,R3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q6)),R3) )
     => ( 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,B4),Q6)),R3)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R3)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q6),Q4) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_3363_unique__quotient__lemma,axiom,
    ! [B: int,Q6: int,R3: int,Q4: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q6)),R3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R3),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q6),Q4) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_3364_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q6: int,R3: int,Q4: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q6)),R3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q4),Q6) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_3365_enat__0__less__mult__iff,axiom,
    ! [Mb: extended_enat,Na: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),times_7803423173614009249d_enat(Mb,Na))
    <=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Mb)
        & aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Na) ) ) ).

% enat_0_less_mult_iff
tff(fact_3366_realpow__pos__nth2,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ? [R4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
          & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R4),aa_nat_nat(suc,Na)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_3367_realpow__pos__nth__unique,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X3),Na) = A )
            & ! [Y4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4)
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y4),Na) = A ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_3368_realpow__pos__nth,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [R4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R4),Na) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_3369_mult__le__cancel__iff2,axiom,
    ! [Z: real,X: 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(aa_rea7340605660300375871l_real(times_times_real,Z),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_3370_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: 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(times_times_rat(Z),X)),aa_rat_rat(times_times_rat(Z),Y))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_3371_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: 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),X)),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,X),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_3372_mult__le__cancel__iff1,axiom,
    ! [Z: real,X: 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(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_3373_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: 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(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_3374_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: 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,X),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,X),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_3375_divides__aux__eq,axiom,
    ! [Q4: nat,R2: nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q4),R2))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_3376_divides__aux__eq,axiom,
    ! [Q4: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R2))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_3377_low__def,axiom,
    ! [X: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% low_def
tff(fact_3378_even__succ__div__exp,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ) ).

% even_succ_div_exp
tff(fact_3379_even__succ__div__exp,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ) ).

% even_succ_div_exp
tff(fact_3380_even__succ__div__exp,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) ) ) ) ).

% even_succ_div_exp
tff(fact_3381_set__decode__Suc,axiom,
    ! [Na: nat,X: nat] :
      ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Na)),nat_set_decode(X))
    <=> aa_set_nat_o(member_nat(Na),nat_set_decode(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% set_decode_Suc
tff(fact_3382_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,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys)) ).

% length_product
tff(fact_3383_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,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_list_o_nat(size_size_list_o,Ys)) ).

% length_product
tff(fact_3384_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,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_list_nat_nat(size_size_list_nat,Ys)) ).

% length_product
tff(fact_3385_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,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_list_int_nat(size_size_list_int,Ys)) ).

% length_product
tff(fact_3386_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,aa_list_o_nat(size_size_list_o,Xs)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys)) ).

% length_product
tff(fact_3387_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,aa_list_o_nat(size_size_list_o,Xs)),aa_list_o_nat(size_size_list_o,Ys)) ).

% length_product
tff(fact_3388_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,aa_list_o_nat(size_size_list_o,Xs)),aa_list_nat_nat(size_size_list_nat,Ys)) ).

% length_product
tff(fact_3389_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,aa_list_o_nat(size_size_list_o,Xs)),aa_list_int_nat(size_size_list_int,Ys)) ).

% length_product
tff(fact_3390_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,aa_list_nat_nat(size_size_list_nat,Xs)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys)) ).

% length_product
tff(fact_3391_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,aa_list_nat_nat(size_size_list_nat,Xs)),aa_list_o_nat(size_size_list_o,Ys)) ).

% length_product
tff(fact_3392_mult__less__iff1,axiom,
    ! [Z: real,X: 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_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ).

% mult_less_iff1
tff(fact_3393_mult__less__iff1,axiom,
    ! [Z: rat,X: 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_rat,aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ) ).

% mult_less_iff1
tff(fact_3394_mult__less__iff1,axiom,
    ! [Z: int,X: 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_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ) ).

% mult_less_iff1
tff(fact_3395_num_Osize__gen_I2_J,axiom,
    ! [X22: num] : aa_num_nat(size_num,aa_num_num(bit0,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_3396_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_3397_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
tff(fact_3398_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,zero_zero_complex),A)
    <=> ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
tff(fact_3399_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),A)
    <=> ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
tff(fact_3400_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A)
    <=> ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
tff(fact_3401_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
    <=> ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
tff(fact_3402_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A)
    <=> ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
tff(fact_3403_dvd__0__right,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),zero_z3403309356797280102nteger) ).

% dvd_0_right
tff(fact_3404_dvd__0__right,axiom,
    ! [A: complex] : aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),zero_zero_complex) ).

% dvd_0_right
tff(fact_3405_dvd__0__right,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),zero_zero_real) ).

% dvd_0_right
tff(fact_3406_dvd__0__right,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),zero_zero_rat) ).

% dvd_0_right
tff(fact_3407_dvd__0__right,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat) ).

% dvd_0_right
tff(fact_3408_dvd__0__right,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),zero_zero_int) ).

% dvd_0_right
tff(fact_3409_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_3410_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(aa_rea7340605660300375871l_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_3411_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(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_3412_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_3413_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_3414_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_3415_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(aa_rea7340605660300375871l_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_3416_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(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_3417_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_3418_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_3419_bits__mod__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).

% bits_mod_0
tff(fact_3420_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_3421_bits__mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% bits_mod_0
tff(fact_3422_mod__self,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),A) = zero_zero_nat ).

% mod_self
tff(fact_3423_mod__self,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),A) = zero_zero_int ).

% mod_self
tff(fact_3424_mod__self,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,A) = zero_z3403309356797280102nteger ).

% mod_self
tff(fact_3425_mod__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),zero_zero_nat) = A ).

% mod_by_0
tff(fact_3426_mod__by__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),zero_zero_int) = A ).

% mod_by_0
tff(fact_3427_mod__by__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,zero_z3403309356797280102nteger) = A ).

% mod_by_0
tff(fact_3428_mod__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).

% mod_0
tff(fact_3429_mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_3430_mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% mod_0
tff(fact_3431_dvd__1__left,axiom,
    ! [K: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K) ).

% dvd_1_left
tff(fact_3432_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_3433_div__dvd__div,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,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ) ).

% div_dvd_div
tff(fact_3434_div__dvd__div,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,aa_int_int(divide_divide_int(B),A)),aa_int_int(divide_divide_int(C),A))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ) ).

% div_dvd_div
tff(fact_3435_div__dvd__div,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,divide6298287555418463151nteger(B,A)),divide6298287555418463151nteger(C,A))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ) ).

% div_dvd_div
tff(fact_3436_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X))
    <=> ( X = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_3437_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_3438_mod__add__self2,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_add_self2
tff(fact_3439_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self2
tff(fact_3440_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_3441_mod__add__self1,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_add_self1
tff(fact_3442_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(B,A),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self1
tff(fact_3443_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
    <=> ( ( K = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Na) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_3444_mod__less,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Na) = Mb ) ) ).

% mod_less
tff(fact_3445_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3446_dvd__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B))
    <=> ( ( C = zero_zero_complex )
        | aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3447_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3448_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3449_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3450_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3451_dvd__mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C))
    <=> ( ( C = zero_zero_complex )
        | aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3452_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3453_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3454_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3455_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3456_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3457_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3458_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3459_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3460_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3461_unit__prod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer) ) ) ).

% unit_prod
tff(fact_3462_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ).

% unit_prod
tff(fact_3463_unit__prod,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int) ) ) ).

% unit_prod
tff(fact_3464_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_3465_dvd__add__times__triv__right__iff,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),aa_complex_complex(plus_plus_complex(B),aa_complex_complex(times_times_complex(C),A)))
    <=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3466_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(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_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_3467_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(plus_plus_rat(B),aa_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_3468_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_3469_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_3470_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_3471_dvd__add__times__triv__left__iff,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(C),A)),B))
    <=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3472_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(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_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_3473_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(plus_plus_rat(aa_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_3474_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_3475_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_3476_mod__mult__self1__is__0,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),B) = zero_zero_nat ).

% mod_mult_self1_is_0
tff(fact_3477_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),B) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_3478_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),B) = zero_z3403309356797280102nteger ).

% mod_mult_self1_is_0
tff(fact_3479_mod__mult__self2__is__0,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = zero_zero_nat ).

% mod_mult_self2_is_0
tff(fact_3480_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_3481_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),B) = zero_z3403309356797280102nteger ).

% mod_mult_self2_is_0
tff(fact_3482_bits__mod__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).

% bits_mod_by_1
tff(fact_3483_bits__mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_3484_bits__mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_3485_mod__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).

% mod_by_1
tff(fact_3486_mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_3487_mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_3488_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_3489_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_3490_dvd__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_3491_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_3492_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_3493_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_3494_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% unit_div
tff(fact_3495_unit__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),one_one_int) ) ) ).

% unit_div
tff(fact_3496_unit__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),one_one_Code_integer) ) ) ).

% unit_div
tff(fact_3497_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)),one_one_nat) ) ).

% unit_div_1_unit
tff(fact_3498_unit__div__1__unit,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(one_one_int),A)),one_one_int) ) ).

% unit_div_1_unit
tff(fact_3499_unit__div__1__unit,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,A)),one_one_Code_integer) ) ).

% unit_div_1_unit
tff(fact_3500_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_3501_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(divide_divide_int(one_one_int),aa_int_int(divide_divide_int(one_one_int),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_3502_unit__div__1__div__1,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( divide6298287555418463151nteger(one_one_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_3503_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_3504_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(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ) ).

% div_add
tff(fact_3505_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_3506_bits__mod__div__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).

% bits_mod_div_trivial
tff(fact_3507_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),B)),B) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_3508_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_3509_mod__div__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).

% mod_div_trivial
tff(fact_3510_mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),B)),B) = zero_zero_int ).

% mod_div_trivial
tff(fact_3511_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_3512_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_3513_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self4
tff(fact_3514_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_3515_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_3516_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self3
tff(fact_3517_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_3518_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_3519_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self2
tff(fact_3520_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_3521_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_3522_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self1
tff(fact_3523_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_3524_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_imp_mod_0
tff(fact_3525_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(modulo_modulo_int(B),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_3526_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
tff(fact_3527_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_3528_set__decode__zero,axiom,
    nat_set_decode(zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_3529_set__encode__inverse,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( nat_set_decode(aa_set_nat_nat(nat_set_encode,A2)) = A2 ) ) ).

% set_encode_inverse
tff(fact_3530_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_3531_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_3532_unit__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_3533_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) ) ) ).

% unit_mult_div_div
tff(fact_3534_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(divide_divide_int(one_one_int),A)) = aa_int_int(divide_divide_int(B),A) ) ) ).

% unit_mult_div_div
tff(fact_3535_unit__mult__div__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(one_one_Code_integer,A)) = divide6298287555418463151nteger(B,A) ) ) ).

% unit_mult_div_div
tff(fact_3536_even__Suc,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,Na))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ) ).

% even_Suc
tff(fact_3537_even__Suc__Suc__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ) ).

% even_Suc_Suc_iff
tff(fact_3538_pow__divides__pow__iff,axiom,
    ! [Na: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_3539_pow__divides__pow__iff,axiom,
    ! [Na: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_3540_Suc__mod__mult__self4,axiom,
    ! [Na: nat,K: 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,Na),K)),Mb))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Na) ).

% Suc_mod_mult_self4
tff(fact_3541_Suc__mod__mult__self3,axiom,
    ! [K: nat,Na: 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,K),Na)),Mb))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Na) ).

% Suc_mod_mult_self3
tff(fact_3542_Suc__mod__mult__self2,axiom,
    ! [Mb: nat,Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Na) ).

% Suc_mod_mult_self2
tff(fact_3543_Suc__mod__mult__self1,axiom,
    ! [Mb: nat,K: nat,Na: 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,K),Na)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Na) ).

% Suc_mod_mult_self1
tff(fact_3544_even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_3545_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_3546_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_3547_odd__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))
    <=> ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_3548_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_3549_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_3550_odd__Suc__div__two,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Na)),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_3551_even__Suc__div__two,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_3552_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_3553_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,Na: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) != one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),Na))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_3554_set__decode__0,axiom,
    ! [X: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),nat_set_decode(X))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X) ) ).

% set_decode_0
tff(fact_3555_zero__le__power__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3556_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3557_zero__le__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3558_power__less__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3559_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3560_power__less__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3561_power__less__zero__eq,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq
tff(fact_3562_power__less__zero__eq,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq
tff(fact_3563_power__less__zero__eq,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_3564_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_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,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_3565_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_3566_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_3567_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_3568_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_3569_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_3570_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_3571_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_3572_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_3573_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_3574_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_3575_zero__less__power__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3576_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3577_zero__less__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3578_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_3579_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(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_3580_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_3581_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_3582_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(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_3583_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),one_one_Code_integer) ) ) ).

% odd_succ_div_two
tff(fact_3584_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_3585_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(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_3586_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_3587_even__power,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% even_power
tff(fact_3588_even__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% even_power
tff(fact_3589_even__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(A),Na))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% even_power
tff(fact_3590_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_3591_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_3592_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(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_3593_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),one_one_Code_integer) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3594_power__le__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3595_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3596_power__le__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3597_even__succ__mod__exp,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3598_even__succ__mod__exp,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3599_even__succ__mod__exp,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3600_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) ) ).

% real_arch_pow
tff(fact_3601_real__arch__pow__inv,axiom,
    ! [Y: real,X: 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,X),one_one_real)
       => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_3602_int__distrib_I2_J,axiom,
    ! [W2: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z22)) ).

% int_distrib(2)
tff(fact_3603_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)),W2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W2)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W2)) ).

% int_distrib(1)
tff(fact_3604_zdvd__period,axiom,
    ! [A: int,D: int,X: int,T: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),D)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),T))
      <=> 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(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))),T)) ) ) ).

% zdvd_period
tff(fact_3605_zdvd__reduce,axiom,
    ! [K: int,Na: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Na) ) ).

% zdvd_reduce
tff(fact_3606_less__eq__real__def,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
        | ( X = Y ) ) ) ).

% less_eq_real_def
tff(fact_3607_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_3608_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_3609_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_3610_dvd__eq__mod__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_eq_mod_eq_0
tff(fact_3611_dvd__eq__mod__eq__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
    <=> ( aa_int_int(modulo_modulo_int(B),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_3612_dvd__eq__mod__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
    <=> ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_eq_mod_eq_0
tff(fact_3613_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_0_imp_dvd
tff(fact_3614_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_0_imp_dvd
tff(fact_3615_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_0_imp_dvd
tff(fact_3616_dvd__refl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),A) ).

% dvd_refl
tff(fact_3617_dvd__refl,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),A) ).

% dvd_refl
tff(fact_3618_dvd__refl,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),A) ).

% dvd_refl
tff(fact_3619_dvd__trans,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,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_trans
tff(fact_3620_dvd__trans,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,B),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_trans
tff(fact_3621_dvd__trans,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,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_trans
tff(fact_3622_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_iff
tff(fact_3623_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_iff
tff(fact_3624_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_iff
tff(fact_3625_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_3626_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_3627_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_3628_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I)
     => ( aa_int_o(P,K)
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I3)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_ge_induct
tff(fact_3629_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_3630_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_3631_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_3632_mod__greater__zero__iff__not__dvd,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Na))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),Mb) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_3633_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_3634_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_right_eq
tff(fact_3635_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_3636_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_3637_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_left_eq
tff(fact_3638_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_3639_mod__add__cong,axiom,
    ! [A: nat,C: nat,A4: nat,B: nat,B4: 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,A4),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,B4),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,A4),B4)),C) ) ) ) ).

% mod_add_cong
tff(fact_3640_mod__add__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B4: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A4),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B4),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A4),B4)),C) ) ) ) ).

% mod_add_cong
tff(fact_3641_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B4: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A4,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B4,C) )
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A4,B4),C) ) ) ) ).

% mod_add_cong
tff(fact_3642_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_3643_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_eq
tff(fact_3644_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_3645_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A)
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
tff(fact_3646_dvd__0__left,axiom,
    ! [A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,zero_zero_complex),A)
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
tff(fact_3647_dvd__0__left,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),A)
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
tff(fact_3648_dvd__0__left,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A)
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
tff(fact_3649_dvd__0__left,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
tff(fact_3650_dvd__0__left,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A)
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
tff(fact_3651_dvd__field__iff,axiom,
    ! [A: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B)
    <=> ( ( A = zero_zero_complex )
       => ( B = zero_zero_complex ) ) ) ).

% dvd_field_iff
tff(fact_3652_dvd__field__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
    <=> ( ( A = zero_zero_real )
       => ( B = zero_zero_real ) ) ) ).

% dvd_field_iff
tff(fact_3653_dvd__field__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
    <=> ( ( A = zero_zero_rat )
       => ( B = zero_zero_rat ) ) ) ).

% dvd_field_iff
tff(fact_3654_mod__Suc__Suc__eq,axiom,
    ! [Mb: nat,Na: 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),Na)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),Na) ).

% mod_Suc_Suc_eq
tff(fact_3655_mod__Suc__eq,axiom,
    ! [Mb: nat,Na: 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),Na))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Na) ).

% mod_Suc_eq
tff(fact_3656_dvd__productE,axiom,
    ! [P2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Y3) )
           => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X3),A)
             => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Y3),B) ) ) ) ).

% dvd_productE
tff(fact_3657_dvd__productE,axiom,
    ! [P2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,P2),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
     => ~ ! [X3: int,Y3: int] :
            ( ( P2 = aa_int_int(aa_int_fun_int_int(times_times_int,X3),Y3) )
           => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X3),A)
             => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y3),B) ) ) ) ).

% dvd_productE
tff(fact_3658_division__decomp,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(times_times_nat,B),C))
     => ? [B7: nat,C5: nat] :
          ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B7),C5) )
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B7),B)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C5),C) ) ) ).

% division_decomp
tff(fact_3659_division__decomp,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(times_times_int,B),C))
     => ? [B7: int,C5: int] :
          ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B7),C5) )
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B7),B)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C5),C) ) ) ).

% division_decomp
tff(fact_3660_dvdE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ~ ! [K2: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K2) ) ).

% dvdE
tff(fact_3661_dvdE,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),A)
     => ~ ! [K2: complex] : A != aa_complex_complex(times_times_complex(B),K2) ) ).

% dvdE
tff(fact_3662_dvdE,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ~ ! [K2: real] : A != aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K2) ) ).

% dvdE
tff(fact_3663_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ~ ! [K2: rat] : A != aa_rat_rat(times_times_rat(B),K2) ) ).

% dvdE
tff(fact_3664_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ~ ! [K2: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) ) ).

% dvdE
tff(fact_3665_dvdE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ~ ! [K2: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,B),K2) ) ).

% dvdE
tff(fact_3666_dvdI,axiom,
    ! [A: code_integer,B: code_integer,K: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% dvdI
tff(fact_3667_dvdI,axiom,
    ! [A: complex,B: complex,K: complex] :
      ( ( A = aa_complex_complex(times_times_complex(B),K) )
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),A) ) ).

% dvdI
tff(fact_3668_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A) ) ).

% dvdI
tff(fact_3669_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A = aa_rat_rat(times_times_rat(B),K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A) ) ).

% dvdI
tff(fact_3670_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% dvdI
tff(fact_3671_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% dvdI
tff(fact_3672_dvd__def,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
    <=> ? [K3: code_integer] : A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K3) ) ).

% dvd_def
tff(fact_3673_dvd__def,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),A)
    <=> ? [K3: complex] : A = aa_complex_complex(times_times_complex(B),K3) ) ).

% dvd_def
tff(fact_3674_dvd__def,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
    <=> ? [K3: real] : A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K3) ) ).

% dvd_def
tff(fact_3675_dvd__def,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
    <=> ? [K3: rat] : A = aa_rat_rat(times_times_rat(B),K3) ) ).

% dvd_def
tff(fact_3676_dvd__def,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
    <=> ? [K3: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K3) ) ).

% dvd_def
tff(fact_3677_dvd__def,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
    <=> ? [K3: int] : A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K3) ) ).

% dvd_def
tff(fact_3678_dvd__mult,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),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult
tff(fact_3679_dvd__mult,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),C)
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),aa_complex_complex(times_times_complex(B),C)) ) ).

% dvd_mult
tff(fact_3680_dvd__mult,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(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ).

% dvd_mult
tff(fact_3681_dvd__mult,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(times_times_rat(B),C)) ) ).

% dvd_mult
tff(fact_3682_dvd__mult,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(times_times_nat,B),C)) ) ).

% dvd_mult
tff(fact_3683_dvd__mult,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(times_times_int,B),C)) ) ).

% dvd_mult
tff(fact_3684_dvd__mult2,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),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult2
tff(fact_3685_dvd__mult2,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B)
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),aa_complex_complex(times_times_complex(B),C)) ) ).

% dvd_mult2
tff(fact_3686_dvd__mult2,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(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ).

% dvd_mult2
tff(fact_3687_dvd__mult2,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(times_times_rat(B),C)) ) ).

% dvd_mult2
tff(fact_3688_dvd__mult2,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(times_times_nat,B),C)) ) ).

% dvd_mult2
tff(fact_3689_dvd__mult2,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(times_times_int,B),C)) ) ).

% dvd_mult2
tff(fact_3690_dvd__mult__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ).

% dvd_mult_left
tff(fact_3691_dvd__mult__left,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),B)),C)
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),C) ) ).

% dvd_mult_left
tff(fact_3692_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ).

% dvd_mult_left
tff(fact_3693_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ).

% dvd_mult_left
tff(fact_3694_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ).

% dvd_mult_left
tff(fact_3695_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ).

% dvd_mult_left
tff(fact_3696_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% dvd_triv_left
tff(fact_3697_dvd__triv__left,axiom,
    ! [A: complex,B: complex] : aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),aa_complex_complex(times_times_complex(A),B)) ).

% dvd_triv_left
tff(fact_3698_dvd__triv__left,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% dvd_triv_left
tff(fact_3699_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(A),B)) ).

% dvd_triv_left
tff(fact_3700_dvd__triv__left,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(times_times_nat,A),B)) ).

% dvd_triv_left
tff(fact_3701_dvd__triv__left,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(times_times_int,A),B)) ).

% dvd_triv_left
tff(fact_3702_mult__dvd__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: 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,C),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3703_mult__dvd__mono,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B)
     => ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,C),D)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3704_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: real,D: 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,C),D)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3705_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: 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,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3706_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: 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,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3707_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: 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,C),D)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3708_dvd__mult__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ).

% dvd_mult_right
tff(fact_3709_dvd__mult__right,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),B)),C)
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),C) ) ).

% dvd_mult_right
tff(fact_3710_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),C) ) ).

% dvd_mult_right
tff(fact_3711_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),C) ) ).

% dvd_mult_right
tff(fact_3712_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ).

% dvd_mult_right
tff(fact_3713_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ).

% dvd_mult_right
tff(fact_3714_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) ).

% dvd_triv_right
tff(fact_3715_dvd__triv__right,axiom,
    ! [A: complex,B: complex] : aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),aa_complex_complex(times_times_complex(B),A)) ).

% dvd_triv_right
tff(fact_3716_dvd__triv__right,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)) ).

% dvd_triv_right
tff(fact_3717_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),A)) ).

% dvd_triv_right
tff(fact_3718_dvd__triv__right,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(times_times_nat,B),A)) ).

% dvd_triv_right
tff(fact_3719_dvd__triv__right,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(times_times_int,B),A)) ).

% dvd_triv_right
tff(fact_3720_one__dvd,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,one_one_Code_integer),A) ).

% one_dvd
tff(fact_3721_one__dvd,axiom,
    ! [A: complex] : aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,one_one_complex),A) ).

% one_dvd
tff(fact_3722_one__dvd,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,one_one_real),A) ).

% one_dvd
tff(fact_3723_one__dvd,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,one_one_rat),A) ).

% one_dvd
tff(fact_3724_one__dvd,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,one_one_nat),A) ).

% one_dvd
tff(fact_3725_one__dvd,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,one_one_int),A) ).

% one_dvd
tff(fact_3726_unit__imp__dvd,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% unit_imp_dvd
tff(fact_3727_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% unit_imp_dvd
tff(fact_3728_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% unit_imp_dvd
tff(fact_3729_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B: 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,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer) ) ) ).

% dvd_unit_imp_unit
tff(fact_3730_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: 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,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat) ) ) ).

% dvd_unit_imp_unit
tff(fact_3731_dvd__unit__imp__unit,axiom,
    ! [A: int,B: 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,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int) ) ) ).

% dvd_unit_imp_unit
tff(fact_3732_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_3733_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(aa_rea7340605660300375871l_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_3734_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(plus_plus_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_3735_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_3736_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_3737_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_3738_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(aa_rea7340605660300375871l_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_3739_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(plus_plus_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3740_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_3741_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_3742_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_3743_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(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ) ).

% dvd_add
tff(fact_3744_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(plus_plus_rat(B),C)) ) ) ).

% dvd_add
tff(fact_3745_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_3746_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_3747_dvd__div__eq__iff,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,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3748_dvd__div__eq__iff,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(divide_divide_int(A),C) = aa_int_int(divide_divide_int(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3749_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),A)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3750_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
       => ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3751_dvd__div__eq__iff,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(A,C) = divide6298287555418463151nteger(B,C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3752_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
     => ( 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)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3753_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(divide_divide_int(A),C) = aa_int_int(divide_divide_int(B),C) )
     => ( 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)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3754_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),A)
       => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3755_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3756_dvd__div__eq__cancel,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger(A,C) = divide6298287555418463151nteger(B,C) )
     => ( 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)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3757_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% div_div_div_same
tff(fact_3758_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),D)),aa_int_int(divide_divide_int(B),D)) = aa_int_int(divide_divide_int(A),B) ) ) ) ).

% div_div_div_same
tff(fact_3759_div__div__div__same,axiom,
    ! [D: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( divide6298287555418463151nteger(divide6298287555418463151nteger(A,D),divide6298287555418463151nteger(B,D)) = divide6298287555418463151nteger(A,B) ) ) ) ).

% div_div_div_same
tff(fact_3760_mod__less__eq__dividend,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Na)),Mb) ).

% mod_less_eq_dividend
tff(fact_3761_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_3762_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat)
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_3763_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
    <=> ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
tff(fact_3764_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
tff(fact_3765_gcd__nat_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat) ).

% gcd_nat.extremum
tff(fact_3766_zdvd__antisym__nonneg,axiom,
    ! [Mb: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Na)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Na)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Na),Mb)
           => ( Mb = Na ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_3767_zdvd__mult__cancel,axiom,
    ! [K: int,Mb: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Na))
     => ( ( K != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Na) ) ) ).

% zdvd_mult_cancel
tff(fact_3768_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_3769_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(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_3770_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_3771_subset__decode__imp__le,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,nat_set_decode(Mb)),nat_set_decode(Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% subset_decode_imp_le
tff(fact_3772_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_3773_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_3774_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(modulo_modulo_int(A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_3775_unique__euclidean__semiring__numeral__class_Opos__mod__bound,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),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_3776_unique__euclidean__semiring__numeral__class_Opos__mod__bound,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(modulo_modulo_int(A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_3777_unique__euclidean__semiring__numeral__class_Opos__mod__bound,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,B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_3778_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_3779_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = A )
    <=> ( aa_int_int(divide_divide_int(A),B) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_3780_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = A )
    <=> ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_3781_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(B),C) )
     => ~ ! [D4: int] : B != aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),D4)) ) ).

% mod_eqE
tff(fact_3782_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_3783_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_3784_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C)) ).

% div_add1_eq
tff(fact_3785_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_3786_not__is__unit__0,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% not_is_unit_0
tff(fact_3787_not__is__unit__0,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),one_one_nat) ).

% not_is_unit_0
tff(fact_3788_not__is__unit__0,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),one_one_int) ).

% not_is_unit_0
tff(fact_3789_minf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X2),Z2)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S)) ) ) ).

% minf(10)
tff(fact_3790_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z2)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S)) ) ) ).

% minf(10)
tff(fact_3791_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z2)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S)) ) ) ).

% minf(10)
tff(fact_3792_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z2)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S)) ) ) ).

% minf(10)
tff(fact_3793_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z2)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S)) ) ) ).

% minf(10)
tff(fact_3794_minf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X2),Z2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S)) ) ) ).

% minf(9)
tff(fact_3795_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S)) ) ) ).

% minf(9)
tff(fact_3796_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S)) ) ) ).

% minf(9)
tff(fact_3797_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S)) ) ) ).

% minf(9)
tff(fact_3798_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S)) ) ) ).

% minf(9)
tff(fact_3799_pinf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z2),X2)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S)) ) ) ).

% pinf(10)
tff(fact_3800_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X2)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S)) ) ) ).

% pinf(10)
tff(fact_3801_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X2)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S)) ) ) ).

% pinf(10)
tff(fact_3802_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X2)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S)) ) ) ).

% pinf(10)
tff(fact_3803_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X2)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S)) ) ) ).

% pinf(10)
tff(fact_3804_pinf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z2),X2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S)) ) ) ).

% pinf(9)
tff(fact_3805_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S)) ) ) ).

% pinf(9)
tff(fact_3806_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S)) ) ) ).

% pinf(9)
tff(fact_3807_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S)) ) ) ).

% pinf(9)
tff(fact_3808_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S)) ) ) ).

% pinf(9)
tff(fact_3809_dvd__div__eq__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),A)
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = zero_zero_complex )
      <=> ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3810_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3811_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( ( aa_int_int(divide_divide_int(A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3812_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3813_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3814_dvd__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3815_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3816_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3817_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,B),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3818_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3819_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3820_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3821_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3822_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3823_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3824_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3825_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3826_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3827_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_3828_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_3829_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_3830_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_3831_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_3832_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_3833_is__unit__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer) ) ) ).

% is_unit_mult_iff
tff(fact_3834_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat) ) ) ).

% is_unit_mult_iff
tff(fact_3835_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int) ) ) ).

% is_unit_mult_iff
tff(fact_3836_mod__Suc,axiom,
    ! [Mb: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Na) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Na)) = Na,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Na))) ).

% mod_Suc
tff(fact_3837_finite__set__decode,axiom,
    ! [Na: nat] : aa_set_nat_o(finite_finite_nat,nat_set_decode(Na)) ).

% finite_set_decode
tff(fact_3838_mod__induct,axiom,
    ! [P: fun_nat_o,Na: nat,P2: nat,Mb: nat] :
      ( aa_nat_o(P,Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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_3839_dvd__div__mult,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(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_3840_dvd__div__mult,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(times_times_int,aa_int_int(divide_divide_int(B),C)),A) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_3841_dvd__div__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,C)),A) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),C) ) ) ).

% dvd_div_mult
tff(fact_3842_div__mult__swap,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(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_3843_div__mult__swap,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(times_times_int,A),aa_int_int(divide_divide_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_3844_div__mult__swap,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ) ) ).

% div_mult_swap
tff(fact_3845_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_3846_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(divide_divide_int(A),aa_int_int(divide_divide_int(B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_3847_div__div__eq__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( divide6298287555418463151nteger(A,divide6298287555418463151nteger(B,C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_3848_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_3849_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)
     => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_3850_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A)
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_3851_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_3852_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(divide_divide_int(B),C)) ) ).

% dvd_mult_imp_div
tff(fact_3853_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C)) ) ).

% dvd_mult_imp_div
tff(fact_3854_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),C)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_3855_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),C)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(C),D)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_3856_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),C)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),divide6298287555418463151nteger(C,D)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_3857_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Mb: nat,Na: nat] :
      ( ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(P,M),zero_zero_nat)
     => ( ! [M: nat,N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( aa_nat_o(aa_nat_fun_nat_o(P,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))
             => aa_nat_o(aa_nat_fun_nat_o(P,M),N) ) )
       => aa_nat_o(aa_nat_fun_nat_o(P,Mb),Na) ) ) ).

% gcd_nat_induct
tff(fact_3858_mod__less__divisor,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Na)),Na) ) ).

% mod_less_divisor
tff(fact_3859_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_3860_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(divide_divide_int(C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_3861_dvd__div__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(C,B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_3862_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_3863_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_3864_div__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_3865_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_3866_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(divide_divide_int(B),A) = aa_int_int(divide_divide_int(C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_3867_unit__div__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(C,A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_3868_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_3869_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(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3870_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_3871_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_3872_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(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3873_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_3874_mod__Suc__le__divisor,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,Na))),Na) ).

% mod_Suc_le_divisor
tff(fact_3875_mod__eq__0D,axiom,
    ! [Mb: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),D) = zero_zero_nat )
     => ? [Q2: nat] : Mb = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q2) ) ).

% mod_eq_0D
tff(fact_3876_le__imp__power__dvd,axiom,
    ! [Mb: nat,Na: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => 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),Na)) ) ).

% le_imp_power_dvd
tff(fact_3877_le__imp__power__dvd,axiom,
    ! [Mb: nat,Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => 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),Na)) ) ).

% le_imp_power_dvd
tff(fact_3878_le__imp__power__dvd,axiom,
    ! [Mb: nat,Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ).

% le_imp_power_dvd
tff(fact_3879_le__imp__power__dvd,axiom,
    ! [Mb: nat,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => 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),Na)) ) ).

% le_imp_power_dvd
tff(fact_3880_le__imp__power__dvd,axiom,
    ! [Mb: nat,Na: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Na)) ) ).

% le_imp_power_dvd
tff(fact_3881_power__le__dvd,axiom,
    ! [A: code_integer,Na: nat,B: code_integer,Mb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
       => 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_3882_power__le__dvd,axiom,
    ! [A: nat,Na: 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),Na)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
       => 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_3883_power__le__dvd,axiom,
    ! [A: real,Na: nat,B: real,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3884_power__le__dvd,axiom,
    ! [A: int,Na: nat,B: int,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Na)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
       => 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_3885_power__le__dvd,axiom,
    ! [A: complex,Na: nat,B: complex,Mb: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Na)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3886_dvd__power__le,axiom,
    ! [X: code_integer,Y: code_integer,Na: nat,Mb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_3887_dvd__power__le,axiom,
    ! [X: nat,Y: nat,Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_3888_dvd__power__le,axiom,
    ! [X: real,Y: real,Na: nat,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_3889_dvd__power__le,axiom,
    ! [X: int,Y: int,Na: nat,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(X),Na)),aa_nat_int(power_power_int(Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_3890_dvd__power__le,axiom,
    ! [X: complex,Y: complex,Na: nat,Mb: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(X),Na)),aa_nat_complex(power_power_complex(Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_3891_nat__mod__eq__iff,axiom,
    ! [X: nat,Na: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Na) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_3892_nat__dvd__not__less,axiom,
    ! [Mb: nat,Na: 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),Na)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),Mb) ) ) ).

% nat_dvd_not_less
tff(fact_3893_dvd__pos__nat,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb) ) ) ).

% dvd_pos_nat
tff(fact_3894_zdvd__imp__le,axiom,
    ! [Z: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Na) ) ) ).

% zdvd_imp_le
tff(fact_3895_zdvd__not__zless,axiom,
    ! [Mb: int,Na: 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_int,Mb),Na)
       => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Na),Mb) ) ) ).

% zdvd_not_zless
tff(fact_3896_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D4: nat,X3: 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),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),D4) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D4) ) ) ) ).

% bezout_add_nat
tff(fact_3897_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X: 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),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)),D) ) )
         => ? [X3: nat,Y3: nat] :
              ( 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),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Y3)),D) )
                | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_3898_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_3899_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(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(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_3900_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
         => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_3901_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_3902_mod2__eq__if,axiom,
    ! [A: int] :
      aa_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_3903_mod2__eq__if,axiom,
    ! [A: code_integer] :
      modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,one_one_Code_integer) ).

% mod2_eq_if
tff(fact_3904_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_3905_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_3906_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(modulo_modulo_int(A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_3907_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_3908_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_3909_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(modulo_modulo_int(A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_3910_cong__exp__iff__simps_I2_J,axiom,
    ! [Na: num,Q4: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,Q4)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_3911_cong__exp__iff__simps_I2_J,axiom,
    ! [Na: num,Q4: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) = zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,Q4)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_3912_cong__exp__iff__simps_I2_J,axiom,
    ! [Na: num,Q4: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Na)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) = zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Na),numera6620942414471956472nteger(Q4)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_3913_cong__exp__iff__simps_I1_J,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_3914_cong__exp__iff__simps_I1_J,axiom,
    ! [Na: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_3915_cong__exp__iff__simps_I1_J,axiom,
    ! [Na: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(Na),numera6620942414471956472nteger(one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_3916_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_3917_mult__div__mod__eq,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(divide_divide_int(A),B))),aa_int_int(modulo_modulo_int(A),B)) = A ).

% mult_div_mod_eq
tff(fact_3918_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_3919_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_3920_mod__mult__div__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(divide_divide_int(A),B))) = A ).

% mod_mult_div_eq
tff(fact_3921_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_3922_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_3923_mod__div__mult__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_3924_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_3925_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_3926_div__mult__mod__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),B)),aa_int_int(modulo_modulo_int(A),B)) = A ).

% div_mult_mod_eq
tff(fact_3927_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_3928_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_3929_mod__div__decomp,axiom,
    ! [A: int,B: int] : A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),B)),aa_int_int(modulo_modulo_int(A),B)) ).

% mod_div_decomp
tff(fact_3930_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_3931_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_3932_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),B)),aa_int_int(modulo_modulo_int(A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_3933_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_3934_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_3935_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(divide_divide_int(A),B))),aa_int_int(modulo_modulo_int(A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_3936_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_3937_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_3938_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B),C))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(modulo_modulo_int(B),C))),C)) ).

% div_mult1_eq
tff(fact_3939_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_3940_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C2: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2) ) ) ).

% unit_dvdE
tff(fact_3941_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) ) ) ).

% unit_dvdE
tff(fact_3942_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [C2: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) ) ) ).

% unit_dvdE
tff(fact_3943_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_3944_unity__coeff__ex,axiom,
    ! [P: fun_complex_o,L: complex] :
      ( ? [X4: complex] : aa_complex_o(P,aa_complex_complex(times_times_complex(L),X4))
    <=> ? [X4: complex] :
          ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,L),aa_complex_complex(plus_plus_complex(X4),zero_zero_complex))
          & aa_complex_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_3945_unity__coeff__ex,axiom,
    ! [P: fun_real_o,L: real] :
      ( ? [X4: real] : aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X4))
    <=> ? [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),zero_zero_real))
          & aa_real_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_3946_unity__coeff__ex,axiom,
    ! [P: fun_rat_o,L: rat] :
      ( ? [X4: rat] : aa_rat_o(P,aa_rat_rat(times_times_rat(L),X4))
    <=> ? [X4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),aa_rat_rat(plus_plus_rat(X4),zero_zero_rat))
          & aa_rat_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_3947_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_3948_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_3949_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3950_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
       => ( ( aa_int_int(divide_divide_int(B),A) = C )
        <=> ( B = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3951_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
       => ( ( divide6298287555418463151nteger(B,A) = C )
        <=> ( B = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3952_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3953_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),C)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3954_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3955_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3956_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(divide_divide_int(B),C))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3957_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3958_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_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,C),D)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D),C) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3959_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_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,C),D)
           => ( ( aa_int_int(divide_divide_int(B),A) = aa_int_int(divide_divide_int(D),C) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3960_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
           => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(D,C) )
            <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3961_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3962_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(divide_divide_int(A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3963_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3964_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_3965_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_3966_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_3967_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_3968_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_3969_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ) ) ).

% unit_div_mult_swap
tff(fact_3970_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_3971_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),C) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_3972_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),C) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),B) ) ) ).

% unit_div_commute
tff(fact_3973_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_3974_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_3975_div__mult__unit2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ) ).

% div_mult_unit2
tff(fact_3976_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_3977_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( A = aa_int_int(divide_divide_int(C),B) )
      <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_3978_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( A = divide6298287555418463151nteger(C,B) )
      <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_3979_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = C )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_3980_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(divide_divide_int(A),B) = C )
      <=> ( A = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_3981_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(A,B) = C )
      <=> ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_3982_mod__le__divisor,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Na)),Na) ) ).

% mod_le_divisor
tff(fact_3983_is__unit__power__iff,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        | ( Na = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3984_is__unit__power__iff,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        | ( Na = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3985_is__unit__power__iff,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Na)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        | ( Na = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3986_div__less__mono,axiom,
    ! [A2: nat,B2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Na) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),Na) = zero_zero_nat )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),Na)) ) ) ) ) ).

% div_less_mono
tff(fact_3987_mod__eq__nat1E,axiom,
    ! [Mb: nat,Q4: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),Q4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
       => ~ ! [S2: nat] : Mb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q4),S2)) ) ) ).

% mod_eq_nat1E
tff(fact_3988_mod__eq__nat2E,axiom,
    ! [Mb: nat,Q4: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),Q4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
       => ~ ! [S2: nat] : Na != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q4),S2)) ) ) ).

% mod_eq_nat2E
tff(fact_3989_nat__mod__eq__lemma,axiom,
    ! [X: nat,Na: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
       => ? [Q2: nat] : X = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q2)) ) ) ).

% nat_mod_eq_lemma
tff(fact_3990_dvd__imp__le,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ) ).

% dvd_imp_le
tff(fact_3991_div__mod__decomp,axiom,
    ! [A2: nat,Na: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Na)),Na)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Na)) ).

% div_mod_decomp
tff(fact_3992_mod__mult2__eq,axiom,
    ! [Mb: nat,Na: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na)),Q4))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Na)) ).

% mod_mult2_eq
tff(fact_3993_dvd__mult__cancel,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Na) ) ) ).

% dvd_mult_cancel
tff(fact_3994_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Na) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_3995_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D4: nat,X3: 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),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),D4) ) ) ) ).

% bezout_add_strong_nat
tff(fact_3996_even__zero,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_z3403309356797280102nteger) ).

% even_zero
tff(fact_3997_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_3998_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_3999_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [B3: nat] :
              ( ( B3 != zero_zero_nat )
             => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B3),one_one_nat)
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B3 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B3) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B3) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4000_is__unitE,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [B3: int] :
              ( ( B3 != zero_zero_int )
             => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B3),one_one_int)
               => ( ( aa_int_int(divide_divide_int(one_one_int),A) = B3 )
                 => ( ( aa_int_int(divide_divide_int(one_one_int),B3) = A )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B3) = one_one_int )
                     => ( aa_int_int(divide_divide_int(C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4001_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B3: code_integer] :
              ( ( B3 != zero_z3403309356797280102nteger )
             => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B3),one_one_Code_integer)
               => ( ( divide6298287555418463151nteger(one_one_Code_integer,A) = B3 )
                 => ( ( divide6298287555418463151nteger(one_one_Code_integer,B3) = A )
                   => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B3) = one_one_Code_integer )
                     => ( divide6298287555418463151nteger(C,A) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4002_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4003_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(divide_divide_int(one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4004_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4005_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4006_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) = aa_int_int(divide_divide_int(one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4007_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4008_odd__even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)) ) ) ).

% odd_even_add
tff(fact_4009_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_4010_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_4011_dvd__power__iff,axiom,
    ! [X: code_integer,Mb: nat,Na: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(X),Na))
      <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),one_one_Code_integer)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ) ) ).

% dvd_power_iff
tff(fact_4012_dvd__power__iff,axiom,
    ! [X: nat,Mb: nat,Na: nat] :
      ( ( X != 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,X),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Na))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ) ) ).

% dvd_power_iff
tff(fact_4013_dvd__power__iff,axiom,
    ! [X: int,Mb: nat,Na: nat] :
      ( ( X != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(X),Mb)),aa_nat_int(power_power_int(X),Na))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ) ) ).

% dvd_power_iff
tff(fact_4014_dvd__power,axiom,
    ! [Na: nat,X: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( X = one_one_Code_integer ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),aa_nat_Code_integer(power_8256067586552552935nteger(X),Na)) ) ).

% dvd_power
tff(fact_4015_dvd__power,axiom,
    ! [Na: nat,X: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( X = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),aa_nat_rat(power_power_rat(X),Na)) ) ).

% dvd_power
tff(fact_4016_dvd__power,axiom,
    ! [Na: nat,X: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( X = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Na)) ) ).

% dvd_power
tff(fact_4017_dvd__power,axiom,
    ! [Na: nat,X: real] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( X = one_one_real ) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na)) ) ).

% dvd_power
tff(fact_4018_dvd__power,axiom,
    ! [Na: nat,X: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( X = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),aa_nat_int(power_power_int(X),Na)) ) ).

% dvd_power
tff(fact_4019_dvd__power,axiom,
    ! [Na: nat,X: complex] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( X = one_one_complex ) )
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,X),aa_nat_complex(power_power_complex(X),Na)) ) ).

% dvd_power
tff(fact_4020_split__mod,axiom,
    ! [P: fun_nat_o,Mb: nat,Na: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Na))
    <=> ( ( ( Na = zero_zero_nat )
         => aa_nat_o(P,Mb) )
        & ( ( Na != zero_zero_nat )
         => ! [I2: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Na)
             => ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),I2)),J3) )
               => aa_nat_o(P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_4021_dvd__mult__cancel1,axiom,
    ! [Mb: nat,Na: 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),Na)),Mb)
      <=> ( Na = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_4022_dvd__mult__cancel2,axiom,
    ! [Mb: nat,Na: 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,Na),Mb)),Mb)
      <=> ( Na = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_4023_power__dvd__imp__le,axiom,
    ! [I: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ) ).

% power_dvd_imp_le
tff(fact_4024_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_4025_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_4026_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(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4027_product__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys)))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys),Na) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys)))) ) ) ).

% product_nth
tff(fact_4028_product__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_list_o_nat(size_size_list_o,Ys)))
     => ( nth_Pr4606735188037164562VEBT_o(product_VEBT_VEBT_o(Xs,Ys),Na) = produc8721562602347293563VEBT_o(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_list_o_nat(size_size_list_o,Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_list_o_nat(size_size_list_o,Ys)))) ) ) ).

% product_nth
tff(fact_4029_product__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_list_nat_nat(size_size_list_nat,Ys)))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys),Na) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_list_nat_nat(size_size_list_nat,Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_list_nat_nat(size_size_list_nat,Ys)))) ) ) ).

% product_nth
tff(fact_4030_product__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs)),aa_list_int_nat(size_size_list_int,Ys)))
     => ( nth_Pr6837108013167703752BT_int(produc7292646706713671643BT_int(Xs,Ys),Na) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_list_int_nat(size_size_list_int,Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_list_int_nat(size_size_list_int,Ys)))) ) ) ).

% product_nth
tff(fact_4031_product__nth,axiom,
    ! [Na: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_o_nat(size_size_list_o,Xs)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys)))
     => ( nth_Pr6777367263587873994T_VEBT(product_o_VEBT_VEBT(Xs,Ys),Na) = produc2982872950893828659T_VEBT(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys)))) ) ) ).

% product_nth
tff(fact_4032_product__nth,axiom,
    ! [Na: nat,Xs: list_o,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_o_nat(size_size_list_o,Xs)),aa_list_o_nat(size_size_list_o,Ys)))
     => ( nth_Product_prod_o_o(product_o_o(Xs,Ys),Na) = product_Pair_o_o(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_list_o_nat(size_size_list_o,Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_list_o_nat(size_size_list_o,Ys)))) ) ) ).

% product_nth
tff(fact_4033_product__nth,axiom,
    ! [Na: nat,Xs: list_o,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_o_nat(size_size_list_o,Xs)),aa_list_nat_nat(size_size_list_nat,Ys)))
     => ( nth_Pr5826913651314560976_o_nat(product_o_nat(Xs,Ys),Na) = product_Pair_o_nat(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_list_nat_nat(size_size_list_nat,Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_list_nat_nat(size_size_list_nat,Ys)))) ) ) ).

% product_nth
tff(fact_4034_product__nth,axiom,
    ! [Na: nat,Xs: list_o,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_o_nat(size_size_list_o,Xs)),aa_list_int_nat(size_size_list_int,Ys)))
     => ( nth_Pr1649062631805364268_o_int(product_o_int(Xs,Ys),Na) = product_Pair_o_int(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_list_int_nat(size_size_list_int,Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_list_int_nat(size_size_list_int,Ys)))) ) ) ).

% product_nth
tff(fact_4035_product__nth,axiom,
    ! [Na: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys)))
     => ( nth_Pr744662078594809490T_VEBT(produc7156399406898700509T_VEBT(Xs,Ys),Na) = produc599794634098209291T_VEBT(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Ys)))) ) ) ).

% product_nth
tff(fact_4036_product__nth,axiom,
    ! [Na: nat,Xs: list_nat,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_list_o_nat(size_size_list_o,Ys)))
     => ( nth_Pr112076138515278198_nat_o(product_nat_o(Xs,Ys),Na) = product_Pair_nat_o(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_list_o_nat(size_size_list_o,Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_list_o_nat(size_size_list_o,Ys)))) ) ) ).

% product_nth
tff(fact_4037_power__mono__odd,axiom,
    ! [Na: 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))),Na)
     => ( 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(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na)) ) ) ).

% power_mono_odd
tff(fact_4038_power__mono__odd,axiom,
    ! [Na: 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))),Na)
     => ( 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),Na)),aa_nat_rat(power_power_rat(B),Na)) ) ) ).

% power_mono_odd
tff(fact_4039_power__mono__odd,axiom,
    ! [Na: 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))),Na)
     => ( 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),Na)),aa_nat_int(power_power_int(B),Na)) ) ) ).

% power_mono_odd
tff(fact_4040_Suc__times__mod__eq,axiom,
    ! [Mb: nat,Na: 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),Na))),Mb) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_4041_odd__pos,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% odd_pos
tff(fact_4042_dvd__power__iff__le,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ) ).

% dvd_power_iff_le
tff(fact_4043_even__unset__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(Mb,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_4044_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_4045_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_4046_even__set__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(Mb,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_4047_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_4048_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_4049_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_4050_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_4051_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(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B)
       => ( aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_4052_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,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),B)
       => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) = modulo364778990260209775nteger(A,B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_4053_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_4054_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_4055_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_4056_div__exp__mod__exp__eq,axiom,
    ! [A: nat,Na: 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))),Na))),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,Na),Mb)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% div_exp_mod_exp_eq
tff(fact_4057_div__exp__mod__exp__eq,axiom,
    ! [A: int,Na: nat,Mb: nat] : aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Mb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% div_exp_mod_exp_eq
tff(fact_4058_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,Na: nat,Mb: nat] : modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb)) = divide6298287555418463151nteger(modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Mb))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) ).

% div_exp_mod_exp_eq
tff(fact_4059_oddE,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ~ ! [B3: code_integer] : A != plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B3),one_one_Code_integer) ) ).

% oddE
tff(fact_4060_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_4061_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_4062_zero__le__even__power,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ).

% zero_le_even_power
tff(fact_4063_zero__le__even__power,axiom,
    ! [Na: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na)) ) ).

% zero_le_even_power
tff(fact_4064_zero__le__even__power,axiom,
    ! [Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Na)) ) ).

% zero_le_even_power
tff(fact_4065_zero__le__odd__power,axiom,
    ! [Na: nat,A: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ).

% zero_le_odd_power
tff(fact_4066_zero__le__odd__power,axiom,
    ! [Na: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ).

% zero_le_odd_power
tff(fact_4067_zero__le__odd__power,axiom,
    ! [Na: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Na))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_odd_power
tff(fact_4068_zero__le__power__eq,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq
tff(fact_4069_zero__le__power__eq,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq
tff(fact_4070_zero__le__power__eq,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq
tff(fact_4071_verit__le__mono__div,axiom,
    ! [A2: nat,B2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Na)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),Na) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),Na)) ) ) ).

% verit_le_mono_div
tff(fact_4072_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A2))
      <=> ~ aa_set_nat_o(member_nat(zero_zero_nat),A2) ) ) ).

% even_set_encode_iff
tff(fact_4073_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_4074_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(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))) = aa_int_int(divide_divide_int(A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_4075_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,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),B)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))) = divide6298287555418463151nteger(A,B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_4076_zero__less__power__eq,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))
    <=> ( ( Na = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq
tff(fact_4077_zero__less__power__eq,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na))
    <=> ( ( Na = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq
tff(fact_4078_zero__less__power__eq,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Na))
    <=> ( ( Na = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq
tff(fact_4079_mod__double__modulus,axiom,
    ! [Mb: code_integer,X: 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),X)
       => ( ( modulo364778990260209775nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb)) = modulo364778990260209775nteger(X,Mb) )
          | ( modulo364778990260209775nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb)) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(X,Mb),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_4080_mod__double__modulus,axiom,
    ! [Mb: nat,X: 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),X)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Mb) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Mb)),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_4081_mod__double__modulus,axiom,
    ! [Mb: int,X: 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),X)
       => ( ( aa_int_int(modulo_modulo_int(X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(modulo_modulo_int(X),Mb) )
          | ( aa_int_int(modulo_modulo_int(X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(X),Mb)),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_4082_unset__bit__Suc,axiom,
    ! [Na: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,Na)),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,Na),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_4083_unset__bit__Suc,axiom,
    ! [Na: nat,A: code_integer] : bit_se8260200283734997820nteger(aa_nat_nat(suc,Na),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(Na,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_4084_unset__bit__Suc,axiom,
    ! [Na: nat,A: int] : aa_int_int(bit_se4203085406695923979it_int(aa_nat_nat(suc,Na)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se4203085406695923979it_int(Na),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_4085_set__bit__Suc,axiom,
    ! [Na: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,Na)),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,Na),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_4086_set__bit__Suc,axiom,
    ! [Na: nat,A: code_integer] : bit_se2793503036327961859nteger(aa_nat_nat(suc,Na),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(Na,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_4087_set__bit__Suc,axiom,
    ! [Na: nat,A: int] : aa_int_int(bit_se7879613467334960850it_int(aa_nat_nat(suc,Na)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se7879613467334960850it_int(Na),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_4088_power__le__zero__eq,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_4089_power__le__zero__eq,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_4090_power__le__zero__eq,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_4091_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,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)))
         => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),one_one_Code_integer) = divide6298287555418463151nteger(A,B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_4092_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_4093_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(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))
         => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))),one_one_int) = aa_int_int(divide_divide_int(A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_4094_flip__bit__Suc,axiom,
    ! [Na: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,Na)),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,Na),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_4095_flip__bit__Suc,axiom,
    ! [Na: nat,A: code_integer] : bit_se1345352211410354436nteger(aa_nat_nat(suc,Na),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(Na,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_4096_flip__bit__Suc,axiom,
    ! [Na: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,Na)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(Na),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_4097_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: 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,Q4),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,Q4),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_4098_signed__take__bit__Suc,axiom,
    ! [Na: nat,A: code_integer] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,Na),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(Na,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_4099_signed__take__bit__Suc,axiom,
    ! [Na: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Na)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(Na),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_4100_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4101_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,Mb: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(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))),Na)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb)
        | ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4102_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,Mb: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb)
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4103_even__flip__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(Mb,A))
    <=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_4104_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_4105_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_4106_one__mod__2__pow__eq,axiom,
    ! [Na: nat] : modulo364778990260209775nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% one_mod_2_pow_eq
tff(fact_4107_one__mod__2__pow__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% one_mod_2_pow_eq
tff(fact_4108_one__mod__2__pow__eq,axiom,
    ! [Na: nat] : aa_int_int(modulo_modulo_int(one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% one_mod_2_pow_eq
tff(fact_4109_set__encode__insert,axiom,
    ! [A2: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(Na),A2)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(Na,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),aa_set_nat_nat(nat_set_encode,A2)) ) ) ) ).

% set_encode_insert
tff(fact_4110_insert__absorb2,axiom,
    ! [X: nat,A2: set_nat] : insert_nat(X,insert_nat(X,A2)) = insert_nat(X,A2) ).

% insert_absorb2
tff(fact_4111_insert__absorb2,axiom,
    ! [X: int,A2: set_int] : insert_int(X,insert_int(X,A2)) = insert_int(X,A2) ).

% insert_absorb2
tff(fact_4112_insert__absorb2,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(X,insert8211810215607154385at_nat(X,A2)) = insert8211810215607154385at_nat(X,A2) ).

% insert_absorb2
tff(fact_4113_insert__absorb2,axiom,
    ! [X: real,A2: set_real] : insert_real(X,insert_real(X,A2)) = insert_real(X,A2) ).

% insert_absorb2
tff(fact_4114_insert__absorb2,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(X,insert9069300056098147895at_nat(X,A2)) = insert9069300056098147895at_nat(X,A2) ).

% insert_absorb2
tff(fact_4115_insert__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(B,A2))
    <=> ( ( A = B )
        | aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A2) ) ) ).

% insert_iff
tff(fact_4116_insert__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(B,A2))
    <=> ( ( A = B )
        | aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A2) ) ) ).

% insert_iff
tff(fact_4117_insert__iff,axiom,
    ! [A: real,B: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),insert_real(B,A2))
    <=> ( ( A = B )
        | aa_set_real_o(member_real(A),A2) ) ) ).

% insert_iff
tff(fact_4118_insert__iff,axiom,
    ! [A: set_nat,B: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),insert_set_nat(B,A2))
    <=> ( ( A = B )
        | aa_set_set_nat_o(member_set_nat(A),A2) ) ) ).

% insert_iff
tff(fact_4119_insert__iff,axiom,
    ! [A: nat,B: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),insert_nat(B,A2))
    <=> ( ( A = B )
        | aa_set_nat_o(member_nat(A),A2) ) ) ).

% insert_iff
tff(fact_4120_insert__iff,axiom,
    ! [A: int,B: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),insert_int(B,A2))
    <=> ( ( A = B )
        | aa_set_int_o(member_int(A),A2) ) ) ).

% insert_iff
tff(fact_4121_insertCI,axiom,
    ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),B2)
       => ( A = B ) )
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(B,B2)) ) ).

% insertCI
tff(fact_4122_insertCI,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),B2)
       => ( A = B ) )
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(B,B2)) ) ).

% insertCI
tff(fact_4123_insertCI,axiom,
    ! [A: real,B2: set_real,B: real] :
      ( ( ~ aa_set_real_o(member_real(A),B2)
       => ( A = B ) )
     => aa_set_real_o(member_real(A),insert_real(B,B2)) ) ).

% insertCI
tff(fact_4124_insertCI,axiom,
    ! [A: set_nat,B2: set_set_nat,B: set_nat] :
      ( ( ~ aa_set_set_nat_o(member_set_nat(A),B2)
       => ( A = B ) )
     => aa_set_set_nat_o(member_set_nat(A),insert_set_nat(B,B2)) ) ).

% insertCI
tff(fact_4125_insertCI,axiom,
    ! [A: nat,B2: set_nat,B: nat] :
      ( ( ~ aa_set_nat_o(member_nat(A),B2)
       => ( A = B ) )
     => aa_set_nat_o(member_nat(A),insert_nat(B,B2)) ) ).

% insertCI
tff(fact_4126_insertCI,axiom,
    ! [A: int,B2: set_int,B: int] :
      ( ( ~ aa_set_int_o(member_int(A),B2)
       => ( A = B ) )
     => aa_set_int_o(member_int(A),insert_int(B,B2)) ) ).

% insertCI
tff(fact_4127_diff__self,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% diff_self
tff(fact_4128_diff__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% diff_self
tff(fact_4129_diff__self,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% diff_self
tff(fact_4130_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_4131_diff__0__right,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_0_right
tff(fact_4132_diff__0__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_0_right
tff(fact_4133_diff__0__right,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_0_right
tff(fact_4134_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_4135_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_4136_diff__zero,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_zero
tff(fact_4137_diff__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_zero
tff(fact_4138_diff__zero,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_zero
tff(fact_4139_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_4140_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_4141_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_4142_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_4143_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_4144_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_4145_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_4146_add__diff__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).

% add_diff_cancel
tff(fact_4147_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),B) = A ).

% add_diff_cancel
tff(fact_4148_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_4149_diff__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ).

% diff_add_cancel
tff(fact_4150_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ).

% diff_add_cancel
tff(fact_4151_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_4152_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).

% add_diff_cancel_left
tff(fact_4153_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) = aa_rat_rat(minus_minus_rat(A),B) ).

% add_diff_cancel_left
tff(fact_4154_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_4155_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_4156_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_4157_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_4158_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_4159_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_4160_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).

% add_diff_cancel_right
tff(fact_4161_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(minus_minus_rat(A),B) ).

% add_diff_cancel_right
tff(fact_4162_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_4163_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_4164_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_4165_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_4166_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_4167_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_4168_singletonI,axiom,
    ! [A: produc3843707927480180839at_nat] : aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)) ).

% singletonI
tff(fact_4169_singletonI,axiom,
    ! [A: set_nat] : aa_set_set_nat_o(member_set_nat(A),insert_set_nat(A,bot_bot_set_set_nat)) ).

% singletonI
tff(fact_4170_singletonI,axiom,
    ! [A: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)) ).

% singletonI
tff(fact_4171_singletonI,axiom,
    ! [A: real] : aa_set_real_o(member_real(A),insert_real(A,bot_bot_set_real)) ).

% singletonI
tff(fact_4172_singletonI,axiom,
    ! [A: nat] : aa_set_nat_o(member_nat(A),insert_nat(A,bot_bot_set_nat)) ).

% singletonI
tff(fact_4173_singletonI,axiom,
    ! [A: int] : aa_set_int_o(member_int(A),insert_int(A,bot_bot_set_int)) ).

% singletonI
tff(fact_4174_finite__insert,axiom,
    ! [A: real,A2: set_real] :
      ( aa_set_real_o(finite_finite_real,insert_real(A,A2))
    <=> aa_set_real_o(finite_finite_real,A2) ) ).

% finite_insert
tff(fact_4175_finite__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,insert9069300056098147895at_nat(A,A2))
    <=> aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A2) ) ).

% finite_insert
tff(fact_4176_finite__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,insert_nat(A,A2))
    <=> aa_set_nat_o(finite_finite_nat,A2) ) ).

% finite_insert
tff(fact_4177_finite__insert,axiom,
    ! [A: int,A2: set_int] :
      ( aa_set_int_o(finite_finite_int,insert_int(A,A2))
    <=> aa_set_int_o(finite_finite_int,A2) ) ).

% finite_insert
tff(fact_4178_finite__insert,axiom,
    ! [A: complex,A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,insert_complex(A,A2))
    <=> aa_set_complex_o(finite3207457112153483333omplex,A2) ) ).

% finite_insert
tff(fact_4179_finite__insert,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,insert8211810215607154385at_nat(A,A2))
    <=> aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2) ) ).

% finite_insert
tff(fact_4180_insert__subset,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(insert9069300056098147895at_nat(X,A2)),B2)
    <=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),B2)
        & aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),B2) ) ) ).

% insert_subset
tff(fact_4181_insert__subset,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(insert8211810215607154385at_nat(X,A2)),B2)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),B2)
        & aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),B2) ) ) ).

% insert_subset
tff(fact_4182_insert__subset,axiom,
    ! [X: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(insert_real(X,A2)),B2)
    <=> ( aa_set_real_o(member_real(X),B2)
        & aa_set_real_o(ord_less_eq_set_real(A2),B2) ) ) ).

% insert_subset
tff(fact_4183_insert__subset,axiom,
    ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(insert_set_nat(X,A2)),B2)
    <=> ( aa_set_set_nat_o(member_set_nat(X),B2)
        & aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2) ) ) ).

% insert_subset
tff(fact_4184_insert__subset,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,insert_nat(X,A2)),B2)
    <=> ( aa_set_nat_o(member_nat(X),B2)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2) ) ) ).

% insert_subset
tff(fact_4185_insert__subset,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(insert_int(X,A2)),B2)
    <=> ( aa_set_int_o(member_int(X),B2)
        & aa_set_int_o(ord_less_eq_set_int(A2),B2) ) ) ).

% insert_subset
tff(fact_4186_diff__Suc__Suc,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na) ).

% diff_Suc_Suc
tff(fact_4187_Suc__diff__diff,axiom,
    ! [Mb: nat,Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Na)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)),K) ).

% Suc_diff_diff
tff(fact_4188_diff__0__eq__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),Na) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_4189_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_4190_diff__diff__cancel,axiom,
    ! [I: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_4191_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ).

% diff_diff_left
tff(fact_4192_flip__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2159334234014336723it_int(Na),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% flip_bit_negative_int_iff
tff(fact_4193_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_4194_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_4195_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_4196_of__bool__eq_I1_J,axiom,
    zero_n1201886186963655149omplex($false) = zero_zero_complex ).

% of_bool_eq(1)
tff(fact_4197_of__bool__eq_I1_J,axiom,
    zero_n3304061248610475627l_real($false) = zero_zero_real ).

% of_bool_eq(1)
tff(fact_4198_of__bool__eq_I1_J,axiom,
    zero_n2052037380579107095ol_rat($false) = zero_zero_rat ).

% of_bool_eq(1)
tff(fact_4199_of__bool__eq_I1_J,axiom,
    aa_o_nat(zero_n2687167440665602831ol_nat,$false) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_4200_of__bool__eq_I1_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$false) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_4201_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n1201886186963655149omplex((P)) = zero_zero_complex )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_4202_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real((P)) = zero_zero_real )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_4203_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat((P)) = zero_zero_rat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_4204_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(P)) = zero_zero_nat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_4205_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P)) = zero_zero_int )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_4206_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_n3304061248610475627l_real((P))),zero_n3304061248610475627l_real((Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_4207_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_4208_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_4209_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_4210_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n1201886186963655149omplex((P)) = one_one_complex )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4211_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real((P)) = one_one_real )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4212_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat((P)) = one_one_rat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4213_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(P)) = one_one_nat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4214_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P)) = one_one_int )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4215_of__bool__eq_I2_J,axiom,
    zero_n1201886186963655149omplex($true) = one_one_complex ).

% of_bool_eq(2)
tff(fact_4216_of__bool__eq_I2_J,axiom,
    zero_n3304061248610475627l_real($true) = one_one_real ).

% of_bool_eq(2)
tff(fact_4217_of__bool__eq_I2_J,axiom,
    zero_n2052037380579107095ol_rat($true) = one_one_rat ).

% of_bool_eq(2)
tff(fact_4218_of__bool__eq_I2_J,axiom,
    aa_o_nat(zero_n2687167440665602831ol_nat,$true) = one_one_nat ).

% of_bool_eq(2)
tff(fact_4219_of__bool__eq_I2_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$true) = one_one_int ).

% of_bool_eq(2)
tff(fact_4220_signed__take__bit__of__0,axiom,
    ! [Na: nat] : aa_int_int(bit_ri631733984087533419it_int(Na),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_4221_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(aa_rea7340605660300375871l_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_4222_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(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_4223_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_4224_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(aa_rea7340605660300375871l_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_4225_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(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_4226_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_4227_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(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_4228_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(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_4229_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_4230_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_4231_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(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_4232_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(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_4233_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_4234_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_4235_diff__numeral__special_I9_J,axiom,
    aa_complex_complex(minus_minus_complex(one_one_complex),one_one_complex) = zero_zero_complex ).

% diff_numeral_special(9)
tff(fact_4236_diff__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),one_one_real) = zero_zero_real ).

% diff_numeral_special(9)
tff(fact_4237_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_4238_diff__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),one_one_int) = zero_zero_int ).

% diff_numeral_special(9)
tff(fact_4239_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_4240_div__diff,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(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ) ).

% div_diff
tff(fact_4241_div__diff,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(minus_8373710615458151222nteger(A,B),C) = minus_8373710615458151222nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ) ).

% div_diff
tff(fact_4242_singleton__insert__inj__eq_H,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( insert9069300056098147895at_nat(A,A2) = insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat) )
    <=> ( ( A = B )
        & aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4243_singleton__insert__inj__eq_H,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A,A2) = insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat) )
    <=> ( ( A = B )
        & aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4244_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A2: set_real,B: real] :
      ( ( insert_real(A,A2) = insert_real(B,bot_bot_set_real) )
    <=> ( ( A = B )
        & aa_set_real_o(ord_less_eq_set_real(A2),insert_real(B,bot_bot_set_real)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4245_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B: nat] :
      ( ( insert_nat(A,A2) = insert_nat(B,bot_bot_set_nat) )
    <=> ( ( A = B )
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat(B,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4246_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A2: set_int,B: int] :
      ( ( insert_int(A,A2) = insert_int(B,bot_bot_set_int) )
    <=> ( ( A = B )
        & aa_set_int_o(ord_less_eq_set_int(A2),insert_int(B,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4247_singleton__insert__inj__eq,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat) = insert9069300056098147895at_nat(A,A2) )
    <=> ( ( A = B )
        & aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_4248_singleton__insert__inj__eq,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat) = insert8211810215607154385at_nat(A,A2) )
    <=> ( ( A = B )
        & aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_4249_singleton__insert__inj__eq,axiom,
    ! [B: real,A: real,A2: set_real] :
      ( ( insert_real(B,bot_bot_set_real) = insert_real(A,A2) )
    <=> ( ( A = B )
        & aa_set_real_o(ord_less_eq_set_real(A2),insert_real(B,bot_bot_set_real)) ) ) ).

% singleton_insert_inj_eq
tff(fact_4250_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A2: set_nat] :
      ( ( insert_nat(B,bot_bot_set_nat) = insert_nat(A,A2) )
    <=> ( ( A = B )
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat(B,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_4251_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A2: set_int] :
      ( ( insert_int(B,bot_bot_set_int) = insert_int(A,A2) )
    <=> ( ( A = B )
        & aa_set_int_o(ord_less_eq_set_int(A2),insert_int(B,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq
tff(fact_4252_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),zero_n3304061248610475627l_real((P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_4253_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_n2052037380579107095ol_rat((P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_4254_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(zero_n2687167440665602831ol_nat,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_4255_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_4256_zero__less__diff,axiom,
    ! [Na: 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,Na),Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ).

% zero_less_diff
tff(fact_4257_diff__is__0__eq,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% diff_is_0_eq
tff(fact_4258_diff__is__0__eq_H,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_4259_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_n3304061248610475627l_real((P))),one_one_real)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_4260_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_n2052037380579107095ol_rat((P))),one_one_rat)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_4261_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),one_one_nat)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_4262_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),one_one_int)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_4263_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n1201886186963655149omplex(~ (P)) = aa_complex_complex(minus_minus_complex(one_one_complex),zero_n1201886186963655149omplex((P))) ).

% of_bool_not_iff
tff(fact_4264_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n3304061248610475627l_real(~ (P)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),zero_n3304061248610475627l_real((P))) ).

% of_bool_not_iff
tff(fact_4265_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n2052037380579107095ol_rat(~ (P)) = aa_rat_rat(minus_minus_rat(one_one_rat),zero_n2052037380579107095ol_rat((P))) ).

% of_bool_not_iff
tff(fact_4266_of__bool__not__iff,axiom,
    ! [P: $o] : aa_o_int(zero_n2684676970156552555ol_int,~ (P)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_o_int(zero_n2684676970156552555ol_int,(P))) ).

% of_bool_not_iff
tff(fact_4267_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_4268_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_4269_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_4270_diff__Suc__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Na)),one_one_nat) = Na ).

% diff_Suc_1
tff(fact_4271_Suc__0__mod__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_o_nat(zero_n2687167440665602831ol_nat,Na != aa_nat_nat(suc,zero_zero_nat)) ).

% Suc_0_mod_eq
tff(fact_4272_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_4273_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_4274_signed__take__bit__Suc__1,axiom,
    ! [Na: nat] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Na)),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_4275_Suc__pred,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))) = Na ) ) ).

% Suc_pred
tff(fact_4276_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_4277_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_4278_signed__take__bit__Suc__bit0,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Na)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Na),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_bit0
tff(fact_4279_Suc__diff__1,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) = Na ) ) ).

% Suc_diff_1
tff(fact_4280_set__replicate,axiom,
    ! [Na: nat,X: produc3843707927480180839at_nat] :
      ( ( Na != zero_zero_nat )
     => ( set_Pr3765526544606949372at_nat(replic2264142908078655527at_nat(Na,X)) = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ).

% set_replicate
tff(fact_4281_set__replicate,axiom,
    ! [Na: nat,X: vEBT_VEBT] :
      ( ( Na != zero_zero_nat )
     => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(Na,X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ) ) ).

% set_replicate
tff(fact_4282_set__replicate,axiom,
    ! [Na: nat,X: product_prod_nat_nat] :
      ( ( Na != zero_zero_nat )
     => ( set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(Na,X)) = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ).

% set_replicate
tff(fact_4283_set__replicate,axiom,
    ! [Na: nat,X: real] :
      ( ( Na != zero_zero_nat )
     => ( set_real2(replicate_real(Na,X)) = insert_real(X,bot_bot_set_real) ) ) ).

% set_replicate
tff(fact_4284_set__replicate,axiom,
    ! [Na: nat,X: nat] :
      ( ( Na != zero_zero_nat )
     => ( set_nat2(replicate_nat(Na,X)) = insert_nat(X,bot_bot_set_nat) ) ) ).

% set_replicate
tff(fact_4285_set__replicate,axiom,
    ! [Na: nat,X: int] :
      ( ( Na != zero_zero_nat )
     => ( set_int2(replicate_int(Na,X)) = insert_int(X,bot_bot_set_int) ) ) ).

% set_replicate
tff(fact_4286_even__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(A,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)) ) ).

% even_diff
tff(fact_4287_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_4288_of__bool__half__eq__0,axiom,
    ! [B: $o] : divide6298287555418463151nteger(zero_n356916108424825756nteger((B)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% of_bool_half_eq_0
tff(fact_4289_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_4290_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_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_4291_odd__Suc__minus__one,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))) = Na ) ) ).

% odd_Suc_minus_one
tff(fact_4292_even__diff__nat,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)) ) ) ).

% even_diff_nat
tff(fact_4293_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na),one_one_Code_integer))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4294_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),one_one_nat))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4295_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),one_one_int))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4296_one__div__2__pow__eq,axiom,
    ! [Na: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) = zero_n356916108424825756nteger(Na = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4297_one__div__2__pow__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_o_nat(zero_n2687167440665602831ol_nat,Na = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4298_one__div__2__pow__eq,axiom,
    ! [Na: nat] : aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_o_int(zero_n2684676970156552555ol_int,Na = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4299_bits__1__div__exp,axiom,
    ! [Na: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) = zero_n356916108424825756nteger(Na = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4300_bits__1__div__exp,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_o_nat(zero_n2687167440665602831ol_nat,Na = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4301_bits__1__div__exp,axiom,
    ! [Na: nat] : aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_o_int(zero_n2684676970156552555ol_int,Na = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4302_flip__bit__0,axiom,
    ! [A: code_integer] : bit_se1345352211410354436nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_4303_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_4304_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(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_4305_mod__int__unique,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R2))
     => ( aa_int_int(modulo_modulo_int(K),L) = R2 ) ) ).

% mod_int_unique
tff(fact_4306_gcd__nat_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
          & ( B != A ) ) ) ).

% gcd_nat.asym
tff(fact_4307_gcd__nat_Orefl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),A) ).

% gcd_nat.refl
tff(fact_4308_gcd__nat_Otrans,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,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% gcd_nat.trans
tff(fact_4309_gcd__nat_Oeq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ) ).

% gcd_nat.eq_iff
tff(fact_4310_gcd__nat_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),A)
        & ( A != A ) ) ).

% gcd_nat.irrefl
tff(fact_4311_gcd__nat_Oantisym,axiom,
    ! [A: nat,B: 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,B),A)
       => ( A = B ) ) ) ).

% gcd_nat.antisym
tff(fact_4312_gcd__nat_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
          & ( B != C ) )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans
tff(fact_4313_gcd__nat_Ostrict__trans1,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,B),C)
          & ( B != C ) )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans1
tff(fact_4314_gcd__nat_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans2
tff(fact_4315_gcd__nat_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ) ).

% gcd_nat.strict_iff_not
tff(fact_4316_gcd__nat_Oorder__iff__strict,axiom,
    ! [A: nat,B: 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),B)
          & ( A != B ) )
        | ( A = B ) ) ) ).

% gcd_nat.order_iff_strict
tff(fact_4317_gcd__nat_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) ) ) ).

% gcd_nat.strict_iff_order
tff(fact_4318_gcd__nat_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% gcd_nat.strict_implies_order
tff(fact_4319_gcd__nat_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
     => ( A != B ) ) ).

% gcd_nat.strict_implies_not_eq
tff(fact_4320_gcd__nat_Onot__eq__order__implies__strict,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( 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),B)
          & ( A != B ) ) ) ) ).

% gcd_nat.not_eq_order_implies_strict
tff(fact_4321_dvd__antisym,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),Mb)
       => ( Mb = Na ) ) ) ).

% dvd_antisym
tff(fact_4322_dvd__diff__nat,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)) ) ) ).

% dvd_diff_nat
tff(fact_4323_eucl__rel__int,axiom,
    ! [K: int,L: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(divide_divide_int(K),L)),aa_int_int(modulo_modulo_int(K),L))) ).

% eucl_rel_int
tff(fact_4324_mk__disjoint__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A2)
     => ? [B8: set_Pr4329608150637261639at_nat] :
          ( ( A2 = insert9069300056098147895at_nat(A,B8) )
          & ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),B8) ) ) ).

% mk_disjoint_insert
tff(fact_4325_mk__disjoint__insert,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A2)
     => ? [B8: set_Pr1261947904930325089at_nat] :
          ( ( A2 = insert8211810215607154385at_nat(A,B8) )
          & ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),B8) ) ) ).

% mk_disjoint_insert
tff(fact_4326_mk__disjoint__insert,axiom,
    ! [A: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),A2)
     => ? [B8: set_real] :
          ( ( A2 = insert_real(A,B8) )
          & ~ aa_set_real_o(member_real(A),B8) ) ) ).

% mk_disjoint_insert
tff(fact_4327_mk__disjoint__insert,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A2)
     => ? [B8: set_set_nat] :
          ( ( A2 = insert_set_nat(A,B8) )
          & ~ aa_set_set_nat_o(member_set_nat(A),B8) ) ) ).

% mk_disjoint_insert
tff(fact_4328_mk__disjoint__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),A2)
     => ? [B8: set_nat] :
          ( ( A2 = insert_nat(A,B8) )
          & ~ aa_set_nat_o(member_nat(A),B8) ) ) ).

% mk_disjoint_insert
tff(fact_4329_mk__disjoint__insert,axiom,
    ! [A: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => ? [B8: set_int] :
          ( ( A2 = insert_int(A,B8) )
          & ~ aa_set_int_o(member_int(A),B8) ) ) ).

% mk_disjoint_insert
tff(fact_4330_insert__commute,axiom,
    ! [X: nat,Y: nat,A2: set_nat] : insert_nat(X,insert_nat(Y,A2)) = insert_nat(Y,insert_nat(X,A2)) ).

% insert_commute
tff(fact_4331_insert__commute,axiom,
    ! [X: int,Y: int,A2: set_int] : insert_int(X,insert_int(Y,A2)) = insert_int(Y,insert_int(X,A2)) ).

% insert_commute
tff(fact_4332_insert__commute,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(X,insert8211810215607154385at_nat(Y,A2)) = insert8211810215607154385at_nat(Y,insert8211810215607154385at_nat(X,A2)) ).

% insert_commute
tff(fact_4333_insert__commute,axiom,
    ! [X: real,Y: real,A2: set_real] : insert_real(X,insert_real(Y,A2)) = insert_real(Y,insert_real(X,A2)) ).

% insert_commute
tff(fact_4334_insert__commute,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(X,insert9069300056098147895at_nat(Y,A2)) = insert9069300056098147895at_nat(Y,insert9069300056098147895at_nat(X,A2)) ).

% insert_commute
tff(fact_4335_insert__eq__iff,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A2)
     => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),B2)
       => ( ( insert9069300056098147895at_nat(A,A2) = insert9069300056098147895at_nat(B,B2) )
        <=> $ite(
              A = B,
              A2 = B2,
              ? [C6: set_Pr4329608150637261639at_nat] :
                ( ( A2 = insert9069300056098147895at_nat(B,C6) )
                & ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),C6)
                & ( B2 = insert9069300056098147895at_nat(A,C6) )
                & ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),C6) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_4336_insert__eq__iff,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A2)
     => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),B2)
       => ( ( insert8211810215607154385at_nat(A,A2) = insert8211810215607154385at_nat(B,B2) )
        <=> $ite(
              A = B,
              A2 = B2,
              ? [C6: set_Pr1261947904930325089at_nat] :
                ( ( A2 = insert8211810215607154385at_nat(B,C6) )
                & ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),C6)
                & ( B2 = insert8211810215607154385at_nat(A,C6) )
                & ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),C6) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_4337_insert__eq__iff,axiom,
    ! [A: real,A2: set_real,B: real,B2: set_real] :
      ( ~ aa_set_real_o(member_real(A),A2)
     => ( ~ aa_set_real_o(member_real(B),B2)
       => ( ( insert_real(A,A2) = insert_real(B,B2) )
        <=> $ite(
              A = B,
              A2 = B2,
              ? [C6: set_real] :
                ( ( A2 = insert_real(B,C6) )
                & ~ aa_set_real_o(member_real(B),C6)
                & ( B2 = insert_real(A,C6) )
                & ~ aa_set_real_o(member_real(A),C6) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_4338_insert__eq__iff,axiom,
    ! [A: set_nat,A2: set_set_nat,B: set_nat,B2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(A),A2)
     => ( ~ aa_set_set_nat_o(member_set_nat(B),B2)
       => ( ( insert_set_nat(A,A2) = insert_set_nat(B,B2) )
        <=> $ite(
              A = B,
              A2 = B2,
              ? [C6: set_set_nat] :
                ( ( A2 = insert_set_nat(B,C6) )
                & ~ aa_set_set_nat_o(member_set_nat(B),C6)
                & ( B2 = insert_set_nat(A,C6) )
                & ~ aa_set_set_nat_o(member_set_nat(A),C6) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_4339_insert__eq__iff,axiom,
    ! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A),A2)
     => ( ~ aa_set_nat_o(member_nat(B),B2)
       => ( ( insert_nat(A,A2) = insert_nat(B,B2) )
        <=> $ite(
              A = B,
              A2 = B2,
              ? [C6: set_nat] :
                ( ( A2 = insert_nat(B,C6) )
                & ~ aa_set_nat_o(member_nat(B),C6)
                & ( B2 = insert_nat(A,C6) )
                & ~ aa_set_nat_o(member_nat(A),C6) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_4340_insert__eq__iff,axiom,
    ! [A: int,A2: set_int,B: int,B2: set_int] :
      ( ~ aa_set_int_o(member_int(A),A2)
     => ( ~ aa_set_int_o(member_int(B),B2)
       => ( ( insert_int(A,A2) = insert_int(B,B2) )
        <=> $ite(
              A = B,
              A2 = B2,
              ? [C6: set_int] :
                ( ( A2 = insert_int(B,C6) )
                & ~ aa_set_int_o(member_int(B),C6)
                & ( B2 = insert_int(A,C6) )
                & ~ aa_set_int_o(member_int(A),C6) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_4341_insert__absorb,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A2)
     => ( insert9069300056098147895at_nat(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_4342_insert__absorb,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A2)
     => ( insert8211810215607154385at_nat(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_4343_insert__absorb,axiom,
    ! [A: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),A2)
     => ( insert_real(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_4344_insert__absorb,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A2)
     => ( insert_set_nat(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_4345_insert__absorb,axiom,
    ! [A: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),A2)
     => ( insert_nat(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_4346_insert__absorb,axiom,
    ! [A: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => ( insert_int(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_4347_insert__ident,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A2)
     => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),B2)
       => ( ( insert9069300056098147895at_nat(X,A2) = insert9069300056098147895at_nat(X,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_4348_insert__ident,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A2)
     => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),B2)
       => ( ( insert8211810215607154385at_nat(X,A2) = insert8211810215607154385at_nat(X,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_4349_insert__ident,axiom,
    ! [X: real,A2: set_real,B2: set_real] :
      ( ~ aa_set_real_o(member_real(X),A2)
     => ( ~ aa_set_real_o(member_real(X),B2)
       => ( ( insert_real(X,A2) = insert_real(X,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_4350_insert__ident,axiom,
    ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(X),A2)
     => ( ~ aa_set_set_nat_o(member_set_nat(X),B2)
       => ( ( insert_set_nat(X,A2) = insert_set_nat(X,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_4351_insert__ident,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(X),A2)
     => ( ~ aa_set_nat_o(member_nat(X),B2)
       => ( ( insert_nat(X,A2) = insert_nat(X,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_4352_insert__ident,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      ( ~ aa_set_int_o(member_int(X),A2)
     => ( ~ aa_set_int_o(member_int(X),B2)
       => ( ( insert_int(X,A2) = insert_int(X,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_4353_Set_Oset__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A2)
     => ~ ! [B8: set_Pr4329608150637261639at_nat] :
            ( ( A2 = insert9069300056098147895at_nat(X,B8) )
           => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),B8) ) ) ).

% Set.set_insert
tff(fact_4354_Set_Oset__insert,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A2)
     => ~ ! [B8: set_Pr1261947904930325089at_nat] :
            ( ( A2 = insert8211810215607154385at_nat(X,B8) )
           => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),B8) ) ) ).

% Set.set_insert
tff(fact_4355_Set_Oset__insert,axiom,
    ! [X: real,A2: set_real] :
      ( aa_set_real_o(member_real(X),A2)
     => ~ ! [B8: set_real] :
            ( ( A2 = insert_real(X,B8) )
           => aa_set_real_o(member_real(X),B8) ) ) ).

% Set.set_insert
tff(fact_4356_Set_Oset__insert,axiom,
    ! [X: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(X),A2)
     => ~ ! [B8: set_set_nat] :
            ( ( A2 = insert_set_nat(X,B8) )
           => aa_set_set_nat_o(member_set_nat(X),B8) ) ) ).

% Set.set_insert
tff(fact_4357_Set_Oset__insert,axiom,
    ! [X: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(X),A2)
     => ~ ! [B8: set_nat] :
            ( ( A2 = insert_nat(X,B8) )
           => aa_set_nat_o(member_nat(X),B8) ) ) ).

% Set.set_insert
tff(fact_4358_Set_Oset__insert,axiom,
    ! [X: int,A2: set_int] :
      ( aa_set_int_o(member_int(X),A2)
     => ~ ! [B8: set_int] :
            ( ( A2 = insert_int(X,B8) )
           => aa_set_int_o(member_int(X),B8) ) ) ).

% Set.set_insert
tff(fact_4359_insertI2,axiom,
    ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),B2)
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(B,B2)) ) ).

% insertI2
tff(fact_4360_insertI2,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),B2)
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(B,B2)) ) ).

% insertI2
tff(fact_4361_insertI2,axiom,
    ! [A: real,B2: set_real,B: real] :
      ( aa_set_real_o(member_real(A),B2)
     => aa_set_real_o(member_real(A),insert_real(B,B2)) ) ).

% insertI2
tff(fact_4362_insertI2,axiom,
    ! [A: set_nat,B2: set_set_nat,B: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),B2)
     => aa_set_set_nat_o(member_set_nat(A),insert_set_nat(B,B2)) ) ).

% insertI2
tff(fact_4363_insertI2,axiom,
    ! [A: nat,B2: set_nat,B: nat] :
      ( aa_set_nat_o(member_nat(A),B2)
     => aa_set_nat_o(member_nat(A),insert_nat(B,B2)) ) ).

% insertI2
tff(fact_4364_insertI2,axiom,
    ! [A: int,B2: set_int,B: int] :
      ( aa_set_int_o(member_int(A),B2)
     => aa_set_int_o(member_int(A),insert_int(B,B2)) ) ).

% insertI2
tff(fact_4365_insertI1,axiom,
    ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(A,B2)) ).

% insertI1
tff(fact_4366_insertI1,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(A,B2)) ).

% insertI1
tff(fact_4367_insertI1,axiom,
    ! [A: real,B2: set_real] : aa_set_real_o(member_real(A),insert_real(A,B2)) ).

% insertI1
tff(fact_4368_insertI1,axiom,
    ! [A: set_nat,B2: set_set_nat] : aa_set_set_nat_o(member_set_nat(A),insert_set_nat(A,B2)) ).

% insertI1
tff(fact_4369_insertI1,axiom,
    ! [A: nat,B2: set_nat] : aa_set_nat_o(member_nat(A),insert_nat(A,B2)) ).

% insertI1
tff(fact_4370_insertI1,axiom,
    ! [A: int,B2: set_int] : aa_set_int_o(member_int(A),insert_int(A,B2)) ).

% insertI1
tff(fact_4371_insertE,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(B,A2))
     => ( ( A != B )
       => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A2) ) ) ).

% insertE
tff(fact_4372_insertE,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(B,A2))
     => ( ( A != B )
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A2) ) ) ).

% insertE
tff(fact_4373_insertE,axiom,
    ! [A: real,B: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),insert_real(B,A2))
     => ( ( A != B )
       => aa_set_real_o(member_real(A),A2) ) ) ).

% insertE
tff(fact_4374_insertE,axiom,
    ! [A: set_nat,B: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),insert_set_nat(B,A2))
     => ( ( A != B )
       => aa_set_set_nat_o(member_set_nat(A),A2) ) ) ).

% insertE
tff(fact_4375_insertE,axiom,
    ! [A: nat,B: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),insert_nat(B,A2))
     => ( ( A != B )
       => aa_set_nat_o(member_nat(A),A2) ) ) ).

% insertE
tff(fact_4376_insertE,axiom,
    ! [A: int,B: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),insert_int(B,A2))
     => ( ( A != B )
       => aa_set_int_o(member_int(A),A2) ) ) ).

% insertE
tff(fact_4377_of__bool__eq__iff,axiom,
    ! [P2: $o,Q4: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(P2)) = aa_o_nat(zero_n2687167440665602831ol_nat,(Q4)) )
    <=> ( (P2)
      <=> (Q4) ) ) ).

% of_bool_eq_iff
tff(fact_4378_of__bool__eq__iff,axiom,
    ! [P2: $o,Q4: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P2)) = aa_o_int(zero_n2684676970156552555ol_int,(Q4)) )
    <=> ( (P2)
      <=> (Q4) ) ) ).

% of_bool_eq_iff
tff(fact_4379_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),K)),J) ).

% diff_commute
tff(fact_4380_diff__right__commute,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) ).

% diff_right_commute
tff(fact_4381_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) ).

% diff_right_commute
tff(fact_4382_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_4383_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_4384_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4385_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4386_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_4387_full__exhaustive__int_H_Ocases,axiom,
    ! [X: produc3043004577601930755nt_int] :
      ~ ! [F2: fun_Pr7763680713840462443e_term,D4: int,I3: int] : X != aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,F2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D4),I3)) ).

% full_exhaustive_int'.cases
tff(fact_4388_exhaustive__int_H_Ocases,axiom,
    ! [X: produc4788131091073775919nt_int] :
      ~ ! [F2: fun_in5730379987311621695e_term,D4: int,I3: int] : X != aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,F2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D4),I3)) ).

% exhaustive_int'.cases
tff(fact_4389_small__lazy_H_Ocases,axiom,
    ! [X: product_prod_int_int] :
      ~ ! [D4: int,I3: int] : X != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,D4),I3) ).

% small_lazy'.cases
tff(fact_4390_unique__quotient,axiom,
    ! [A: int,B: int,Q4: int,R2: int,Q6: int,R3: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R2))
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q6),R3))
       => ( Q4 = Q6 ) ) ) ).

% unique_quotient
tff(fact_4391_unique__remainder,axiom,
    ! [A: int,B: int,Q4: int,R2: int,Q6: int,R3: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R2))
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q6),R3))
       => ( R2 = R3 ) ) ) ).

% unique_remainder
tff(fact_4392_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n1201886186963655149omplex(( (P)
        & (Q) )) = aa_complex_complex(times_times_complex(zero_n1201886186963655149omplex((P))),zero_n1201886186963655149omplex((Q))) ).

% of_bool_conj
tff(fact_4393_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n3304061248610475627l_real(( (P)
        & (Q) )) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_n3304061248610475627l_real((P))),zero_n3304061248610475627l_real((Q))) ).

% of_bool_conj
tff(fact_4394_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n2052037380579107095ol_rat(( (P)
        & (Q) )) = aa_rat_rat(times_times_rat(zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q))) ).

% of_bool_conj
tff(fact_4395_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,
        ( (P)
        & (Q) )) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q))) ).

% of_bool_conj
tff(fact_4396_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,
        ( (P)
        & (Q) )) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q))) ).

% of_bool_conj
tff(fact_4397_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(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ) ) ).

% diff_mono
tff(fact_4398_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(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D)) ) ) ).

% diff_mono
tff(fact_4399_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_4400_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(aa_rea7340605660300375871l_real(minus_minus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)) ) ).

% diff_left_mono
tff(fact_4401_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(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B)) ) ).

% diff_left_mono
tff(fact_4402_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_4403_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(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) ) ).

% diff_right_mono
tff(fact_4404_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(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C)) ) ).

% diff_right_mono
tff(fact_4405_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_4406_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_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_4407_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( 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_4408_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_4409_eq__iff__diff__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = B )
    <=> ( aa_complex_complex(minus_minus_complex(A),B) = zero_zero_complex ) ) ).

% eq_iff_diff_eq_0
tff(fact_4410_eq__iff__diff__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_4411_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_rat(minus_minus_rat(A),B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_4412_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_4413_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(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ) ) ).

% diff_strict_mono
tff(fact_4414_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(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D)) ) ) ).

% diff_strict_mono
tff(fact_4415_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_4416_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_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_4417_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( 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_4418_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_4419_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(aa_rea7340605660300375871l_real(minus_minus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)) ) ).

% diff_strict_left_mono
tff(fact_4420_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(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B)) ) ).

% diff_strict_left_mono
tff(fact_4421_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_4422_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(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) ) ).

% diff_strict_right_mono
tff(fact_4423_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(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C)) ) ).

% diff_strict_right_mono
tff(fact_4424_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_4425_right__diff__distrib_H,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(minus_minus_complex(B),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),B)),aa_complex_complex(times_times_complex(A),C)) ).

% right_diff_distrib'
tff(fact_4426_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib'
tff(fact_4427_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib'
tff(fact_4428_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% right_diff_distrib'
tff(fact_4429_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib'
tff(fact_4430_left__diff__distrib_H,axiom,
    ! [B: complex,C: complex,A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(B),C)),A) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(B),A)),aa_complex_complex(times_times_complex(C),A)) ).

% left_diff_distrib'
tff(fact_4431_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)) ).

% left_diff_distrib'
tff(fact_4432_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),C)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(B),A)),aa_rat_rat(times_times_rat(C),A)) ).

% left_diff_distrib'
tff(fact_4433_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)) ).

% left_diff_distrib'
tff(fact_4434_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)) ).

% left_diff_distrib'
tff(fact_4435_right__diff__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(minus_minus_complex(B),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),B)),aa_complex_complex(times_times_complex(A),C)) ).

% right_diff_distrib
tff(fact_4436_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib
tff(fact_4437_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib
tff(fact_4438_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib
tff(fact_4439_left__diff__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),B)),C) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)) ).

% left_diff_distrib
tff(fact_4440_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% left_diff_distrib
tff(fact_4441_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% left_diff_distrib
tff(fact_4442_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% left_diff_distrib
tff(fact_4443_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D)) ).

% add_diff_add
tff(fact_4444_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),aa_rat_rat(minus_minus_rat(C),D)) ).

% add_diff_add
tff(fact_4445_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_4446_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4447_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(minus_minus_rat(A2),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4448_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4449_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = C )
    <=> ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) ) ) ).

% diff_eq_eq
tff(fact_4450_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = C )
    <=> ( A = aa_rat_rat(plus_plus_rat(C),B) ) ) ).

% diff_eq_eq
tff(fact_4451_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_4452_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_4453_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A = aa_rat_rat(minus_minus_rat(C),B) )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = C ) ) ).

% eq_diff_eq
tff(fact_4454_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_4455_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ).

% add_diff_eq
tff(fact_4456_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) ).

% add_diff_eq
tff(fact_4457_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_4458_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).

% diff_diff_eq2
tff(fact_4459_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),B) ).

% diff_diff_eq2
tff(fact_4460_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_4461_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).

% diff_add_eq
tff(fact_4462_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),B) ).

% diff_add_eq
tff(fact_4463_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_4464_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4465_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4466_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_4467_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) = A )
     => ( C = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ) ) ).

% add_implies_diff
tff(fact_4468_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(C),B) = A )
     => ( C = aa_rat_rat(minus_minus_rat(A),B) ) ) ).

% add_implies_diff
tff(fact_4469_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_4470_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_4471_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% diff_diff_eq
tff(fact_4472_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% diff_diff_eq
tff(fact_4473_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_4474_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_4475_diff__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ).

% diff_divide_distrib
tff(fact_4476_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ).

% diff_divide_distrib
tff(fact_4477_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_4478_singletonD,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(B),insert_set_nat(A,bot_bot_set_set_nat))
     => ( B = A ) ) ).

% singletonD
tff(fact_4479_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_4480_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_4481_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_4482_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_4483_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_4484_singleton__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(B),insert_set_nat(A,bot_bot_set_set_nat))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_4485_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_4486_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_4487_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_4488_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_4489_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_4490_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_4491_doubleton__eq__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( insert_real(A,insert_real(B,bot_bot_set_real)) = insert_real(C,insert_real(D,bot_bot_set_real)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_4492_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_4493_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_4494_insert__not__empty,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A,A2) != bot_bo228742789529271731at_nat ).

% insert_not_empty
tff(fact_4495_insert__not__empty,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A,A2) != bot_bo2099793752762293965at_nat ).

% insert_not_empty
tff(fact_4496_insert__not__empty,axiom,
    ! [A: real,A2: set_real] : insert_real(A,A2) != bot_bot_set_real ).

% insert_not_empty
tff(fact_4497_insert__not__empty,axiom,
    ! [A: nat,A2: set_nat] : insert_nat(A,A2) != bot_bot_set_nat ).

% insert_not_empty
tff(fact_4498_insert__not__empty,axiom,
    ! [A: int,A2: set_int] : insert_int(A,A2) != bot_bot_set_int ).

% insert_not_empty
tff(fact_4499_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_4500_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_4501_singleton__inject,axiom,
    ! [A: real,B: real] :
      ( ( insert_real(A,bot_bot_set_real) = insert_real(B,bot_bot_set_real) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_4502_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_4503_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_4504_finite_OinsertI,axiom,
    ! [A2: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => aa_set_real_o(finite_finite_real,insert_real(A,A2)) ) ).

% finite.insertI
tff(fact_4505_finite_OinsertI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A2)
     => aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,insert9069300056098147895at_nat(A,A2)) ) ).

% finite.insertI
tff(fact_4506_finite_OinsertI,axiom,
    ! [A2: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => aa_set_nat_o(finite_finite_nat,insert_nat(A,A2)) ) ).

% finite.insertI
tff(fact_4507_finite_OinsertI,axiom,
    ! [A2: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => aa_set_int_o(finite_finite_int,insert_int(A,A2)) ) ).

% finite.insertI
tff(fact_4508_finite_OinsertI,axiom,
    ! [A2: set_complex,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => aa_set_complex_o(finite3207457112153483333omplex,insert_complex(A,A2)) ) ).

% finite.insertI
tff(fact_4509_finite_OinsertI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,insert8211810215607154385at_nat(A,A2)) ) ).

% finite.insertI
tff(fact_4510_insert__subsetI,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,X5: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A2)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X5),A2)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(insert9069300056098147895at_nat(X,X5)),A2) ) ) ).

% insert_subsetI
tff(fact_4511_insert__subsetI,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A2)
     => ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(X5),A2)
       => aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(insert8211810215607154385at_nat(X,X5)),A2) ) ) ).

% insert_subsetI
tff(fact_4512_insert__subsetI,axiom,
    ! [X: real,A2: set_real,X5: set_real] :
      ( aa_set_real_o(member_real(X),A2)
     => ( aa_set_real_o(ord_less_eq_set_real(X5),A2)
       => aa_set_real_o(ord_less_eq_set_real(insert_real(X,X5)),A2) ) ) ).

% insert_subsetI
tff(fact_4513_insert__subsetI,axiom,
    ! [X: set_nat,A2: set_set_nat,X5: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(X),A2)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(X5),A2)
       => aa_set_set_nat_o(ord_le6893508408891458716et_nat(insert_set_nat(X,X5)),A2) ) ) ).

% insert_subsetI
tff(fact_4514_insert__subsetI,axiom,
    ! [X: nat,A2: set_nat,X5: set_nat] :
      ( aa_set_nat_o(member_nat(X),A2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X5),A2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,insert_nat(X,X5)),A2) ) ) ).

% insert_subsetI
tff(fact_4515_insert__subsetI,axiom,
    ! [X: int,A2: set_int,X5: set_int] :
      ( aa_set_int_o(member_int(X),A2)
     => ( aa_set_int_o(ord_less_eq_set_int(X5),A2)
       => aa_set_int_o(ord_less_eq_set_int(insert_int(X,X5)),A2) ) ) ).

% insert_subsetI
tff(fact_4516_insert__mono,axiom,
    ! [C4: set_nat,D5: set_nat,A: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C4),D5)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,insert_nat(A,C4)),insert_nat(A,D5)) ) ).

% insert_mono
tff(fact_4517_insert__mono,axiom,
    ! [C4: set_Pr1261947904930325089at_nat,D5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(C4),D5)
     => aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(insert8211810215607154385at_nat(A,C4)),insert8211810215607154385at_nat(A,D5)) ) ).

% insert_mono
tff(fact_4518_insert__mono,axiom,
    ! [C4: set_real,D5: set_real,A: real] :
      ( aa_set_real_o(ord_less_eq_set_real(C4),D5)
     => aa_set_real_o(ord_less_eq_set_real(insert_real(A,C4)),insert_real(A,D5)) ) ).

% insert_mono
tff(fact_4519_insert__mono,axiom,
    ! [C4: set_Pr4329608150637261639at_nat,D5: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C4),D5)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(insert9069300056098147895at_nat(A,C4)),insert9069300056098147895at_nat(A,D5)) ) ).

% insert_mono
tff(fact_4520_insert__mono,axiom,
    ! [C4: set_int,D5: set_int,A: int] :
      ( aa_set_int_o(ord_less_eq_set_int(C4),D5)
     => aa_set_int_o(ord_less_eq_set_int(insert_int(A,C4)),insert_int(A,D5)) ) ).

% insert_mono
tff(fact_4521_subset__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A2)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),insert9069300056098147895at_nat(X,B2))
      <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),B2) ) ) ).

% subset_insert
tff(fact_4522_subset__insert,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A2)
     => ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),insert8211810215607154385at_nat(X,B2))
      <=> aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),B2) ) ) ).

% subset_insert
tff(fact_4523_subset__insert,axiom,
    ! [X: real,A2: set_real,B2: set_real] :
      ( ~ aa_set_real_o(member_real(X),A2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),insert_real(X,B2))
      <=> aa_set_real_o(ord_less_eq_set_real(A2),B2) ) ) ).

% subset_insert
tff(fact_4524_subset__insert,axiom,
    ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(X),A2)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),insert_set_nat(X,B2))
      <=> aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2) ) ) ).

% subset_insert
tff(fact_4525_subset__insert,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(X),A2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat(X,B2))
      <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2) ) ) ).

% subset_insert
tff(fact_4526_subset__insert,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      ( ~ aa_set_int_o(member_int(X),A2)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),insert_int(X,B2))
      <=> aa_set_int_o(ord_less_eq_set_int(A2),B2) ) ) ).

% subset_insert
tff(fact_4527_subset__insertI,axiom,
    ! [B2: set_nat,A: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),insert_nat(A,B2)) ).

% subset_insertI
tff(fact_4528_subset__insertI,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(B2),insert8211810215607154385at_nat(A,B2)) ).

% subset_insertI
tff(fact_4529_subset__insertI,axiom,
    ! [B2: set_real,A: real] : aa_set_real_o(ord_less_eq_set_real(B2),insert_real(A,B2)) ).

% subset_insertI
tff(fact_4530_subset__insertI,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B2),insert9069300056098147895at_nat(A,B2)) ).

% subset_insertI
tff(fact_4531_subset__insertI,axiom,
    ! [B2: set_int,A: int] : aa_set_int_o(ord_less_eq_set_int(B2),insert_int(A,B2)) ).

% subset_insertI
tff(fact_4532_subset__insertI2,axiom,
    ! [A2: set_nat,B2: set_nat,B: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat(B,B2)) ) ).

% subset_insertI2
tff(fact_4533_subset__insertI2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),B2)
     => aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),insert8211810215607154385at_nat(B,B2)) ) ).

% subset_insertI2
tff(fact_4534_subset__insertI2,axiom,
    ! [A2: set_real,B2: set_real,B: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
     => aa_set_real_o(ord_less_eq_set_real(A2),insert_real(B,B2)) ) ).

% subset_insertI2
tff(fact_4535_subset__insertI2,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),B2)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),insert9069300056098147895at_nat(B,B2)) ) ).

% subset_insertI2
tff(fact_4536_subset__insertI2,axiom,
    ! [A2: set_int,B2: set_int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => aa_set_int_o(ord_less_eq_set_int(A2),insert_int(B,B2)) ) ).

% subset_insertI2
tff(fact_4537_dvd__diff,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),Z)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),minus_8373710615458151222nteger(Y,Z)) ) ) ).

% dvd_diff
tff(fact_4538_dvd__diff,axiom,
    ! [X: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X),Y)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X),Z)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z)) ) ) ).

% dvd_diff
tff(fact_4539_dvd__diff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),Z)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),aa_rat_rat(minus_minus_rat(Y),Z)) ) ) ).

% dvd_diff
tff(fact_4540_dvd__diff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),Z)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ) ) ).

% dvd_diff
tff(fact_4541_zero__induct__lemma,axiom,
    ! [P: fun_nat_o,K: nat,I: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I)) ) ) ).

% zero_induct_lemma
tff(fact_4542_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_4543_diffs0__imp__equal,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Mb) = zero_zero_nat )
       => ( Mb = Na ) ) ) ).

% diffs0_imp_equal
tff(fact_4544_diff__less__mono2,axiom,
    ! [Mb: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
     => ( 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),Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Mb)) ) ) ).

% diff_less_mono2
tff(fact_4545_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Na)),K) ) ).

% less_imp_diff_less
tff(fact_4546_dvd__minus__self,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Mb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Na) ) ) ).

% dvd_minus_self
tff(fact_4547_eq__diff__iff,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K) )
        <=> ( Mb = Na ) ) ) ) ).

% eq_diff_iff
tff(fact_4548_le__diff__iff,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ) ) ).

% le_diff_iff
tff(fact_4549_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_4550_diff__le__mono,axiom,
    ! [Mb: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => 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,Na),L)) ) ).

% diff_le_mono
tff(fact_4551_diff__le__self,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)),Mb) ).

% diff_le_self
tff(fact_4552_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_4553_diff__le__mono2,axiom,
    ! [Mb: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Mb)) ) ).

% diff_le_mono2
tff(fact_4554_dvd__diffD,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb) ) ) ) ).

% dvd_diffD
tff(fact_4555_dvd__diffD1,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na) ) ) ) ).

% dvd_diffD1
tff(fact_4556_less__eq__dvd__minus,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Na)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Mb)) ) ) ).

% less_eq_dvd_minus
tff(fact_4557_Nat_Odiff__cancel,axiom,
    ! [K: nat,Mb: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na) ).

% Nat.diff_cancel
tff(fact_4558_diff__cancel2,axiom,
    ! [Mb: nat,K: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na) ).

% diff_cancel2
tff(fact_4559_diff__add__inverse,axiom,
    ! [Na: 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,Na),Mb)),Na) = Mb ).

% diff_add_inverse
tff(fact_4560_diff__add__inverse2,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)),Na) = Mb ).

% diff_add_inverse2
tff(fact_4561_diff__mult__distrib2,axiom,
    ! [K: nat,Mb: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) ).

% diff_mult_distrib2
tff(fact_4562_diff__mult__distrib,axiom,
    ! [Mb: nat,Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)) ).

% diff_mult_distrib
tff(fact_4563_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D4: nat,X3: 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(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)) = D4 )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)) = D4 ) ) ) ).

% bezout1_nat
tff(fact_4564_eucl__rel__int__by0,axiom,
    ! [K: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K,zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K)) ).

% eucl_rel_int_by0
tff(fact_4565_div__int__unique,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R2))
     => ( aa_int_int(divide_divide_int(K),L) = Q4 ) ) ).

% div_int_unique
tff(fact_4566_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),zero_n3304061248610475627l_real((P))) ).

% zero_less_eq_of_bool
tff(fact_4567_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),zero_n2052037380579107095ol_rat((P))) ).

% zero_less_eq_of_bool
tff(fact_4568_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_4569_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_4570_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_n3304061248610475627l_real((P))),one_one_real) ).

% of_bool_less_eq_one
tff(fact_4571_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_n2052037380579107095ol_rat((P))),one_one_rat) ).

% of_bool_less_eq_one
tff(fact_4572_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_4573_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_4574_split__of__bool__asm,axiom,
    ! [P: fun_complex_o,P2: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_complex_o(P,one_one_complex) )
          | ( ~ (P2)
            & ~ aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool_asm
tff(fact_4575_split__of__bool__asm,axiom,
    ! [P: fun_real_o,P2: $o] :
      ( aa_real_o(P,zero_n3304061248610475627l_real((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_real_o(P,one_one_real) )
          | ( ~ (P2)
            & ~ aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool_asm
tff(fact_4576_split__of__bool__asm,axiom,
    ! [P: fun_rat_o,P2: $o] :
      ( aa_rat_o(P,zero_n2052037380579107095ol_rat((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_rat_o(P,one_one_rat) )
          | ( ~ (P2)
            & ~ aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool_asm
tff(fact_4577_split__of__bool__asm,axiom,
    ! [P: fun_nat_o,P2: $o] :
      ( aa_nat_o(P,aa_o_nat(zero_n2687167440665602831ol_nat,(P2)))
    <=> ~ ( ( (P2)
            & ~ aa_nat_o(P,one_one_nat) )
          | ( ~ (P2)
            & ~ aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool_asm
tff(fact_4578_split__of__bool__asm,axiom,
    ! [P: fun_int_o,P2: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> ~ ( ( (P2)
            & ~ aa_int_o(P,one_one_int) )
          | ( ~ (P2)
            & ~ aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool_asm
tff(fact_4579_split__of__bool,axiom,
    ! [P: fun_complex_o,P2: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P2)))
    <=> ( ( (P2)
         => aa_complex_o(P,one_one_complex) )
        & ( ~ (P2)
         => aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool
tff(fact_4580_split__of__bool,axiom,
    ! [P: fun_real_o,P2: $o] :
      ( aa_real_o(P,zero_n3304061248610475627l_real((P2)))
    <=> ( ( (P2)
         => aa_real_o(P,one_one_real) )
        & ( ~ (P2)
         => aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool
tff(fact_4581_split__of__bool,axiom,
    ! [P: fun_rat_o,P2: $o] :
      ( aa_rat_o(P,zero_n2052037380579107095ol_rat((P2)))
    <=> ( ( (P2)
         => aa_rat_o(P,one_one_rat) )
        & ( ~ (P2)
         => aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool
tff(fact_4582_split__of__bool,axiom,
    ! [P: fun_nat_o,P2: $o] :
      ( aa_nat_o(P,aa_o_nat(zero_n2687167440665602831ol_nat,(P2)))
    <=> ( ( (P2)
         => aa_nat_o(P,one_one_nat) )
        & ( ~ (P2)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool
tff(fact_4583_split__of__bool,axiom,
    ! [P: fun_int_o,P2: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> ( ( (P2)
         => aa_int_o(P,one_one_int) )
        & ( ~ (P2)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool
tff(fact_4584_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n1201886186963655149omplex((P2)) = $ite((P2),one_one_complex,zero_zero_complex) ).

% of_bool_def
tff(fact_4585_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n3304061248610475627l_real((P2)) = $ite((P2),one_one_real,zero_zero_real) ).

% of_bool_def
tff(fact_4586_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n2052037380579107095ol_rat((P2)) = $ite((P2),one_one_rat,zero_zero_rat) ).

% of_bool_def
tff(fact_4587_of__bool__def,axiom,
    ! [P2: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,(P2)) = $ite((P2),one_one_nat,zero_zero_nat) ).

% of_bool_def
tff(fact_4588_of__bool__def,axiom,
    ! [P2: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,(P2)) = $ite((P2),one_one_int,zero_zero_int) ).

% of_bool_def
tff(fact_4589_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(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real) ) ).

% le_iff_diff_le_0
tff(fact_4590_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(minus_minus_rat(A),B)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_4591_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_4592_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(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real) ) ).

% less_iff_diff_less_0
tff(fact_4593_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(minus_minus_rat(A),B)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_4594_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_4595_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_4596_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_4597_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_4598_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_4599_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_4600_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_4601_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_4602_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_4603_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_4604_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_4605_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(aa_rea7340605660300375871l_real(minus_minus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ) ).

% le_diff_eq
tff(fact_4606_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(minus_minus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),C) ) ).

% le_diff_eq
tff(fact_4607_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_4608_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(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% diff_le_eq
tff(fact_4609_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(minus_minus_rat(A),B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% diff_le_eq
tff(fact_4610_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_4611_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,Na: real,J: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Na),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Na)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Na),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4612_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,Na: rat,J: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Na),aa_rat_rat(plus_plus_rat(J),K))
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),Na)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Na),aa_rat_rat(plus_plus_rat(J),K))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4613_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,Na: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Na)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4614_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,Na: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Na)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4615_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,Na: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_4616_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,Na: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),aa_rat_rat(minus_minus_rat(Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_4617_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_4618_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_4619_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(aa_rea7340605660300375871l_real(minus_minus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ) ).

% less_diff_eq
tff(fact_4620_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(minus_minus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),C) ) ).

% less_diff_eq
tff(fact_4621_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_4622_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% diff_less_eq
tff(fact_4623_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% diff_less_eq
tff(fact_4624_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_4625_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(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4626_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(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4627_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_4628_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_4629_mult__diff__mult,axiom,
    ! [X: complex,Y: complex,A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Y)),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),aa_complex_complex(minus_minus_complex(Y),B))),aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),A)),B)) ).

% mult_diff_mult
tff(fact_4630_mult__diff__mult,axiom,
    ! [X: real,Y: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A)),B)) ).

% mult_diff_mult
tff(fact_4631_mult__diff__mult,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Y)),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),aa_rat_rat(minus_minus_rat(Y),B))),aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X),A)),B)) ).

% mult_diff_mult
tff(fact_4632_mult__diff__mult,axiom,
    ! [X: int,Y: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A)),B)) ).

% mult_diff_mult
tff(fact_4633_square__diff__square__factored,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),X)),aa_complex_complex(times_times_complex(Y),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(X),Y)),aa_complex_complex(minus_minus_complex(X),Y)) ).

% square_diff_square_factored
tff(fact_4634_square__diff__square__factored,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) ).

% square_diff_square_factored
tff(fact_4635_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(X),Y)),aa_rat_rat(minus_minus_rat(X),Y)) ).

% square_diff_square_factored
tff(fact_4636_square__diff__square__factored,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) ).

% square_diff_square_factored
tff(fact_4637_eq__add__iff2,axiom,
    ! [A: complex,E2: complex,C: complex,B: complex,D: complex] :
      ( ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),E2)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(B),E2)),D) )
    <=> ( C = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_4638_eq__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D) )
    <=> ( C = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_4639_eq__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D) )
    <=> ( C = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_4640_eq__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D) )
    <=> ( C = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_4641_eq__add__iff1,axiom,
    ! [A: complex,E2: complex,C: complex,B: complex,D: complex] :
      ( ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),E2)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(B),E2)),D) )
    <=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_4642_eq__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_4643_eq__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_4644_eq__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_4645_finite_Ocases,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A)
     => ( ( A != bot_bo228742789529271731at_nat )
       => ~ ! [A5: set_Pr4329608150637261639at_nat] :
              ( ? [A3: produc3843707927480180839at_nat] : A = insert9069300056098147895at_nat(A3,A5)
             => ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A5) ) ) ) ).

% finite.cases
tff(fact_4646_finite_Ocases,axiom,
    ! [A: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ( A != bot_bot_set_complex )
       => ~ ! [A5: set_complex] :
              ( ? [A3: complex] : A = insert_complex(A3,A5)
             => ~ aa_set_complex_o(finite3207457112153483333omplex,A5) ) ) ) ).

% finite.cases
tff(fact_4647_finite_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A)
     => ( ( A != bot_bo2099793752762293965at_nat )
       => ~ ! [A5: set_Pr1261947904930325089at_nat] :
              ( ? [A3: product_prod_nat_nat] : A = insert8211810215607154385at_nat(A3,A5)
             => ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A5) ) ) ) ).

% finite.cases
tff(fact_4648_finite_Ocases,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( ( A != bot_bot_set_real )
       => ~ ! [A5: set_real] :
              ( ? [A3: real] : A = insert_real(A3,A5)
             => ~ aa_set_real_o(finite_finite_real,A5) ) ) ) ).

% finite.cases
tff(fact_4649_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A5: set_nat] :
              ( ? [A3: nat] : A = insert_nat(A3,A5)
             => ~ aa_set_nat_o(finite_finite_nat,A5) ) ) ) ).

% finite.cases
tff(fact_4650_finite_Ocases,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ~ ! [A5: set_int] :
              ( ? [A3: int] : A = insert_int(A3,A5)
             => ~ aa_set_int_o(finite_finite_int,A5) ) ) ) ).

% finite.cases
tff(fact_4651_finite_Osimps,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A)
    <=> ( ( A = bot_bo228742789529271731at_nat )
        | ? [A7: set_Pr4329608150637261639at_nat,A6: produc3843707927480180839at_nat] :
            ( ( A = insert9069300056098147895at_nat(A6,A7) )
            & aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A7) ) ) ) ).

% finite.simps
tff(fact_4652_finite_Osimps,axiom,
    ! [A: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
    <=> ( ( A = bot_bot_set_complex )
        | ? [A7: set_complex,A6: complex] :
            ( ( A = insert_complex(A6,A7) )
            & aa_set_complex_o(finite3207457112153483333omplex,A7) ) ) ) ).

% finite.simps
tff(fact_4653_finite_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A)
    <=> ( ( A = bot_bo2099793752762293965at_nat )
        | ? [A7: set_Pr1261947904930325089at_nat,A6: product_prod_nat_nat] :
            ( ( A = insert8211810215607154385at_nat(A6,A7) )
            & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A7) ) ) ) ).

% finite.simps
tff(fact_4654_finite_Osimps,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(finite_finite_real,A)
    <=> ( ( A = bot_bot_set_real )
        | ? [A7: set_real,A6: real] :
            ( ( A = insert_real(A6,A7) )
            & aa_set_real_o(finite_finite_real,A7) ) ) ) ).

% finite.simps
tff(fact_4655_finite_Osimps,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
    <=> ( ( A = bot_bot_set_nat )
        | ? [A7: set_nat,A6: nat] :
            ( ( A = insert_nat(A6,A7) )
            & aa_set_nat_o(finite_finite_nat,A7) ) ) ) ).

% finite.simps
tff(fact_4656_finite_Osimps,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
    <=> ( ( A = bot_bot_set_int )
        | ? [A7: set_int,A6: int] :
            ( ( A = insert_int(A6,A7) )
            & aa_set_int_o(finite_finite_int,A7) ) ) ) ).

% finite.simps
tff(fact_4657_finite__induct,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F3)
     => ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
       => ( ! [X3: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
             => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),F4)
               => ( aa_set5844364881195198285_nat_o(P,F4)
                 => aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(X3,F4)) ) ) )
         => aa_set5844364881195198285_nat_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_4658_finite__induct,axiom,
    ! [F3: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
     => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
       => ( ! [X3: set_nat,F4: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
             => ( ~ aa_set_set_nat_o(member_set_nat(X3),F4)
               => ( aa_set_set_nat_o(P,F4)
                 => aa_set_set_nat_o(P,insert_set_nat(X3,F4)) ) ) )
         => aa_set_set_nat_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_4659_finite__induct,axiom,
    ! [F3: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F3)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X3: complex,F4: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,F4)
             => ( ~ aa_set_complex_o(member_complex(X3),F4)
               => ( aa_set_complex_o(P,F4)
                 => aa_set_complex_o(P,insert_complex(X3,F4)) ) ) )
         => aa_set_complex_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_4660_finite__induct,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
     => ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
       => ( ! [X3: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
             => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),F4)
               => ( aa_set3702741264012733747_nat_o(P,F4)
                 => aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(X3,F4)) ) ) )
         => aa_set3702741264012733747_nat_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_4661_finite__induct,axiom,
    ! [F3: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F3)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X3: real,F4: set_real] :
              ( aa_set_real_o(finite_finite_real,F4)
             => ( ~ aa_set_real_o(member_real(X3),F4)
               => ( aa_set_real_o(P,F4)
                 => aa_set_real_o(P,insert_real(X3,F4)) ) ) )
         => aa_set_real_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_4662_finite__induct,axiom,
    ! [F3: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X3: nat,F4: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,F4)
             => ( ~ aa_set_nat_o(member_nat(X3),F4)
               => ( aa_set_nat_o(P,F4)
                 => aa_set_nat_o(P,insert_nat(X3,F4)) ) ) )
         => aa_set_nat_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_4663_finite__induct,axiom,
    ! [F3: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X3: int,F4: set_int] :
              ( aa_set_int_o(finite_finite_int,F4)
             => ( ~ aa_set_int_o(member_int(X3),F4)
               => ( aa_set_int_o(P,F4)
                 => aa_set_int_o(P,insert_int(X3,F4)) ) ) )
         => aa_set_int_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_4664_finite__ne__induct,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F3)
     => ( ( F3 != bot_bo228742789529271731at_nat )
       => ( ! [X3: produc3843707927480180839at_nat] : aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat))
         => ( ! [X3: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
                ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
               => ( ( F4 != bot_bo228742789529271731at_nat )
                 => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),F4)
                   => ( aa_set5844364881195198285_nat_o(P,F4)
                     => aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(X3,F4)) ) ) ) )
           => aa_set5844364881195198285_nat_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_4665_finite__ne__induct,axiom,
    ! [F3: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
     => ( ( F3 != bot_bot_set_set_nat )
       => ( ! [X3: set_nat] : aa_set_set_nat_o(P,insert_set_nat(X3,bot_bot_set_set_nat))
         => ( ! [X3: set_nat,F4: set_set_nat] :
                ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
               => ( ( F4 != bot_bot_set_set_nat )
                 => ( ~ aa_set_set_nat_o(member_set_nat(X3),F4)
                   => ( aa_set_set_nat_o(P,F4)
                     => aa_set_set_nat_o(P,insert_set_nat(X3,F4)) ) ) ) )
           => aa_set_set_nat_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_4666_finite__ne__induct,axiom,
    ! [F3: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F3)
     => ( ( F3 != bot_bot_set_complex )
       => ( ! [X3: complex] : aa_set_complex_o(P,insert_complex(X3,bot_bot_set_complex))
         => ( ! [X3: complex,F4: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F4)
               => ( ( F4 != bot_bot_set_complex )
                 => ( ~ aa_set_complex_o(member_complex(X3),F4)
                   => ( aa_set_complex_o(P,F4)
                     => aa_set_complex_o(P,insert_complex(X3,F4)) ) ) ) )
           => aa_set_complex_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_4667_finite__ne__induct,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
     => ( ( F3 != bot_bo2099793752762293965at_nat )
       => ( ! [X3: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat))
         => ( ! [X3: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
               => ( ( F4 != bot_bo2099793752762293965at_nat )
                 => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),F4)
                   => ( aa_set3702741264012733747_nat_o(P,F4)
                     => aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(X3,F4)) ) ) ) )
           => aa_set3702741264012733747_nat_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_4668_finite__ne__induct,axiom,
    ! [F3: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F3)
     => ( ( F3 != bot_bot_set_real )
       => ( ! [X3: real] : aa_set_real_o(P,insert_real(X3,bot_bot_set_real))
         => ( ! [X3: real,F4: set_real] :
                ( aa_set_real_o(finite_finite_real,F4)
               => ( ( F4 != bot_bot_set_real )
                 => ( ~ aa_set_real_o(member_real(X3),F4)
                   => ( aa_set_real_o(P,F4)
                     => aa_set_real_o(P,insert_real(X3,F4)) ) ) ) )
           => aa_set_real_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_4669_finite__ne__induct,axiom,
    ! [F3: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( ( F3 != bot_bot_set_nat )
       => ( ! [X3: nat] : aa_set_nat_o(P,insert_nat(X3,bot_bot_set_nat))
         => ( ! [X3: nat,F4: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F4)
               => ( ( F4 != bot_bot_set_nat )
                 => ( ~ aa_set_nat_o(member_nat(X3),F4)
                   => ( aa_set_nat_o(P,F4)
                     => aa_set_nat_o(P,insert_nat(X3,F4)) ) ) ) )
           => aa_set_nat_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_4670_finite__ne__induct,axiom,
    ! [F3: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( ( F3 != bot_bot_set_int )
       => ( ! [X3: int] : aa_set_int_o(P,insert_int(X3,bot_bot_set_int))
         => ( ! [X3: int,F4: set_int] :
                ( aa_set_int_o(finite_finite_int,F4)
               => ( ( F4 != bot_bot_set_int )
                 => ( ~ aa_set_int_o(member_int(X3),F4)
                   => ( aa_set_int_o(P,F4)
                     => aa_set_int_o(P,insert_int(X3,F4)) ) ) ) )
           => aa_set_int_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_4671_infinite__finite__induct,axiom,
    ! [P: fun_se7788542033685991124_nat_o,A2: set_Pr4329608150637261639at_nat] :
      ( ! [A5: set_Pr4329608150637261639at_nat] :
          ( ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A5)
         => aa_set5844364881195198285_nat_o(P,A5) )
     => ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
       => ( ! [X3: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
             => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),F4)
               => ( aa_set5844364881195198285_nat_o(P,F4)
                 => aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(X3,F4)) ) ) )
         => aa_set5844364881195198285_nat_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_4672_infinite__finite__induct,axiom,
    ! [P: fun_set_set_nat_o,A2: set_set_nat] :
      ( ! [A5: set_set_nat] :
          ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A5)
         => aa_set_set_nat_o(P,A5) )
     => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
       => ( ! [X3: set_nat,F4: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
             => ( ~ aa_set_set_nat_o(member_set_nat(X3),F4)
               => ( aa_set_set_nat_o(P,F4)
                 => aa_set_set_nat_o(P,insert_set_nat(X3,F4)) ) ) )
         => aa_set_set_nat_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_4673_infinite__finite__induct,axiom,
    ! [P: fun_set_complex_o,A2: set_complex] :
      ( ! [A5: set_complex] :
          ( ~ aa_set_complex_o(finite3207457112153483333omplex,A5)
         => aa_set_complex_o(P,A5) )
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X3: complex,F4: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,F4)
             => ( ~ aa_set_complex_o(member_complex(X3),F4)
               => ( aa_set_complex_o(P,F4)
                 => aa_set_complex_o(P,insert_complex(X3,F4)) ) ) )
         => aa_set_complex_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_4674_infinite__finite__induct,axiom,
    ! [P: fun_se4782206882984618170_nat_o,A2: set_Pr1261947904930325089at_nat] :
      ( ! [A5: set_Pr1261947904930325089at_nat] :
          ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A5)
         => aa_set3702741264012733747_nat_o(P,A5) )
     => ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
       => ( ! [X3: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
             => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),F4)
               => ( aa_set3702741264012733747_nat_o(P,F4)
                 => aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(X3,F4)) ) ) )
         => aa_set3702741264012733747_nat_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_4675_infinite__finite__induct,axiom,
    ! [P: fun_set_real_o,A2: set_real] :
      ( ! [A5: set_real] :
          ( ~ aa_set_real_o(finite_finite_real,A5)
         => aa_set_real_o(P,A5) )
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X3: real,F4: set_real] :
              ( aa_set_real_o(finite_finite_real,F4)
             => ( ~ aa_set_real_o(member_real(X3),F4)
               => ( aa_set_real_o(P,F4)
                 => aa_set_real_o(P,insert_real(X3,F4)) ) ) )
         => aa_set_real_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_4676_infinite__finite__induct,axiom,
    ! [P: fun_set_nat_o,A2: set_nat] :
      ( ! [A5: set_nat] :
          ( ~ aa_set_nat_o(finite_finite_nat,A5)
         => aa_set_nat_o(P,A5) )
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X3: nat,F4: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,F4)
             => ( ~ aa_set_nat_o(member_nat(X3),F4)
               => ( aa_set_nat_o(P,F4)
                 => aa_set_nat_o(P,insert_nat(X3,F4)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_4677_infinite__finite__induct,axiom,
    ! [P: fun_set_int_o,A2: set_int] :
      ( ! [A5: set_int] :
          ( ~ aa_set_int_o(finite_finite_int,A5)
         => aa_set_int_o(P,A5) )
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X3: int,F4: set_int] :
              ( aa_set_int_o(finite_finite_int,F4)
             => ( ~ aa_set_int_o(member_int(X3),F4)
               => ( aa_set_int_o(P,F4)
                 => aa_set_int_o(P,insert_int(X3,F4)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_4678_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(K),L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_4679_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,L),aa_int_int(modulo_modulo_int(K),L)) ) ).

% neg_mod_bound
tff(fact_4680_subset__singleton__iff,axiom,
    ! [X5: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X5),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))
    <=> ( ( X5 = bot_bo228742789529271731at_nat )
        | ( X5 = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_4681_subset__singleton__iff,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(X5),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))
    <=> ( ( X5 = bot_bo2099793752762293965at_nat )
        | ( X5 = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_4682_subset__singleton__iff,axiom,
    ! [X5: set_real,A: real] :
      ( aa_set_real_o(ord_less_eq_set_real(X5),insert_real(A,bot_bot_set_real))
    <=> ( ( X5 = bot_bot_set_real )
        | ( X5 = insert_real(A,bot_bot_set_real) ) ) ) ).

% subset_singleton_iff
tff(fact_4683_subset__singleton__iff,axiom,
    ! [X5: set_nat,A: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X5),insert_nat(A,bot_bot_set_nat))
    <=> ( ( X5 = bot_bot_set_nat )
        | ( X5 = insert_nat(A,bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_4684_subset__singleton__iff,axiom,
    ! [X5: set_int,A: int] :
      ( aa_set_int_o(ord_less_eq_set_int(X5),insert_int(A,bot_bot_set_int))
    <=> ( ( X5 = bot_bot_set_int )
        | ( X5 = insert_int(A,bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_4685_subset__singletonD,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))
     => ( ( A2 = bot_bo228742789529271731at_nat )
        | ( A2 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ) ).

% subset_singletonD
tff(fact_4686_subset__singletonD,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))
     => ( ( A2 = bot_bo2099793752762293965at_nat )
        | ( A2 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ) ).

% subset_singletonD
tff(fact_4687_subset__singletonD,axiom,
    ! [A2: set_real,X: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),insert_real(X,bot_bot_set_real))
     => ( ( A2 = bot_bot_set_real )
        | ( A2 = insert_real(X,bot_bot_set_real) ) ) ) ).

% subset_singletonD
tff(fact_4688_subset__singletonD,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat(X,bot_bot_set_nat))
     => ( ( A2 = bot_bot_set_nat )
        | ( A2 = insert_nat(X,bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_4689_subset__singletonD,axiom,
    ! [A2: set_int,X: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),insert_int(X,bot_bot_set_int))
     => ( ( A2 = bot_bot_set_int )
        | ( A2 = insert_int(X,bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_4690_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))) ).

% dvd_minus_mod
tff(fact_4691_dvd__minus__mod,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(modulo_modulo_int(A),B))) ).

% dvd_minus_mod
tff(fact_4692_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B))) ).

% dvd_minus_mod
tff(fact_4693_diff__less__Suc,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)),aa_nat_nat(suc,Mb)) ).

% diff_less_Suc
tff(fact_4694_Suc__diff__Suc,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na) ) ) ).

% Suc_diff_Suc
tff(fact_4695_diff__less,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)),Mb) ) ) ).

% diff_less
tff(fact_4696_Suc__diff__le,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)) ) ) ).

% Suc_diff_le
tff(fact_4697_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_4698_less__diff__iff,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ) ) ).

% less_diff_iff
tff(fact_4699_diff__add__0,axiom,
    ! [Na: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Mb)) = zero_zero_nat ).

% diff_add_0
tff(fact_4700_add__diff__inverse__nat,axiom,
    ! [Mb: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)) = Mb ) ) ).

% add_diff_inverse_nat
tff(fact_4701_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ).

% less_diff_conv
tff(fact_4702_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) = K )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_4703_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_4704_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_4705_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_4706_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ).

% le_diff_conv
tff(fact_4707_diff__Suc__eq__diff__pred,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Na) ).

% diff_Suc_eq_diff_pred
tff(fact_4708_mod__geq,axiom,
    ! [Mb: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)),Na) ) ) ).

% mod_geq
tff(fact_4709_mod__if,axiom,
    ! [Mb: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na),Mb,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)),Na)) ).

% mod_if
tff(fact_4710_le__mod__geq,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)),Na) ) ) ).

% le_mod_geq
tff(fact_4711_mod__eq__dvd__iff__nat,axiom,
    ! [Na: nat,Mb: nat,Q4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),Q4) )
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_4712_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4713_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4714_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: 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),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D))
    <=> aa_int_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)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4715_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E2)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4716_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4717_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: 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),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D))
    <=> aa_int_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)),E2)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4718_less__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D)) ) ).

% less_add_iff2
tff(fact_4719_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D)) ) ).

% less_add_iff2
tff(fact_4720_less__add__iff2,axiom,
    ! [A: int,E2: 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),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D))
    <=> aa_int_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)),E2)),D)) ) ).

% less_add_iff2
tff(fact_4721_less__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E2)),C)),D) ) ).

% less_add_iff1
tff(fact_4722_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C)),D) ) ).

% less_add_iff1
tff(fact_4723_less__add__iff1,axiom,
    ! [A: int,E2: 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),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D))
    <=> aa_int_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)),E2)),C)),D) ) ).

% less_add_iff1
tff(fact_4724_divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(X),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4725_divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4726_divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(X),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4727_diff__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(X),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4728_diff__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4729_diff__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(Y),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4730_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(W2),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Z)),aa_complex_complex(times_times_complex(W2),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4731_diff__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4732_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W2),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(W2),Y))),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4733_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: complex,B: complex,Z: complex] :
      aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = $ite(Z = zero_zero_complex,A,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_4734_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_4735_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      aa_rat_rat(minus_minus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_4736_square__diff__one__factored,axiom,
    ! [X: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),X)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(X),one_one_complex)),aa_complex_complex(minus_minus_complex(X),one_one_complex)) ).

% square_diff_one_factored
tff(fact_4737_square__diff__one__factored,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)) ).

% square_diff_one_factored
tff(fact_4738_square__diff__one__factored,axiom,
    ! [X: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),X)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(X),one_one_rat)),aa_rat_rat(minus_minus_rat(X),one_one_rat)) ).

% square_diff_one_factored
tff(fact_4739_square__diff__one__factored,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)) ).

% square_diff_one_factored
tff(fact_4740_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: fun_set_complex_o,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X3: complex,S5: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S5)
             => ( ! [Y4: complex] :
                    ( aa_set_complex_o(member_complex(Y4),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,Y4)),aa_complex_rat(F,X3)) )
               => ( aa_set_complex_o(P,S5)
                 => aa_set_complex_o(P,insert_complex(X3,S5)) ) ) )
         => aa_set_complex_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_4741_finite__ranking__induct,axiom,
    ! [S3: set_real,P: fun_set_real_o,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X3: real,S5: set_real] :
              ( aa_set_real_o(finite_finite_real,S5)
             => ( ! [Y4: real] :
                    ( aa_set_real_o(member_real(Y4),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,Y4)),aa_real_rat(F,X3)) )
               => ( aa_set_real_o(P,S5)
                 => aa_set_real_o(P,insert_real(X3,S5)) ) ) )
         => aa_set_real_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_4742_finite__ranking__induct,axiom,
    ! [S3: set_nat,P: fun_set_nat_o,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X3: nat,S5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S5)
             => ( ! [Y4: nat] :
                    ( aa_set_nat_o(member_nat(Y4),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Y4)),aa_nat_rat(F,X3)) )
               => ( aa_set_nat_o(P,S5)
                 => aa_set_nat_o(P,insert_nat(X3,S5)) ) ) )
         => aa_set_nat_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_4743_finite__ranking__induct,axiom,
    ! [S3: set_int,P: fun_set_int_o,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X3: int,S5: set_int] :
              ( aa_set_int_o(finite_finite_int,S5)
             => ( ! [Y4: int] :
                    ( aa_set_int_o(member_int(Y4),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,Y4)),aa_int_rat(F,X3)) )
               => ( aa_set_int_o(P,S5)
                 => aa_set_int_o(P,insert_int(X3,S5)) ) ) )
         => aa_set_int_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_4744_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: fun_set_complex_o,F: fun_complex_num] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X3: complex,S5: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S5)
             => ( ! [Y4: complex] :
                    ( aa_set_complex_o(member_complex(Y4),S5)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_complex_num(F,Y4)),aa_complex_num(F,X3)) )
               => ( aa_set_complex_o(P,S5)
                 => aa_set_complex_o(P,insert_complex(X3,S5)) ) ) )
         => aa_set_complex_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_4745_finite__ranking__induct,axiom,
    ! [S3: set_real,P: fun_set_real_o,F: fun_real_num] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X3: real,S5: set_real] :
              ( aa_set_real_o(finite_finite_real,S5)
             => ( ! [Y4: real] :
                    ( aa_set_real_o(member_real(Y4),S5)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_real_num(F,Y4)),aa_real_num(F,X3)) )
               => ( aa_set_real_o(P,S5)
                 => aa_set_real_o(P,insert_real(X3,S5)) ) ) )
         => aa_set_real_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_4746_finite__ranking__induct,axiom,
    ! [S3: set_nat,P: fun_set_nat_o,F: fun_nat_num] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X3: nat,S5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S5)
             => ( ! [Y4: nat] :
                    ( aa_set_nat_o(member_nat(Y4),S5)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Y4)),aa_nat_num(F,X3)) )
               => ( aa_set_nat_o(P,S5)
                 => aa_set_nat_o(P,insert_nat(X3,S5)) ) ) )
         => aa_set_nat_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_4747_finite__ranking__induct,axiom,
    ! [S3: set_int,P: fun_set_int_o,F: fun_int_num] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X3: int,S5: set_int] :
              ( aa_set_int_o(finite_finite_int,S5)
             => ( ! [Y4: int] :
                    ( aa_set_int_o(member_int(Y4),S5)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,Y4)),aa_int_num(F,X3)) )
               => ( aa_set_int_o(P,S5)
                 => aa_set_int_o(P,insert_int(X3,S5)) ) ) )
         => aa_set_int_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_4748_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: fun_set_complex_o,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X3: complex,S5: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S5)
             => ( ! [Y4: complex] :
                    ( aa_set_complex_o(member_complex(Y4),S5)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,Y4)),aa_complex_nat(F,X3)) )
               => ( aa_set_complex_o(P,S5)
                 => aa_set_complex_o(P,insert_complex(X3,S5)) ) ) )
         => aa_set_complex_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_4749_finite__ranking__induct,axiom,
    ! [S3: set_real,P: fun_set_real_o,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X3: real,S5: set_real] :
              ( aa_set_real_o(finite_finite_real,S5)
             => ( ! [Y4: real] :
                    ( aa_set_real_o(member_real(Y4),S5)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,Y4)),aa_real_nat(F,X3)) )
               => ( aa_set_real_o(P,S5)
                 => aa_set_real_o(P,insert_real(X3,S5)) ) ) )
         => aa_set_real_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_4750_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [B3: real,A5: set_real] :
              ( aa_set_real_o(finite_finite_real,A5)
             => ( ! [X2: real] :
                    ( aa_set_real_o(member_real(X2),A5)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,B3),X2) )
               => ( aa_set_real_o(P,A5)
                 => aa_set_real_o(P,insert_real(B3,A5)) ) ) )
         => aa_set_real_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4751_finite__linorder__min__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B3: rat,A5: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A5)
             => ( ! [X2: rat] :
                    ( aa_set_rat_o(member_rat(X2),A5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B3),X2) )
               => ( aa_set_rat_o(P,A5)
                 => aa_set_rat_o(P,insert_rat(B3,A5)) ) ) )
         => aa_set_rat_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4752_finite__linorder__min__induct,axiom,
    ! [A2: set_num,P: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( aa_set_num_o(P,bot_bot_set_num)
       => ( ! [B3: num,A5: set_num] :
              ( aa_set_num_o(finite_finite_num,A5)
             => ( ! [X2: num] :
                    ( aa_set_num_o(member_num(X2),A5)
                   => aa_num_o(aa_num_fun_num_o(ord_less_num,B3),X2) )
               => ( aa_set_num_o(P,A5)
                 => aa_set_num_o(P,insert_num(B3,A5)) ) ) )
         => aa_set_num_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4753_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B3: nat,A5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A5)
             => ( ! [X2: nat] :
                    ( aa_set_nat_o(member_nat(X2),A5)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B3),X2) )
               => ( aa_set_nat_o(P,A5)
                 => aa_set_nat_o(P,insert_nat(B3,A5)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4754_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B3: int,A5: set_int] :
              ( aa_set_int_o(finite_finite_int,A5)
             => ( ! [X2: int] :
                    ( aa_set_int_o(member_int(X2),A5)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,B3),X2) )
               => ( aa_set_int_o(P,A5)
                 => aa_set_int_o(P,insert_int(B3,A5)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4755_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [B3: real,A5: set_real] :
              ( aa_set_real_o(finite_finite_real,A5)
             => ( ! [X2: real] :
                    ( aa_set_real_o(member_real(X2),A5)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B3) )
               => ( aa_set_real_o(P,A5)
                 => aa_set_real_o(P,insert_real(B3,A5)) ) ) )
         => aa_set_real_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4756_finite__linorder__max__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B3: rat,A5: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A5)
             => ( ! [X2: rat] :
                    ( aa_set_rat_o(member_rat(X2),A5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),B3) )
               => ( aa_set_rat_o(P,A5)
                 => aa_set_rat_o(P,insert_rat(B3,A5)) ) ) )
         => aa_set_rat_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4757_finite__linorder__max__induct,axiom,
    ! [A2: set_num,P: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( aa_set_num_o(P,bot_bot_set_num)
       => ( ! [B3: num,A5: set_num] :
              ( aa_set_num_o(finite_finite_num,A5)
             => ( ! [X2: num] :
                    ( aa_set_num_o(member_num(X2),A5)
                   => aa_num_o(aa_num_fun_num_o(ord_less_num,X2),B3) )
               => ( aa_set_num_o(P,A5)
                 => aa_set_num_o(P,insert_num(B3,A5)) ) ) )
         => aa_set_num_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4758_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B3: nat,A5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A5)
             => ( ! [X2: nat] :
                    ( aa_set_nat_o(member_nat(X2),A5)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),B3) )
               => ( aa_set_nat_o(P,A5)
                 => aa_set_nat_o(P,insert_nat(B3,A5)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4759_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B3: int,A5: set_int] :
              ( aa_set_int_o(finite_finite_int,A5)
             => ( ! [X2: int] :
                    ( aa_set_int_o(member_int(X2),A5)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,X2),B3) )
               => ( aa_set_int_o(P,A5)
                 => aa_set_int_o(P,insert_int(B3,A5)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4760_inf__period_I4_J,axiom,
    ! [D: code_integer,D5: code_integer,T: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D5)
     => ! [X2: code_integer,K4: code_integer] :
          ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,T))
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X2,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D5)),T)) ) ) ).

% inf_period(4)
tff(fact_4761_inf__period_I4_J,axiom,
    ! [D: complex,D5: complex,T: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,D),D5)
     => ! [X2: complex,K4: complex] :
          ( ~ aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,D),aa_complex_complex(plus_plus_complex(X2),T))
        <=> ~ aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,D),aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(X2),aa_complex_complex(times_times_complex(K4),D5))),T)) ) ) ).

% inf_period(4)
tff(fact_4762_inf__period_I4_J,axiom,
    ! [D: real,D5: real,T: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D5)
     => ! [X2: real,K4: real] :
          ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),T))
        <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D5))),T)) ) ) ).

% inf_period(4)
tff(fact_4763_inf__period_I4_J,axiom,
    ! [D: rat,D5: rat,T: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D5)
     => ! [X2: rat,K4: rat] :
          ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),T))
        <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K4),D5))),T)) ) ) ).

% inf_period(4)
tff(fact_4764_inf__period_I4_J,axiom,
    ! [D: int,D5: int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D5)
     => ! [X2: 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,X2),T))
        <=> ~ 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,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5))),T)) ) ) ).

% inf_period(4)
tff(fact_4765_inf__period_I3_J,axiom,
    ! [D: code_integer,D5: code_integer,T: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D5)
     => ! [X2: code_integer,K4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,T))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X2,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D5)),T)) ) ) ).

% inf_period(3)
tff(fact_4766_inf__period_I3_J,axiom,
    ! [D: complex,D5: complex,T: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,D),D5)
     => ! [X2: complex,K4: complex] :
          ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,D),aa_complex_complex(plus_plus_complex(X2),T))
        <=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,D),aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(X2),aa_complex_complex(times_times_complex(K4),D5))),T)) ) ) ).

% inf_period(3)
tff(fact_4767_inf__period_I3_J,axiom,
    ! [D: real,D5: real,T: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D5)
     => ! [X2: real,K4: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),T))
        <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D5))),T)) ) ) ).

% inf_period(3)
tff(fact_4768_inf__period_I3_J,axiom,
    ! [D: rat,D5: rat,T: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D5)
     => ! [X2: rat,K4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),T))
        <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K4),D5))),T)) ) ) ).

% inf_period(3)
tff(fact_4769_inf__period_I3_J,axiom,
    ! [D: int,D5: int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D5)
     => ! [X2: 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,X2),T))
        <=> 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,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5))),T)) ) ) ).

% inf_period(3)
tff(fact_4770_finite__subset__induct,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F3)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(F3),A2)
       => ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
         => ( ! [A3: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
                ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
               => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),A2)
                 => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),F4)
                   => ( aa_set5844364881195198285_nat_o(P,F4)
                     => aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(A3,F4)) ) ) ) )
           => aa_set5844364881195198285_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_4771_finite__subset__induct,axiom,
    ! [F3: set_set_nat,A2: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F3),A2)
       => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
         => ( ! [A3: set_nat,F4: set_set_nat] :
                ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
               => ( aa_set_set_nat_o(member_set_nat(A3),A2)
                 => ( ~ aa_set_set_nat_o(member_set_nat(A3),F4)
                   => ( aa_set_set_nat_o(P,F4)
                     => aa_set_set_nat_o(P,insert_set_nat(A3,F4)) ) ) ) )
           => aa_set_set_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_4772_finite__subset__induct,axiom,
    ! [F3: set_complex,A2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(F3),A2)
       => ( aa_set_complex_o(P,bot_bot_set_complex)
         => ( ! [A3: complex,F4: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F4)
               => ( aa_set_complex_o(member_complex(A3),A2)
                 => ( ~ aa_set_complex_o(member_complex(A3),F4)
                   => ( aa_set_complex_o(P,F4)
                     => aa_set_complex_o(P,insert_complex(A3,F4)) ) ) ) )
           => aa_set_complex_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_4773_finite__subset__induct,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
     => ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(F3),A2)
       => ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
         => ( ! [A3: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
               => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),A2)
                 => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),F4)
                   => ( aa_set3702741264012733747_nat_o(P,F4)
                     => aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(A3,F4)) ) ) ) )
           => aa_set3702741264012733747_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_4774_finite__subset__induct,axiom,
    ! [F3: set_real,A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F3)
     => ( aa_set_real_o(ord_less_eq_set_real(F3),A2)
       => ( aa_set_real_o(P,bot_bot_set_real)
         => ( ! [A3: real,F4: set_real] :
                ( aa_set_real_o(finite_finite_real,F4)
               => ( aa_set_real_o(member_real(A3),A2)
                 => ( ~ aa_set_real_o(member_real(A3),F4)
                   => ( aa_set_real_o(P,F4)
                     => aa_set_real_o(P,insert_real(A3,F4)) ) ) ) )
           => aa_set_real_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_4775_finite__subset__induct,axiom,
    ! [F3: set_nat,A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F3),A2)
       => ( aa_set_nat_o(P,bot_bot_set_nat)
         => ( ! [A3: nat,F4: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F4)
               => ( aa_set_nat_o(member_nat(A3),A2)
                 => ( ~ aa_set_nat_o(member_nat(A3),F4)
                   => ( aa_set_nat_o(P,F4)
                     => aa_set_nat_o(P,insert_nat(A3,F4)) ) ) ) )
           => aa_set_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_4776_finite__subset__induct,axiom,
    ! [F3: set_int,A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( aa_set_int_o(ord_less_eq_set_int(F3),A2)
       => ( aa_set_int_o(P,bot_bot_set_int)
         => ( ! [A3: int,F4: set_int] :
                ( aa_set_int_o(finite_finite_int,F4)
               => ( aa_set_int_o(member_int(A3),A2)
                 => ( ~ aa_set_int_o(member_int(A3),F4)
                   => ( aa_set_int_o(P,F4)
                     => aa_set_int_o(P,insert_int(A3,F4)) ) ) ) )
           => aa_set_int_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_4777_finite__subset__induct_H,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F3)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(F3),A2)
       => ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
         => ( ! [A3: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
                ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
               => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),A2)
                 => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(F4),A2)
                   => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),F4)
                     => ( aa_set5844364881195198285_nat_o(P,F4)
                       => aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(A3,F4)) ) ) ) ) )
           => aa_set5844364881195198285_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4778_finite__subset__induct_H,axiom,
    ! [F3: set_set_nat,A2: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F3),A2)
       => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
         => ( ! [A3: set_nat,F4: set_set_nat] :
                ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
               => ( aa_set_set_nat_o(member_set_nat(A3),A2)
                 => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F4),A2)
                   => ( ~ aa_set_set_nat_o(member_set_nat(A3),F4)
                     => ( aa_set_set_nat_o(P,F4)
                       => aa_set_set_nat_o(P,insert_set_nat(A3,F4)) ) ) ) ) )
           => aa_set_set_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4779_finite__subset__induct_H,axiom,
    ! [F3: set_complex,A2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(F3),A2)
       => ( aa_set_complex_o(P,bot_bot_set_complex)
         => ( ! [A3: complex,F4: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F4)
               => ( aa_set_complex_o(member_complex(A3),A2)
                 => ( aa_set_complex_o(ord_le211207098394363844omplex(F4),A2)
                   => ( ~ aa_set_complex_o(member_complex(A3),F4)
                     => ( aa_set_complex_o(P,F4)
                       => aa_set_complex_o(P,insert_complex(A3,F4)) ) ) ) ) )
           => aa_set_complex_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4780_finite__subset__induct_H,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
     => ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(F3),A2)
       => ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
         => ( ! [A3: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
               => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),A2)
                 => ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(F4),A2)
                   => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),F4)
                     => ( aa_set3702741264012733747_nat_o(P,F4)
                       => aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(A3,F4)) ) ) ) ) )
           => aa_set3702741264012733747_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4781_finite__subset__induct_H,axiom,
    ! [F3: set_real,A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F3)
     => ( aa_set_real_o(ord_less_eq_set_real(F3),A2)
       => ( aa_set_real_o(P,bot_bot_set_real)
         => ( ! [A3: real,F4: set_real] :
                ( aa_set_real_o(finite_finite_real,F4)
               => ( aa_set_real_o(member_real(A3),A2)
                 => ( aa_set_real_o(ord_less_eq_set_real(F4),A2)
                   => ( ~ aa_set_real_o(member_real(A3),F4)
                     => ( aa_set_real_o(P,F4)
                       => aa_set_real_o(P,insert_real(A3,F4)) ) ) ) ) )
           => aa_set_real_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4782_finite__subset__induct_H,axiom,
    ! [F3: set_nat,A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F3),A2)
       => ( aa_set_nat_o(P,bot_bot_set_nat)
         => ( ! [A3: nat,F4: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F4)
               => ( aa_set_nat_o(member_nat(A3),A2)
                 => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F4),A2)
                   => ( ~ aa_set_nat_o(member_nat(A3),F4)
                     => ( aa_set_nat_o(P,F4)
                       => aa_set_nat_o(P,insert_nat(A3,F4)) ) ) ) ) )
           => aa_set_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4783_finite__subset__induct_H,axiom,
    ! [F3: set_int,A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( aa_set_int_o(ord_less_eq_set_int(F3),A2)
       => ( aa_set_int_o(P,bot_bot_set_int)
         => ( ! [A3: int,F4: set_int] :
                ( aa_set_int_o(finite_finite_int,F4)
               => ( aa_set_int_o(member_int(A3),A2)
                 => ( aa_set_int_o(ord_less_eq_set_int(F4),A2)
                   => ( ~ aa_set_int_o(member_int(A3),F4)
                     => ( aa_set_int_o(P,F4)
                       => aa_set_int_o(P,insert_int(A3,F4)) ) ) ) ) )
           => aa_set_int_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4784_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% minus_mult_div_eq_mod
tff(fact_4785_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(divide_divide_int(A),B))) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_mult_div_eq_mod
tff(fact_4786_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = modulo364778990260209775nteger(A,B) ).

% minus_mult_div_eq_mod
tff(fact_4787_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_4788_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(modulo_modulo_int(A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(divide_divide_int(A),B)) ).

% minus_mod_eq_mult_div
tff(fact_4789_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)) ).

% minus_mod_eq_mult_div
tff(fact_4790_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4791_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(modulo_modulo_int(A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4792_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4793_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% minus_div_mult_eq_mod
tff(fact_4794_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),B)) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_div_mult_eq_mod
tff(fact_4795_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = modulo364778990260209775nteger(A,B) ).

% minus_div_mult_eq_mod
tff(fact_4796_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q4: int] :
      ( ( L != zero_zero_int )
     => ( ( K = aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L) )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_4797_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K),L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_4798_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(K),L)),zero_zero_int) ) ).

% neg_mod_sign
tff(fact_4799_neg__mod__conj,axiom,
    ! [B: int,A: 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_eq_int,aa_int_int(modulo_modulo_int(A),B)),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% neg_mod_conj
tff(fact_4800_pos__mod__conj,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(modulo_modulo_int(A),B))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A),B)),B) ) ) ).

% pos_mod_conj
tff(fact_4801_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(modulo_modulo_int(I),K) = I )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_4802_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( ( L = zero_zero_int )
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L) ) ) ).

% mod_int_pos_iff
tff(fact_4803_diff__Suc__less,axiom,
    ! [Na: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,I))),Na) ) ).

% diff_Suc_less
tff(fact_4804_zdiv__mono__strict,axiom,
    ! [A2: int,B2: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => ( ( aa_int_int(modulo_modulo_int(A2),Na) = zero_zero_int )
         => ( ( aa_int_int(modulo_modulo_int(B2),Na) = zero_zero_int )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(A2),Na)),aa_int_int(divide_divide_int(B2),Na)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_4805_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) )
          | ? [D6: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D6) )
              & ~ aa_nat_o(P,D6) ) ) ) ).

% nat_diff_split_asm
tff(fact_4806_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) )
        & ! [D6: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D6) )
           => aa_nat_o(P,D6) ) ) ) ).

% nat_diff_split
tff(fact_4807_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ) ).

% less_diff_conv2
tff(fact_4808_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),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)),Na)) = 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)),Na)) ) ) ).

% nat_diff_add_eq2
tff(fact_4809_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),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)),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb)),Na) ) ) ).

% nat_diff_add_eq1
tff(fact_4810_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),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)),Na))
      <=> 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)),Na)) ) ) ).

% nat_le_add_iff2
tff(fact_4811_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),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)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb)),Na) ) ) ).

% nat_le_add_iff1
tff(fact_4812_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),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)),Na) )
      <=> ( 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)),Na) ) ) ) ).

% nat_eq_add_iff2
tff(fact_4813_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),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)),Na) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb) = Na ) ) ) ).

% nat_eq_add_iff1
tff(fact_4814_dvd__minus__add,axiom,
    ! [Q4: nat,Na: nat,R2: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q4),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q4),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,Na),Q4))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),Mb)),Q4))) ) ) ) ).

% dvd_minus_add
tff(fact_4815_set__update__subset__insert,axiom,
    ! [Xs: list_P6011104703257516679at_nat,I: nat,X: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,I,X))),insert8211810215607154385at_nat(X,set_Pr5648618587558075414at_nat(Xs))) ).

% set_update_subset_insert
tff(fact_4816_set__update__subset__insert,axiom,
    ! [Xs: list_real,I: nat,X: real] : aa_set_real_o(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,X))),insert_real(X,set_real2(Xs))) ).

% set_update_subset_insert
tff(fact_4817_set__update__subset__insert,axiom,
    ! [Xs: list_P5464809261938338413at_nat,I: nat,X: produc3843707927480180839at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(set_Pr3765526544606949372at_nat(list_u4696772448584712917at_nat(Xs,I,X))),insert9069300056098147895at_nat(X,set_Pr3765526544606949372at_nat(Xs))) ).

% set_update_subset_insert
tff(fact_4818_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,X)),insert_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))) ).

% set_update_subset_insert
tff(fact_4819_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,X: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(list_update_nat(Xs,I,X))),insert_nat(X,set_nat2(Xs))) ).

% set_update_subset_insert
tff(fact_4820_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,X: int] : aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X))),insert_int(X,set_int2(Xs))) ).

% set_update_subset_insert
tff(fact_4821_mod__nat__eqI,axiom,
    ! [R2: nat,Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R2),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Mb)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),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),Na) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_4822_exp__div__exp__eq,axiom,
    ! [Mb: nat,Na: nat] :
      divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
          zero_n356916108424825756nteger(( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb) != zero_z3403309356797280102nteger )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb) ))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na))) ).

% exp_div_exp_eq
tff(fact_4823_exp__div__exp__eq,axiom,
    ! [Mb: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != zero_zero_nat )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),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),Na))) ).

% exp_div_exp_eq
tff(fact_4824_exp__div__exp__eq,axiom,
    ! [Mb: nat,Na: nat] :
      aa_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))),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb) != zero_zero_int )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),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),Na))) ).

% exp_div_exp_eq
tff(fact_4825_frac__le__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real) ) ) ) ).

% frac_le_eq
tff(fact_4826_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W2),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(W2),Y))),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_4827_frac__less__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real) ) ) ) ).

% frac_less_eq
tff(fact_4828_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W2),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(W2),Y))),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_4829_signed__take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Na),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% signed_take_bit_int_less_exp
tff(fact_4830_power__diff,axiom,
    ! [A: complex,Na: nat,Mb: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
       => ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Na)) ) ) ) ).

% power_diff
tff(fact_4831_power__diff,axiom,
    ! [A: nat,Na: nat,Mb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),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),Na)) = 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),Na)) ) ) ) ).

% power_diff
tff(fact_4832_power__diff,axiom,
    ! [A: int,Na: nat,Mb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
       => ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)) = aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Na)) ) ) ) ).

% power_diff
tff(fact_4833_power__diff,axiom,
    ! [A: real,Na: nat,Mb: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ) ).

% power_diff
tff(fact_4834_power__diff,axiom,
    ! [A: rat,Na: nat,Mb: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Na)) ) ) ) ).

% power_diff
tff(fact_4835_power__diff,axiom,
    ! [A: code_integer,Na: nat,Mb: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)) = divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Mb),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ) ) ) ).

% power_diff
tff(fact_4836_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int)
       => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_4837_Suc__diff__eq__diff__pred,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_4838_Suc__pred_H,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( Na = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_4839_div__geq,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)),Na)) ) ) ) ).

% div_geq
tff(fact_4840_div__if,axiom,
    ! [Mb: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
        | ( Na = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)),Na)) ) ).

% div_if
tff(fact_4841_set__replicate__Suc,axiom,
    ! [Na: nat,X: produc3843707927480180839at_nat] : set_Pr3765526544606949372at_nat(replic2264142908078655527at_nat(aa_nat_nat(suc,Na),X)) = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ).

% set_replicate_Suc
tff(fact_4842_set__replicate__Suc,axiom,
    ! [Na: nat,X: vEBT_VEBT] : set_VEBT_VEBT2(replicate_VEBT_VEBT(aa_nat_nat(suc,Na),X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ).

% set_replicate_Suc
tff(fact_4843_set__replicate__Suc,axiom,
    ! [Na: nat,X: product_prod_nat_nat] : set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(aa_nat_nat(suc,Na),X)) = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ).

% set_replicate_Suc
tff(fact_4844_set__replicate__Suc,axiom,
    ! [Na: nat,X: real] : set_real2(replicate_real(aa_nat_nat(suc,Na),X)) = insert_real(X,bot_bot_set_real) ).

% set_replicate_Suc
tff(fact_4845_set__replicate__Suc,axiom,
    ! [Na: nat,X: nat] : set_nat2(replicate_nat(aa_nat_nat(suc,Na),X)) = insert_nat(X,bot_bot_set_nat) ).

% set_replicate_Suc
tff(fact_4846_set__replicate__Suc,axiom,
    ! [Na: nat,X: int] : set_int2(replicate_int(aa_nat_nat(suc,Na),X)) = insert_int(X,bot_bot_set_int) ).

% set_replicate_Suc
tff(fact_4847_add__eq__if,axiom,
    ! [Mb: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na) = $ite(Mb = zero_zero_nat,Na,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Na))) ).

% add_eq_if
tff(fact_4848_set__replicate__conv__if,axiom,
    ! [Na: nat,X: produc3843707927480180839at_nat] :
      set_Pr3765526544606949372at_nat(replic2264142908078655527at_nat(Na,X)) = $ite(Na = zero_zero_nat,bot_bo228742789529271731at_nat,insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) ).

% set_replicate_conv_if
tff(fact_4849_set__replicate__conv__if,axiom,
    ! [Na: nat,X: vEBT_VEBT] :
      set_VEBT_VEBT2(replicate_VEBT_VEBT(Na,X)) = $ite(Na = zero_zero_nat,bot_bo8194388402131092736T_VEBT,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)) ).

% set_replicate_conv_if
tff(fact_4850_set__replicate__conv__if,axiom,
    ! [Na: nat,X: product_prod_nat_nat] :
      set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(Na,X)) = $ite(Na = zero_zero_nat,bot_bo2099793752762293965at_nat,insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) ).

% set_replicate_conv_if
tff(fact_4851_set__replicate__conv__if,axiom,
    ! [Na: nat,X: real] :
      set_real2(replicate_real(Na,X)) = $ite(Na = zero_zero_nat,bot_bot_set_real,insert_real(X,bot_bot_set_real)) ).

% set_replicate_conv_if
tff(fact_4852_set__replicate__conv__if,axiom,
    ! [Na: nat,X: nat] :
      set_nat2(replicate_nat(Na,X)) = $ite(Na = zero_zero_nat,bot_bot_set_nat,insert_nat(X,bot_bot_set_nat)) ).

% set_replicate_conv_if
tff(fact_4853_set__replicate__conv__if,axiom,
    ! [Na: nat,X: int] :
      set_int2(replicate_int(Na,X)) = $ite(Na = zero_zero_nat,bot_bot_set_int,insert_int(X,bot_bot_set_int)) ).

% set_replicate_conv_if
tff(fact_4854_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),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)),Na))
      <=> 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)),Na)) ) ) ).

% nat_less_add_iff2
tff(fact_4855_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),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)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb)),Na) ) ) ).

% nat_less_add_iff1
tff(fact_4856_mult__eq__if,axiom,
    ! [Mb: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na) = $ite(Mb = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Na))) ).

% mult_eq_if
tff(fact_4857_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),V)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),S)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,V),U))),S))),V) ) ) ) ).

% scaling_mono
tff(fact_4858_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),V)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),R2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R2),S)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(U),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(R2),aa_rat_rat(minus_minus_rat(V),U))),S))),V) ) ) ) ).

% scaling_mono
tff(fact_4859_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Na: 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))),Na) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Mb)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4860_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Na: nat,Mb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Mb)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4861_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_ri631733984087533419it_int(Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_4862_signed__take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Na),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_4863_power__diff__power__eq,axiom,
    ! [A: nat,Mb: nat,Na: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),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),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Mb)))) ) ) ).

% power_diff_power_eq
tff(fact_4864_power__diff__power__eq,axiom,
    ! [A: int,Mb: nat,Na: nat] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)),aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Mb)))) ) ) ).

% power_diff_power_eq
tff(fact_4865_power__diff__power__eq,axiom,
    ! [A: code_integer,Mb: nat,Na: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Mb),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)),divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Mb)))) ) ) ).

% power_diff_power_eq
tff(fact_4866_split__zmod,axiom,
    ! [P: fun_int_o,Na: int,K: int] :
      ( aa_int_o(P,aa_int_int(modulo_modulo_int(Na),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,Na) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I2: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J3) ) )
             => aa_int_o(P,J3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I2: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J3) ) )
             => aa_int_o(P,J3) ) ) ) ) ).

% split_zmod
tff(fact_4867_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_int(modulo_modulo_int(A),B) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_4868_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2) )
     => ( 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),B)
         => ( aa_int_int(modulo_modulo_int(A),B) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_4869_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_4870_power__eq__if,axiom,
    ! [P2: real,Mb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),Mb) = $ite(Mb = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P2),aa_nat_real(aa_real_fun_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_4871_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(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_4872_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_4873_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_4874_power__minus__mult,axiom,
    ! [Na: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),Na) ) ) ).

% power_minus_mult
tff(fact_4875_power__minus__mult,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) ) ) ).

% power_minus_mult
tff(fact_4876_power__minus__mult,axiom,
    ! [Na: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),Na) ) ) ).

% power_minus_mult
tff(fact_4877_power__minus__mult,axiom,
    ! [Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) ) ) ).

% power_minus_mult
tff(fact_4878_power__minus__mult,axiom,
    ! [Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_int(power_power_int(A),Na) ) ) ).

% power_minus_mult
tff(fact_4879_diff__le__diff__pow,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Na))) ) ).

% diff_le_diff_pow
tff(fact_4880_le__div__geq,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)),Na)) ) ) ) ).

% le_div_geq
tff(fact_4881_bits__induct,axiom,
    ! [P: fun_Code_integer_o,A: code_integer] :
      ( ! [A3: code_integer] :
          ( ( divide6298287555418463151nteger(A3,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(zero_n356916108424825756nteger((B3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A3)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = A3 )
             => aa_Code_integer_o(P,plus_p5714425477246183910nteger(zero_n356916108424825756nteger((B3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A3))) ) )
       => aa_Code_integer_o(P,A) ) ) ).

% bits_induct
tff(fact_4882_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_4883_bits__induct,axiom,
    ! [P: fun_int_o,A: int] :
      ( ! [A3: int] :
          ( ( aa_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(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_4884_verit__le__mono__div__int,axiom,
    ! [A2: int,B2: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,
              aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A2),Na)),
                $ite(aa_int_int(modulo_modulo_int(B2),Na) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(divide_divide_int(B2),Na)) ) ) ).

% verit_le_mono_div_int
tff(fact_4885_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(divide_divide_int(Na),K)),aa_int_int(modulo_modulo_int(Na),K))
      <=> ! [I2: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
              & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I2),J3) ) ) ) ).

% split_pos_lemma
tff(fact_4886_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(divide_divide_int(Na),K)),aa_int_int(modulo_modulo_int(Na),K))
      <=> ! [I2: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
              & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I2),J3) ) ) ) ).

% split_neg_lemma
tff(fact_4887_int__power__div__base,axiom,
    ! [Mb: nat,K: 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),K)
       => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(K),Mb)),K) = aa_nat_int(power_power_int(K),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_4888_set__decode__plus__power__2,axiom,
    ! [Na: nat,Z: nat] :
      ( ~ aa_set_nat_o(member_nat(Na),nat_set_decode(Z))
     => ( nat_set_decode(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),Z)) = insert_nat(Na,nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_4889_exp__mod__exp,axiom,
    ! [Mb: nat,Na: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb)) ).

% exp_mod_exp
tff(fact_4890_exp__mod__exp,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na))),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_4891_exp__mod__exp,axiom,
    ! [Mb: nat,Na: nat] : aa_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))),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) ).

% exp_mod_exp
tff(fact_4892_power2__diff,axiom,
    ! [X: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_4893_power2__diff,axiom,
    ! [X: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_4894_power2__diff,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_4895_power2__diff,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_4896_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),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_4897_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),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_4898_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Na: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Mb)))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4899_even__mod__4__div__2,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% even_mod_4_div_2
tff(fact_4900_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R2))
    <=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q4)),R2) )
        & $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) ),
              Q4 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_4901_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,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)))
         => ( minus_8373710615458151222nteger(modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)),B) = modulo364778990260209775nteger(A,B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4902_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_4903_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(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4904_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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))),Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% even_mask_div_iff'
tff(fact_4905_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(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))),Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% even_mask_div_iff'
tff(fact_4906_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% even_mask_div_iff'
tff(fact_4907_even__mask__div__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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))),Na)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ) ).

% even_mask_div_iff
tff(fact_4908_even__mask__div__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(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))),Na)))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ) ).

% even_mask_div_iff
tff(fact_4909_even__mask__div__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ) ).

% even_mask_div_iff
tff(fact_4910_divmod__step__eq,axiom,
    ! [L: num,Q4: nat,R2: nat] :
      unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q4),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))),Q4)),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))),Q4)),R2)) ).

% divmod_step_eq
tff(fact_4911_divmod__step__eq,axiom,
    ! [L: num,Q4: int,R2: int] :
      aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),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))),Q4)),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))),Q4)),R2)) ).

% divmod_step_eq
tff(fact_4912_divmod__step__eq,axiom,
    ! [L: num,Q4: code_integer,R2: code_integer] :
      unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q4,R2)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R2),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q4),one_one_Code_integer),minus_8373710615458151222nteger(R2,numera6620942414471956472nteger(L))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q4),R2)) ).

% divmod_step_eq
tff(fact_4913_inrange,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Na)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,vEBT_VEBT_set_vebt(T)),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),one_one_nat))) ) ).

% inrange
tff(fact_4914_Divides_Oadjust__div__eq,axiom,
    ! [Q4: int,R2: int] : adjust_div(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q4),aa_o_int(zero_n2684676970156552555ol_int,R2 != zero_zero_int)) ).

% Divides.adjust_div_eq
tff(fact_4915_signed__take__bit__rec,axiom,
    ! [Na: nat,A: code_integer] :
      bit_ri6519982836138164636nteger(Na,A) = $ite(Na = zero_zero_nat,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_4916_signed__take__bit__rec,axiom,
    ! [Na: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(Na),A) = $ite(Na = zero_zero_nat,aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_4917_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: 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,Q4),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,Q4),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_4918_diff__shunt__var,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat(X,Y) = bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(X),Y) ) ).

% diff_shunt_var
tff(fact_4919_diff__shunt__var,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( minus_minus_set_real(X,Y) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(X),Y) ) ).

% diff_shunt_var
tff(fact_4920_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( minus_minus_set_nat(X,Y) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y) ) ).

% diff_shunt_var
tff(fact_4921_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( minus_minus_set_int(X,Y) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(X),Y) ) ).

% diff_shunt_var
tff(fact_4922_take__bit__rec,axiom,
    ! [Na: nat,A: code_integer] :
      bit_se1745604003318907178nteger(Na,A) = $ite(Na = 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,Na),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))),numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_4923_take__bit__rec,axiom,
    ! [Na: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A) = $ite(Na = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_4924_take__bit__rec,axiom,
    ! [Na: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Na),A) = $ite(Na = zero_zero_nat,zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_4925_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_4926_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_4927_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_4928_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).

% add.inverse_inverse
tff(fact_4929_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_4930_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_4931_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_4932_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_4933_Diff__cancel,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A2,A2) = bot_bo2099793752762293965at_nat ).

% Diff_cancel
tff(fact_4934_Diff__cancel,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,A2) = bot_bot_set_real ).

% Diff_cancel
tff(fact_4935_Diff__cancel,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,A2) = bot_bot_set_int ).

% Diff_cancel
tff(fact_4936_Diff__cancel,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(A2,A2) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_4937_empty__Diff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(bot_bo2099793752762293965at_nat,A2) = bot_bo2099793752762293965at_nat ).

% empty_Diff
tff(fact_4938_empty__Diff,axiom,
    ! [A2: set_real] : minus_minus_set_real(bot_bot_set_real,A2) = bot_bot_set_real ).

% empty_Diff
tff(fact_4939_empty__Diff,axiom,
    ! [A2: set_int] : minus_minus_set_int(bot_bot_set_int,A2) = bot_bot_set_int ).

% empty_Diff
tff(fact_4940_empty__Diff,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(bot_bot_set_nat,A2) = bot_bot_set_nat ).

% empty_Diff
tff(fact_4941_Diff__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A2,bot_bo2099793752762293965at_nat) = A2 ).

% Diff_empty
tff(fact_4942_Diff__empty,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,bot_bot_set_real) = A2 ).

% Diff_empty
tff(fact_4943_Diff__empty,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,bot_bot_set_int) = A2 ).

% Diff_empty
tff(fact_4944_Diff__empty,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(A2,bot_bot_set_nat) = A2 ).

% Diff_empty
tff(fact_4945_finite__Diff2,axiom,
    ! [B2: set_int,A2: set_int] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(finite_finite_int,minus_minus_set_int(A2,B2))
      <=> aa_set_int_o(finite_finite_int,A2) ) ) ).

% finite_Diff2
tff(fact_4946_finite__Diff2,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B2))
      <=> aa_set_complex_o(finite3207457112153483333omplex,A2) ) ) ).

% finite_Diff2
tff(fact_4947_finite__Diff2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A2,B2))
      <=> aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2) ) ) ).

% finite_Diff2
tff(fact_4948_finite__Diff2,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A2,B2))
      <=> aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% finite_Diff2
tff(fact_4949_finite__Diff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => aa_set_int_o(finite_finite_int,minus_minus_set_int(A2,B2)) ) ).

% finite_Diff
tff(fact_4950_finite__Diff,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B2)) ) ).

% finite_Diff
tff(fact_4951_finite__Diff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A2,B2)) ) ).

% finite_Diff
tff(fact_4952_finite__Diff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A2,B2)) ) ).

% finite_Diff
tff(fact_4953_Compl__subset__Compl__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(A2)),uminus1532241313380277803et_int(B2))
    <=> aa_set_int_o(ord_less_eq_set_int(B2),A2) ) ).

% Compl_subset_Compl_iff
tff(fact_4954_Compl__anti__mono,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(B2)),uminus1532241313380277803et_int(A2)) ) ).

% Compl_anti_mono
tff(fact_4955_Diff__insert0,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A2)
     => ( minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(X,B2)) = minus_3314409938677909166at_nat(A2,B2) ) ) ).

% Diff_insert0
tff(fact_4956_Diff__insert0,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A2)
     => ( minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(X,B2)) = minus_1356011639430497352at_nat(A2,B2) ) ) ).

% Diff_insert0
tff(fact_4957_Diff__insert0,axiom,
    ! [X: real,A2: set_real,B2: set_real] :
      ( ~ aa_set_real_o(member_real(X),A2)
     => ( minus_minus_set_real(A2,insert_real(X,B2)) = minus_minus_set_real(A2,B2) ) ) ).

% Diff_insert0
tff(fact_4958_Diff__insert0,axiom,
    ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(X),A2)
     => ( minus_2163939370556025621et_nat(A2,insert_set_nat(X,B2)) = minus_2163939370556025621et_nat(A2,B2) ) ) ).

% Diff_insert0
tff(fact_4959_Diff__insert0,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      ( ~ aa_set_int_o(member_int(X),A2)
     => ( minus_minus_set_int(A2,insert_int(X,B2)) = minus_minus_set_int(A2,B2) ) ) ).

% Diff_insert0
tff(fact_4960_Diff__insert0,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(X),A2)
     => ( minus_minus_set_nat(A2,insert_nat(X,B2)) = minus_minus_set_nat(A2,B2) ) ) ).

% Diff_insert0
tff(fact_4961_insert__Diff1,axiom,
    ! [X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),B2)
     => ( minus_3314409938677909166at_nat(insert9069300056098147895at_nat(X,A2),B2) = minus_3314409938677909166at_nat(A2,B2) ) ) ).

% insert_Diff1
tff(fact_4962_insert__Diff1,axiom,
    ! [X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),B2)
     => ( minus_1356011639430497352at_nat(insert8211810215607154385at_nat(X,A2),B2) = minus_1356011639430497352at_nat(A2,B2) ) ) ).

% insert_Diff1
tff(fact_4963_insert__Diff1,axiom,
    ! [X: real,B2: set_real,A2: set_real] :
      ( aa_set_real_o(member_real(X),B2)
     => ( minus_minus_set_real(insert_real(X,A2),B2) = minus_minus_set_real(A2,B2) ) ) ).

% insert_Diff1
tff(fact_4964_insert__Diff1,axiom,
    ! [X: set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(X),B2)
     => ( minus_2163939370556025621et_nat(insert_set_nat(X,A2),B2) = minus_2163939370556025621et_nat(A2,B2) ) ) ).

% insert_Diff1
tff(fact_4965_insert__Diff1,axiom,
    ! [X: int,B2: set_int,A2: set_int] :
      ( aa_set_int_o(member_int(X),B2)
     => ( minus_minus_set_int(insert_int(X,A2),B2) = minus_minus_set_int(A2,B2) ) ) ).

% insert_Diff1
tff(fact_4966_insert__Diff1,axiom,
    ! [X: nat,B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(X),B2)
     => ( minus_minus_set_nat(insert_nat(X,A2),B2) = minus_minus_set_nat(A2,B2) ) ) ).

% insert_Diff1
tff(fact_4967_compl__le__compl__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(X)),uminus1532241313380277803et_int(Y))
    <=> aa_set_int_o(ord_less_eq_set_int(Y),X) ) ).

% compl_le_compl_iff
tff(fact_4968_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_4969_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_4970_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_4971_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_4972_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_4973_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_4974_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_4975_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_4976_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_4977_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_4978_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_4979_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_4980_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_4981_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_4982_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_4983_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_4984_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_4985_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_4986_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_4987_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_4988_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_4989_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_4990_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_4991_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_4992_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_4993_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_4994_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_4995_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_4996_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_4997_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_4998_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_4999_mult__minus__left,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B)) ).

% mult_minus_left
tff(fact_5000_mult__minus__left,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% mult_minus_left
tff(fact_5001_mult__minus__left,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% mult_minus_left
tff(fact_5002_mult__minus__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_left
tff(fact_5003_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_left
tff(fact_5004_minus__mult__minus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(times_times_complex(A),B) ).

% minus_mult_minus
tff(fact_5005_minus__mult__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),B) ).

% minus_mult_minus
tff(fact_5006_minus__mult__minus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) ).

% minus_mult_minus
tff(fact_5007_minus__mult__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(times_times_rat(A),B) ).

% minus_mult_minus
tff(fact_5008_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) ).

% minus_mult_minus
tff(fact_5009_mult__minus__right,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B)) ).

% mult_minus_right
tff(fact_5010_mult__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% mult_minus_right
tff(fact_5011_mult__minus__right,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% mult_minus_right
tff(fact_5012_mult__minus__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_right
tff(fact_5013_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_right
tff(fact_5014_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_5015_minus__add__distrib,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) ).

% minus_add_distrib
tff(fact_5016_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_add_distrib
tff(fact_5017_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_5018_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_5019_minus__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = B ).

% minus_add_cancel
tff(fact_5020_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(plus_plus_rat(A),B)) = B ).

% minus_add_cancel
tff(fact_5021_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_5022_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_5023_add__minus__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B)) = B ).

% add_minus_cancel
tff(fact_5024_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),B)) = B ).

% add_minus_cancel
tff(fact_5025_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_5026_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_5027_minus__diff__eq,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A) ).

% minus_diff_eq
tff(fact_5028_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(minus_minus_rat(B),A) ).

% minus_diff_eq
tff(fact_5029_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_5030_dvd__minus__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),aa_int_int(uminus_uminus_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),Y) ) ).

% dvd_minus_iff
tff(fact_5031_dvd__minus__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X),aa_real_real(uminus_uminus_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X),Y) ) ).

% dvd_minus_iff
tff(fact_5032_dvd__minus__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),aa_rat_rat(uminus_uminus_rat,Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),Y) ) ).

% dvd_minus_iff
tff(fact_5033_dvd__minus__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),Y) ) ).

% dvd_minus_iff
tff(fact_5034_minus__dvd__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(uminus_uminus_int,X)),Y)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),Y) ) ).

% minus_dvd_iff
tff(fact_5035_minus__dvd__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(uminus_uminus_real,X)),Y)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X),Y) ) ).

% minus_dvd_iff
tff(fact_5036_minus__dvd__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(uminus_uminus_rat,X)),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),Y) ) ).

% minus_dvd_iff
tff(fact_5037_minus__dvd__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),Y)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),Y) ) ).

% minus_dvd_iff
tff(fact_5038_Diff__eq__empty__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat(A2,B2) = bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),B2) ) ).

% Diff_eq_empty_iff
tff(fact_5039_Diff__eq__empty__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( minus_minus_set_real(A2,B2) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(A2),B2) ) ).

% Diff_eq_empty_iff
tff(fact_5040_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( minus_minus_set_nat(A2,B2) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2) ) ).

% Diff_eq_empty_iff
tff(fact_5041_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( minus_minus_set_int(A2,B2) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(A2),B2) ) ).

% Diff_eq_empty_iff
tff(fact_5042_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(I),set_or4548717258645045905et_nat(L,U))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,L),I)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_5043_atLeastAtMost__iff,axiom,
    ! [I: set_int,L: set_int,U: set_int] :
      ( member_set_int(I,set_or370866239135849197et_int(L,U))
    <=> ( aa_set_int_o(ord_less_eq_set_int(L),I)
        & aa_set_int_o(ord_less_eq_set_int(I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_5044_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_5045_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_5046_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat(I),set_or1269000886237332187st_nat(L,U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_5047_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( aa_set_int_o(member_int(I),aa_int_set_int(aa_int8639817059373041269et_int(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_5048_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_5049_Icc__eq__Icc,axiom,
    ! [L: set_int,H: set_int,L2: set_int,H2: set_int] :
      ( ( set_or370866239135849197et_int(L,H) = set_or370866239135849197et_int(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_set_int_o(ord_less_eq_set_int(L),H)
          & ~ aa_set_int_o(ord_less_eq_set_int(L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_5050_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_5051_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_5052_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L2: nat,H2: nat] :
      ( ( set_or1269000886237332187st_nat(L,H) = set_or1269000886237332187st_nat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),H)
          & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_5053_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L2: int,H2: int] :
      ( ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),H) = aa_int_set_int(aa_int8639817059373041269et_int(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_5054_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_5055_insert__Diff__single,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A,minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) = insert9069300056098147895at_nat(A,A2) ).

% insert_Diff_single
tff(fact_5056_insert__Diff__single,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A,minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = insert8211810215607154385at_nat(A,A2) ).

% insert_Diff_single
tff(fact_5057_insert__Diff__single,axiom,
    ! [A: real,A2: set_real] : insert_real(A,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = insert_real(A,A2) ).

% insert_Diff_single
tff(fact_5058_insert__Diff__single,axiom,
    ! [A: int,A2: set_int] : insert_int(A,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = insert_int(A,A2) ).

% insert_Diff_single
tff(fact_5059_insert__Diff__single,axiom,
    ! [A: nat,A2: set_nat] : insert_nat(A,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = insert_nat(A,A2) ).

% insert_Diff_single
tff(fact_5060_finite__Diff__insert,axiom,
    ! [A2: set_real,A: real,B2: set_real] :
      ( aa_set_real_o(finite_finite_real,minus_minus_set_real(A2,insert_real(A,B2)))
    <=> aa_set_real_o(finite_finite_real,minus_minus_set_real(A2,B2)) ) ).

% finite_Diff_insert
tff(fact_5061_finite__Diff__insert,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(A,B2)))
    <=> aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,minus_3314409938677909166at_nat(A2,B2)) ) ).

% finite_Diff_insert
tff(fact_5062_finite__Diff__insert,axiom,
    ! [A2: set_int,A: int,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,minus_minus_set_int(A2,insert_int(A,B2)))
    <=> aa_set_int_o(finite_finite_int,minus_minus_set_int(A2,B2)) ) ).

% finite_Diff_insert
tff(fact_5063_finite__Diff__insert,axiom,
    ! [A2: set_complex,A: complex,B2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,insert_complex(A,B2)))
    <=> aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B2)) ) ).

% finite_Diff_insert
tff(fact_5064_finite__Diff__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,B2)))
    <=> aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A2,B2)) ) ).

% finite_Diff_insert
tff(fact_5065_finite__Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A2,insert_nat(A,B2)))
    <=> aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A2,B2)) ) ).

% finite_Diff_insert
tff(fact_5066_take__bit__of__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),zero_zero_nat) = zero_zero_nat ).

% take_bit_of_0
tff(fact_5067_take__bit__of__0,axiom,
    ! [Na: nat] : aa_int_int(bit_se2923211474154528505it_int(Na),zero_zero_int) = zero_zero_int ).

% take_bit_of_0
tff(fact_5068_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,set_or1269000886237332187st_nat(L,U)) ).

% finite_atLeastAtMost
tff(fact_5069_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_5070_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_5071_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_5072_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_5073_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_5074_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_5075_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_5076_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_5077_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_5078_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_5079_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_5080_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_5081_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_5082_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_5083_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_5084_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_5085_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_5086_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_5087_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_5088_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_5089_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_5090_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_5091_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_5092_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_5093_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_5094_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_5095_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_5096_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_5097_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_5098_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_5099_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_5100_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_5101_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_5102_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_5103_add_Oright__inverse,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,A)) = zero_zero_real ).

% add.right_inverse
tff(fact_5104_add_Oright__inverse,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_5105_add_Oright__inverse,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_5106_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_5107_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_5108_ab__left__minus,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_left_minus
tff(fact_5109_ab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_5110_ab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_5111_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% verit_minus_simplify(3)
tff(fact_5112_verit__minus__simplify_I3_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),B) = aa_int_int(uminus_uminus_int,B) ).

% verit_minus_simplify(3)
tff(fact_5113_verit__minus__simplify_I3_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),B) = aa_real_real(uminus_uminus_real,B) ).

% verit_minus_simplify(3)
tff(fact_5114_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% verit_minus_simplify(3)
tff(fact_5115_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% verit_minus_simplify(3)
tff(fact_5116_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_5117_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_5118_diff__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),A) = aa_real_real(uminus_uminus_real,A) ).

% diff_0
tff(fact_5119_diff__0,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),A) = aa_rat_rat(uminus_uminus_rat,A) ).

% diff_0
tff(fact_5120_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% diff_0
tff(fact_5121_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Na: 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,Na))) = 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,Na))) ).

% add_neg_numeral_simps(3)
tff(fact_5122_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_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,Na))) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Na))) ).

% add_neg_numeral_simps(3)
tff(fact_5123_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Na: num] : aa_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,Na))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Na))) ).

% add_neg_numeral_simps(3)
tff(fact_5124_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Na: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Na))) ).

% add_neg_numeral_simps(3)
tff(fact_5125_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_5126_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A) ).

% uminus_add_conv_diff
tff(fact_5127_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(minus_minus_rat(B),A) ).

% uminus_add_conv_diff
tff(fact_5128_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_5129_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_5130_diff__minus__eq__add,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) ).

% diff_minus_eq_add
tff(fact_5131_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(plus_plus_rat(A),B) ).

% diff_minus_eq_add
tff(fact_5132_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_5133_divide__minus1,axiom,
    ! [X: complex] : aa_complex_complex(divide1717551699836669952omplex(X),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,X) ).

% divide_minus1
tff(fact_5134_divide__minus1,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,X) ).

% divide_minus1
tff(fact_5135_divide__minus1,axiom,
    ! [X: rat] : aa_rat_rat(divide_divide_rat(X),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,X) ).

% divide_minus1
tff(fact_5136_atLeastatMost__empty__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( set_or370866239135849197et_int(A,B) = bot_bot_set_set_int )
    <=> ~ aa_set_int_o(ord_less_eq_set_int(A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_5137_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_5138_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_5139_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: 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_5140_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_set_int(aa_int8639817059373041269et_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_5141_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_5142_atLeastatMost__empty__iff2,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_set_int = set_or370866239135849197et_int(A,B) )
    <=> ~ aa_set_int_o(ord_less_eq_set_int(A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_5143_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_5144_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_5145_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_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_5146_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = aa_int_set_int(aa_int8639817059373041269et_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_5147_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_5148_atLeastatMost__subset__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(A,B),set_or370866239135849197et_int(C,D))
    <=> ( ~ aa_set_int_o(ord_less_eq_set_int(A),B)
        | ( aa_set_int_o(ord_less_eq_set_int(C),A)
          & aa_set_int_o(ord_less_eq_set_int(B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_5149_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_5150_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_5151_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_or1269000886237332187st_nat(A,B)),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_5152_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B)),aa_int_set_int(aa_int8639817059373041269et_int(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_5153_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_5154_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_5155_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_5156_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( set_or1269000886237332187st_nat(A,B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_5157_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_5158_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_5159_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% infinite_Icc_iff
tff(fact_5160_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ~ aa_set_real_o(finite_finite_real,set_or1222579329274155063t_real(A,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% infinite_Icc_iff
tff(fact_5161_subset__Compl__singleton,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),uminus935396558254630718at_nat(insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat)))
    <=> ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),A2) ) ).

% subset_Compl_singleton
tff(fact_5162_subset__Compl__singleton,axiom,
    ! [A2: set_set_nat,B: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),uminus613421341184616069et_nat(insert_set_nat(B,bot_bot_set_set_nat)))
    <=> ~ aa_set_set_nat_o(member_set_nat(B),A2) ) ).

% subset_Compl_singleton
tff(fact_5163_subset__Compl__singleton,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),uminus6524753893492686040at_nat(insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat)))
    <=> ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),A2) ) ).

% subset_Compl_singleton
tff(fact_5164_subset__Compl__singleton,axiom,
    ! [A2: set_real,B: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),uminus612125837232591019t_real(insert_real(B,bot_bot_set_real)))
    <=> ~ aa_set_real_o(member_real(B),A2) ) ).

% subset_Compl_singleton
tff(fact_5165_subset__Compl__singleton,axiom,
    ! [A2: set_nat,B: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),uminus5710092332889474511et_nat(insert_nat(B,bot_bot_set_nat)))
    <=> ~ aa_set_nat_o(member_nat(B),A2) ) ).

% subset_Compl_singleton
tff(fact_5166_subset__Compl__singleton,axiom,
    ! [A2: set_int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),uminus1532241313380277803et_int(insert_int(B,bot_bot_set_int)))
    <=> ~ aa_set_int_o(member_int(B),A2) ) ).

% subset_Compl_singleton
tff(fact_5167_take__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,zero_zero_nat),A) = zero_zero_nat ).

% take_bit_0
tff(fact_5168_take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(zero_zero_nat),A) = zero_zero_int ).

% take_bit_0
tff(fact_5169_atLeastAtMost__singleton,axiom,
    ! [A: nat] : set_or1269000886237332187st_nat(A,A) = insert_nat(A,bot_bot_set_nat) ).

% atLeastAtMost_singleton
tff(fact_5170_atLeastAtMost__singleton,axiom,
    ! [A: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),A) = insert_int(A,bot_bot_set_int) ).

% atLeastAtMost_singleton
tff(fact_5171_atLeastAtMost__singleton,axiom,
    ! [A: real] : set_or1222579329274155063t_real(A,A) = insert_real(A,bot_bot_set_real) ).

% atLeastAtMost_singleton
tff(fact_5172_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( set_or1269000886237332187st_nat(A,B) = insert_nat(C,bot_bot_set_nat) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_5173_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B) = insert_int(C,bot_bot_set_int) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_5174_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_5175_take__bit__Suc__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Na)),one_one_nat) = one_one_nat ).

% take_bit_Suc_1
tff(fact_5176_take__bit__Suc__1,axiom,
    ! [Na: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),one_one_int) = one_one_int ).

% take_bit_Suc_1
tff(fact_5177_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_5178_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_5179_add__neg__numeral__special_I8_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = zero_zero_real ).

% add_neg_numeral_special(8)
tff(fact_5180_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_5181_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_5182_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_5183_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_5184_add__neg__numeral__special_I7_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% add_neg_numeral_special(7)
tff(fact_5185_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_5186_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_5187_diff__numeral__special_I12_J,axiom,
    aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% diff_numeral_special(12)
tff(fact_5188_diff__numeral__special_I12_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% diff_numeral_special(12)
tff(fact_5189_diff__numeral__special_I12_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% diff_numeral_special(12)
tff(fact_5190_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_5191_diff__numeral__special_I12_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_5192_mod__minus1__right,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_5193_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_5194_take__bit__of__1__eq__0__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),one_one_nat) = zero_zero_nat )
    <=> ( Na = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_5195_take__bit__of__1__eq__0__iff,axiom,
    ! [Na: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Na),one_one_int) = zero_zero_int )
    <=> ( Na = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_5196_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W2))),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,V),W2)))),Y) ).

% semiring_norm(168)
tff(fact_5197_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2)))),Y) ).

% semiring_norm(168)
tff(fact_5198_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),Y)) = aa_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,V),W2)))),Y) ).

% semiring_norm(168)
tff(fact_5199_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W2)),Y)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2))),Y) ).

% semiring_norm(168)
tff(fact_5200_neg__numeral__le__iff,axiom,
    ! [Mb: num,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),Mb) ) ).

% neg_numeral_le_iff
tff(fact_5201_neg__numeral__le__iff,axiom,
    ! [Mb: num,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),Mb) ) ).

% neg_numeral_le_iff
tff(fact_5202_neg__numeral__le__iff,axiom,
    ! [Mb: num,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),Mb) ) ).

% neg_numeral_le_iff
tff(fact_5203_neg__numeral__le__iff,axiom,
    ! [Mb: num,Na: 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,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),Mb) ) ).

% neg_numeral_le_iff
tff(fact_5204_neg__numeral__less__iff,axiom,
    ! [Mb: num,Na: 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,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Na),Mb) ) ).

% neg_numeral_less_iff
tff(fact_5205_neg__numeral__less__iff,axiom,
    ! [Mb: num,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Na),Mb) ) ).

% neg_numeral_less_iff
tff(fact_5206_neg__numeral__less__iff,axiom,
    ! [Mb: num,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Na),Mb) ) ).

% neg_numeral_less_iff
tff(fact_5207_neg__numeral__less__iff,axiom,
    ! [Mb: num,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Na),Mb) ) ).

% neg_numeral_less_iff
tff(fact_5208_zle__diff1__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% zle_diff1_eq
tff(fact_5209_take__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% take_bit_of_Suc_0
tff(fact_5210_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_5211_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,numera6620942414471956472nteger(Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5212_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_5213_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_5214_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_5215_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_5216_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_5217_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_5218_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W2: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) = A )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5219_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) = A )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5220_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) = A )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5221_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W2: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) != zero_zero_complex,aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) = B,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5222_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5223_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) != zero_zero_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5224_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_5225_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_5226_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_5227_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5228_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_5229_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_5230_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_5231_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_5232_take__bit__of__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),one_one_nat) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% take_bit_of_1
tff(fact_5233_take__bit__of__1,axiom,
    ! [Na: nat] : aa_int_int(bit_se2923211474154528505it_int(Na),one_one_int) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% take_bit_of_1
tff(fact_5234_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_5235_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_5236_add__neg__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_5237_add__neg__numeral__special_I9_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_5238_add__neg__numeral__special_I9_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_5239_even__take__bit__eq,axiom,
    ! [Na: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1745604003318907178nteger(Na,A))
    <=> ( ( Na = zero_zero_nat )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ) ).

% even_take_bit_eq
tff(fact_5240_even__take__bit__eq,axiom,
    ! [Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A))
    <=> ( ( Na = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ) ).

% even_take_bit_eq
tff(fact_5241_even__take__bit__eq,axiom,
    ! [Na: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2923211474154528505it_int(Na),A))
    <=> ( ( Na = zero_zero_nat )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ) ).

% even_take_bit_eq
tff(fact_5242_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_5243_take__bit__Suc__0,axiom,
    ! [A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,zero_zero_nat),A) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_5244_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_5245_take__bit__Suc__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,zero_zero_nat)),A) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_5246_signed__take__bit__0,axiom,
    ! [A: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_5247_signed__take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_ri631733984087533419it_int(zero_zero_nat),A) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_5248_take__bit__of__exp,axiom,
    ! [Mb: nat,Na: 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))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% take_bit_of_exp
tff(fact_5249_take__bit__of__exp,axiom,
    ! [Mb: nat,Na: 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))),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% take_bit_of_exp
tff(fact_5250_take__bit__of__2,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_5251_take__bit__of__2,axiom,
    ! [Na: nat] : aa_int_int(bit_se2923211474154528505it_int(Na),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_5252_minus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),L) = aa_int_int(uminus_uminus_int,L) ).

% minus_int_code(2)
tff(fact_5253_compl__mono,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(X),Y)
     => aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Y)),uminus1532241313380277803et_int(X)) ) ).

% compl_mono
tff(fact_5254_compl__le__swap1,axiom,
    ! [Y: set_int,X: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Y),uminus1532241313380277803et_int(X))
     => aa_set_int_o(ord_less_eq_set_int(X),uminus1532241313380277803et_int(Y)) ) ).

% compl_le_swap1
tff(fact_5255_compl__le__swap2,axiom,
    ! [Y: set_int,X: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Y)),X)
     => aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(X)),Y) ) ).

% compl_le_swap2
tff(fact_5256_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_5257_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_5258_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_5259_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_5260_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_5261_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_5262_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_5263_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_5264_Compl__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] : uminus935396558254630718at_nat(insert9069300056098147895at_nat(X,A2)) = minus_3314409938677909166at_nat(uminus935396558254630718at_nat(A2),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) ).

% Compl_insert
tff(fact_5265_Compl__insert,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(insert8211810215607154385at_nat(X,A2)) = minus_1356011639430497352at_nat(uminus6524753893492686040at_nat(A2),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) ).

% Compl_insert
tff(fact_5266_Compl__insert,axiom,
    ! [X: real,A2: set_real] : uminus612125837232591019t_real(insert_real(X,A2)) = minus_minus_set_real(uminus612125837232591019t_real(A2),insert_real(X,bot_bot_set_real)) ).

% Compl_insert
tff(fact_5267_Compl__insert,axiom,
    ! [X: int,A2: set_int] : uminus1532241313380277803et_int(insert_int(X,A2)) = minus_minus_set_int(uminus1532241313380277803et_int(A2),insert_int(X,bot_bot_set_int)) ).

% Compl_insert
tff(fact_5268_Compl__insert,axiom,
    ! [X: nat,A2: set_nat] : uminus5710092332889474511et_nat(insert_nat(X,A2)) = minus_minus_set_nat(uminus5710092332889474511et_nat(A2),insert_nat(X,bot_bot_set_nat)) ).

% Compl_insert
tff(fact_5269_take__bit__add,axiom,
    ! [Na: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),B))) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% take_bit_add
tff(fact_5270_take__bit__add,axiom,
    ! [Na: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se2923211474154528505it_int(Na),A)),aa_int_int(bit_se2923211474154528505it_int(Na),B))) = aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ).

% take_bit_add
tff(fact_5271_take__bit__tightened,axiom,
    ! [Na: nat,A: nat,B: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
       => ( 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_5272_take__bit__tightened,axiom,
    ! [Na: nat,A: int,B: int,Mb: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Na),A) = aa_int_int(bit_se2923211474154528505it_int(Na),B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
       => ( aa_int_int(bit_se2923211474154528505it_int(Mb),A) = aa_int_int(bit_se2923211474154528505it_int(Mb),B) ) ) ) ).

% take_bit_tightened
tff(fact_5273_take__bit__nat__less__eq__self,axiom,
    ! [Na: 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,Na),Mb)),Mb) ).

% take_bit_nat_less_eq_self
tff(fact_5274_take__bit__tightened__less__eq__nat,axiom,
    ! [Mb: nat,Na: nat,Q4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),Q4)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_5275_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_5276_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_5277_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_5278_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_5279_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_5280_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_5281_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_5282_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_5283_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_5284_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_5285_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_5286_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_5287_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_5288_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_5289_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_5290_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_5291_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_5292_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_5293_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_5294_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_5295_verit__negate__coefficient_I2_J,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(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5296_verit__negate__coefficient_I2_J,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(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5297_verit__negate__coefficient_I2_J,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(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5298_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5299_square__eq__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),A) = aa_complex_complex(times_times_complex(B),B) )
    <=> ( ( A = B )
        | ( A = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).

% square_eq_iff
tff(fact_5300_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),A) = aa_int_int(aa_int_fun_int_int(times_times_int,B),B) )
    <=> ( ( A = B )
        | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ).

% square_eq_iff
tff(fact_5301_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),B) )
    <=> ( ( A = B )
        | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% square_eq_iff
tff(fact_5302_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),A) = aa_rat_rat(times_times_rat(B),B) )
    <=> ( ( A = B )
        | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% square_eq_iff
tff(fact_5303_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),B) )
    <=> ( ( A = B )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ).

% square_eq_iff
tff(fact_5304_minus__mult__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_mult_commute
tff(fact_5305_minus__mult__commute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) ).

% minus_mult_commute
tff(fact_5306_minus__mult__commute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B)) ).

% minus_mult_commute
tff(fact_5307_minus__mult__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_mult_commute
tff(fact_5308_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_mult_commute
tff(fact_5309_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_5310_is__num__normalize_I8_J,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% is_num_normalize(8)
tff(fact_5311_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_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_5312_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_5313_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_5314_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% add.inverse_distrib_swap
tff(fact_5315_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_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_5316_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_5317_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(uminus_uminus_int,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_5318_group__cancel_Oneg1,axiom,
    ! [A2: real,K: real,A: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(uminus_uminus_real,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_5319_group__cancel_Oneg1,axiom,
    ! [A2: rat,K: rat,A: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(uminus_uminus_rat,A2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).

% group_cancel.neg1
tff(fact_5320_group__cancel_Oneg1,axiom,
    ! [A2: code_integer,K: code_integer,A: code_integer] :
      ( ( A2 = plus_p5714425477246183910nteger(K,A) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).

% group_cancel.neg1
tff(fact_5321_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_5322_minus__diff__commute,axiom,
    ! [B: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,B)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),B) ).

% minus_diff_commute
tff(fact_5323_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,B)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_diff_commute
tff(fact_5324_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_5325_minus__divide__right,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) ).

% minus_divide_right
tff(fact_5326_minus__divide__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_divide_right
tff(fact_5327_minus__divide__divide,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ).

% minus_divide_divide
tff(fact_5328_minus__divide__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(divide_divide_rat(A),B) ).

% minus_divide_divide
tff(fact_5329_minus__divide__left,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),B) ).

% minus_divide_left
tff(fact_5330_minus__divide__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_divide_left
tff(fact_5331_Diff__infinite__finite,axiom,
    ! [T3: set_int,S3: set_int] :
      ( aa_set_int_o(finite_finite_int,T3)
     => ( ~ aa_set_int_o(finite_finite_int,S3)
       => ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(S3,T3)) ) ) ).

% Diff_infinite_finite
tff(fact_5332_Diff__infinite__finite,axiom,
    ! [T3: set_complex,S3: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T3)
     => ( ~ aa_set_complex_o(finite3207457112153483333omplex,S3)
       => ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(S3,T3)) ) ) ).

% Diff_infinite_finite
tff(fact_5333_Diff__infinite__finite,axiom,
    ! [T3: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,T3)
     => ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S3)
       => ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(S3,T3)) ) ) ).

% Diff_infinite_finite
tff(fact_5334_Diff__infinite__finite,axiom,
    ! [T3: set_nat,S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( ~ aa_set_nat_o(finite_finite_nat,S3)
       => ~ aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(S3,T3)) ) ) ).

% Diff_infinite_finite
tff(fact_5335_double__diff,axiom,
    ! [A2: set_nat,B2: set_nat,C4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C4)
       => ( minus_minus_set_nat(B2,minus_minus_set_nat(C4,A2)) = A2 ) ) ) ).

% double_diff
tff(fact_5336_double__diff,axiom,
    ! [A2: set_int,B2: set_int,C4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),C4)
       => ( minus_minus_set_int(B2,minus_minus_set_int(C4,A2)) = A2 ) ) ) ).

% double_diff
tff(fact_5337_Diff__subset,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,minus_minus_set_nat(A2,B2)),A2) ).

% Diff_subset
tff(fact_5338_Diff__subset,axiom,
    ! [A2: set_int,B2: set_int] : aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A2,B2)),A2) ).

% Diff_subset
tff(fact_5339_Diff__mono,axiom,
    ! [A2: set_nat,C4: set_nat,D5: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,D5),B2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,minus_minus_set_nat(A2,B2)),minus_minus_set_nat(C4,D5)) ) ) ).

% Diff_mono
tff(fact_5340_Diff__mono,axiom,
    ! [A2: set_int,C4: set_int,D5: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),C4)
     => ( aa_set_int_o(ord_less_eq_set_int(D5),B2)
       => aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A2,B2)),minus_minus_set_int(C4,D5)) ) ) ).

% Diff_mono
tff(fact_5341_insert__Diff__if,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      minus_3314409938677909166at_nat(insert9069300056098147895at_nat(X,A2),B2) = $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),B2),minus_3314409938677909166at_nat(A2,B2),insert9069300056098147895at_nat(X,minus_3314409938677909166at_nat(A2,B2))) ).

% insert_Diff_if
tff(fact_5342_insert__Diff__if,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      minus_1356011639430497352at_nat(insert8211810215607154385at_nat(X,A2),B2) = $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),B2),minus_1356011639430497352at_nat(A2,B2),insert8211810215607154385at_nat(X,minus_1356011639430497352at_nat(A2,B2))) ).

% insert_Diff_if
tff(fact_5343_insert__Diff__if,axiom,
    ! [X: real,A2: set_real,B2: set_real] :
      minus_minus_set_real(insert_real(X,A2),B2) = $ite(aa_set_real_o(member_real(X),B2),minus_minus_set_real(A2,B2),insert_real(X,minus_minus_set_real(A2,B2))) ).

% insert_Diff_if
tff(fact_5344_insert__Diff__if,axiom,
    ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
      minus_2163939370556025621et_nat(insert_set_nat(X,A2),B2) = $ite(aa_set_set_nat_o(member_set_nat(X),B2),minus_2163939370556025621et_nat(A2,B2),insert_set_nat(X,minus_2163939370556025621et_nat(A2,B2))) ).

% insert_Diff_if
tff(fact_5345_insert__Diff__if,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      minus_minus_set_int(insert_int(X,A2),B2) = $ite(aa_set_int_o(member_int(X),B2),minus_minus_set_int(A2,B2),insert_int(X,minus_minus_set_int(A2,B2))) ).

% insert_Diff_if
tff(fact_5346_insert__Diff__if,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      minus_minus_set_nat(insert_nat(X,A2),B2) = $ite(aa_set_nat_o(member_nat(X),B2),minus_minus_set_nat(A2,B2),insert_nat(X,minus_minus_set_nat(A2,B2))) ).

% insert_Diff_if
tff(fact_5347_minus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,K),zero_zero_int) = K ).

% minus_int_code(1)
tff(fact_5348_uminus__int__code_I1_J,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% uminus_int_code(1)
tff(fact_5349_int__distrib_I4_J,axiom,
    ! [W2: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z22)) ).

% int_distrib(4)
tff(fact_5350_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)),W2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W2)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W2)) ).

% int_distrib(3)
tff(fact_5351_psubset__imp__ex__mem,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(A2),B2)
     => ? [B3: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B3),minus_1356011639430497352at_nat(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_5352_psubset__imp__ex__mem,axiom,
    ! [A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_set_real(A2),B2)
     => ? [B3: real] : aa_set_real_o(member_real(B3),minus_minus_set_real(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_5353_psubset__imp__ex__mem,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A2),B2)
     => ? [B3: set_nat] : aa_set_set_nat_o(member_set_nat(B3),minus_2163939370556025621et_nat(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_5354_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B2)
     => ? [B3: int] : aa_set_int_o(member_int(B3),minus_minus_set_int(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_5355_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2)
     => ? [B3: nat] : aa_set_nat_o(member_nat(B3),minus_minus_set_nat(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_5356_zdvd__zdiffD,axiom,
    ! [K: int,Mb: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,Mb),Na))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Na)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Mb) ) ) ).

% zdvd_zdiffD
tff(fact_5357_zmod__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),one_one_int) ) ) ).

% zmod_minus1
tff(fact_5358_take__bit__minus__small__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(uminus_uminus_int,K)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_5359_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(A,B)) ) ).

% infinite_Icc
tff(fact_5360_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_set_real_o(finite_finite_real,set_or1222579329274155063t_real(A,B)) ) ).

% infinite_Icc
tff(fact_5361_atLeastAtMost__singleton_H,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( set_or1269000886237332187st_nat(A,B) = insert_nat(A,bot_bot_set_nat) ) ) ).

% atLeastAtMost_singleton'
tff(fact_5362_atLeastAtMost__singleton_H,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B) = insert_int(A,bot_bot_set_int) ) ) ).

% atLeastAtMost_singleton'
tff(fact_5363_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_5364_take__bit__tightened__less__eq__int,axiom,
    ! [Mb: nat,Na: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Mb),K)),aa_int_int(bit_se2923211474154528505it_int(Na),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_5365_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(bit_se2923211474154528505it_int(Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% take_bit_int_greater_self_iff
tff(fact_5366_not__take__bit__negative,axiom,
    ! [Na: nat,K: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Na),K)),zero_zero_int) ).

% not_take_bit_negative
tff(fact_5367_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [Na: nat,A: int,B: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(Na),A) = aa_int_int(bit_ri631733984087533419it_int(Na),B) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),A) = aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),B) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_5368_signed__take__bit__take__bit,axiom,
    ! [Mb: nat,Na: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Na),A)) = aa_int_int($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb),bit_se2923211474154528505it_int(Na),bit_ri631733984087533419it_int(Mb)),A) ).

% signed_take_bit_take_bit
tff(fact_5369_ex__nat__less,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),Na)
          & aa_nat_o(P,M5) )
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_or1269000886237332187st_nat(zero_zero_nat,Na))
          & aa_nat_o(P,X4) ) ) ).

% ex_nat_less
tff(fact_5370_all__nat__less,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),Na)
         => aa_nat_o(P,M5) )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_or1269000886237332187st_nat(zero_zero_nat,Na))
         => aa_nat_o(P,X4) ) ) ).

% all_nat_less
tff(fact_5371_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Na)) ).

% neg_numeral_le_numeral
tff(fact_5372_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),numera6620942414471956472nteger(Na)) ).

% neg_numeral_le_numeral
tff(fact_5373_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Na)) ).

% neg_numeral_le_numeral
tff(fact_5374_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Na: 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,Na)) ).

% neg_numeral_le_numeral
tff(fact_5375_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Na: 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,Na))) ).

% not_numeral_le_neg_numeral
tff(fact_5376_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) ).

% not_numeral_le_neg_numeral
tff(fact_5377_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Na: 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,Na))) ).

% not_numeral_le_neg_numeral
tff(fact_5378_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Na: 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,Na))) ).

% not_numeral_le_neg_numeral
tff(fact_5379_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na)) ).

% zero_neq_neg_numeral
tff(fact_5380_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) ).

% zero_neq_neg_numeral
tff(fact_5381_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_zero_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)) ).

% zero_neq_neg_numeral
tff(fact_5382_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)) ).

% zero_neq_neg_numeral
tff(fact_5383_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na)) ).

% zero_neq_neg_numeral
tff(fact_5384_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Na: 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,Na))) ).

% not_numeral_less_neg_numeral
tff(fact_5385_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Na: 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,Na))) ).

% not_numeral_less_neg_numeral
tff(fact_5386_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Na: 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,Na))) ).

% not_numeral_less_neg_numeral
tff(fact_5387_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) ).

% not_numeral_less_neg_numeral
tff(fact_5388_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Na: 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,Na)) ).

% neg_numeral_less_numeral
tff(fact_5389_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Na)) ).

% neg_numeral_less_numeral
tff(fact_5390_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Na)) ).

% neg_numeral_less_numeral
tff(fact_5391_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),numera6620942414471956472nteger(Na)) ).

% neg_numeral_less_numeral
tff(fact_5392_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_5393_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_5394_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_5395_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_5396_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_5397_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_5398_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_5399_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_5400_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_5401_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_5402_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_5403_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_5404_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_5405_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_5406_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_5407_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% add_eq_0_iff
tff(fact_5408_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% add_eq_0_iff
tff(fact_5409_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_5410_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_5411_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_5412_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_group_add_class.ab_left_minus
tff(fact_5413_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : aa_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_5414_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_5415_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_5416_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_5417_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,A) = B ) ) ).

% add.inverse_unique
tff(fact_5418_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A) = B ) ) ).

% add.inverse_unique
tff(fact_5419_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_5420_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_5421_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_5422_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5423_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5424_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_5425_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_5426_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_5427_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5428_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5429_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_5430_less__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(4)
tff(fact_5431_less__minus__one__simps_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% less_minus_one_simps(4)
tff(fact_5432_less__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% less_minus_one_simps(4)
tff(fact_5433_less__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% less_minus_one_simps(4)
tff(fact_5434_less__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% less_minus_one_simps(2)
tff(fact_5435_less__minus__one__simps_I2_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) ).

% less_minus_one_simps(2)
tff(fact_5436_less__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).

% less_minus_one_simps(2)
tff(fact_5437_less__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).

% less_minus_one_simps(2)
tff(fact_5438_nonzero__minus__divide__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5439_nonzero__minus__divide__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5440_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5441_nonzero__minus__divide__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5442_nonzero__minus__divide__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5443_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5444_square__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(times_times_complex(X),X) = one_one_complex )
    <=> ( ( X = one_one_complex )
        | ( X = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_5445_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),X) = one_one_int )
    <=> ( ( X = one_one_int )
        | ( X = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_5446_square__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X) = one_one_real )
    <=> ( ( X = one_one_real )
        | ( X = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_5447_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(times_times_rat(X),X) = one_one_rat )
    <=> ( ( X = one_one_rat )
        | ( X = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_5448_square__eq__1__iff,axiom,
    ! [X: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X) = one_one_Code_integer )
    <=> ( ( X = one_one_Code_integer )
        | ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_5449_group__cancel_Osub2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5450_group__cancel_Osub2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5451_group__cancel_Osub2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = aa_rat_rat(plus_plus_rat(K),B) )
     => ( aa_rat_rat(minus_minus_rat(A),B2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5452_group__cancel_Osub2,axiom,
    ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B2 = plus_p5714425477246183910nteger(K,B) )
     => ( minus_8373710615458151222nteger(A,B2) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),minus_8373710615458151222nteger(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_5453_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_5454_diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ).

% diff_conv_add_uminus
tff(fact_5455_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% diff_conv_add_uminus
tff(fact_5456_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_5457_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_5458_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5459_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5460_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_5461_take__bit__unset__bit__eq,axiom,
    ! [Na: nat,Mb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A))) ).

% take_bit_unset_bit_eq
tff(fact_5462_take__bit__unset__bit__eq,axiom,
    ! [Na: nat,Mb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(bit_se4203085406695923979it_int(Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb),aa_int_int(bit_se2923211474154528505it_int(Na),A),aa_int_int(bit_se4203085406695923979it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Na),A))) ).

% take_bit_unset_bit_eq
tff(fact_5463_take__bit__set__bit__eq,axiom,
    ! [Na: nat,Mb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A))) ).

% take_bit_set_bit_eq
tff(fact_5464_take__bit__set__bit__eq,axiom,
    ! [Na: nat,Mb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(bit_se7879613467334960850it_int(Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb),aa_int_int(bit_se2923211474154528505it_int(Na),A),aa_int_int(bit_se7879613467334960850it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Na),A))) ).

% take_bit_set_bit_eq
tff(fact_5465_take__bit__flip__bit__eq,axiom,
    ! [Na: nat,Mb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A))) ).

% take_bit_flip_bit_eq
tff(fact_5466_take__bit__flip__bit__eq,axiom,
    ! [Na: nat,Mb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(bit_se2159334234014336723it_int(Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb),aa_int_int(bit_se2923211474154528505it_int(Na),A),aa_int_int(bit_se2159334234014336723it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Na),A))) ).

% take_bit_flip_bit_eq
tff(fact_5467_dvd__neg__div,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)) ) ) ).

% dvd_neg_div
tff(fact_5468_dvd__neg__div,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ) ) ).

% dvd_neg_div
tff(fact_5469_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) ) ) ).

% dvd_neg_div
tff(fact_5470_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_neg_div
tff(fact_5471_dvd__div__neg,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)) ) ) ).

% dvd_div_neg
tff(fact_5472_dvd__div__neg,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ) ) ).

% dvd_div_neg
tff(fact_5473_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) ) ) ).

% dvd_div_neg
tff(fact_5474_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_div_neg
tff(fact_5475_subset__Compl__self__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),uminus6524753893492686040at_nat(A2))
    <=> ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% subset_Compl_self_eq
tff(fact_5476_subset__Compl__self__eq,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),uminus612125837232591019t_real(A2))
    <=> ( A2 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
tff(fact_5477_subset__Compl__self__eq,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),uminus5710092332889474511et_nat(A2))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_5478_subset__Compl__self__eq,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),uminus1532241313380277803et_int(A2))
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_5479_Diff__insert__absorb,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A2)
     => ( minus_3314409938677909166at_nat(insert9069300056098147895at_nat(X,A2),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5480_Diff__insert__absorb,axiom,
    ! [X: set_nat,A2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(X),A2)
     => ( minus_2163939370556025621et_nat(insert_set_nat(X,A2),insert_set_nat(X,bot_bot_set_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5481_Diff__insert__absorb,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A2)
     => ( minus_1356011639430497352at_nat(insert8211810215607154385at_nat(X,A2),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5482_Diff__insert__absorb,axiom,
    ! [X: real,A2: set_real] :
      ( ~ aa_set_real_o(member_real(X),A2)
     => ( minus_minus_set_real(insert_real(X,A2),insert_real(X,bot_bot_set_real)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5483_Diff__insert__absorb,axiom,
    ! [X: int,A2: set_int] :
      ( ~ aa_set_int_o(member_int(X),A2)
     => ( minus_minus_set_int(insert_int(X,A2),insert_int(X,bot_bot_set_int)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5484_Diff__insert__absorb,axiom,
    ! [X: nat,A2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(X),A2)
     => ( minus_minus_set_nat(insert_nat(X,A2),insert_nat(X,bot_bot_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5485_Diff__insert2,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(A,B2)) = minus_3314409938677909166at_nat(minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)),B2) ).

% Diff_insert2
tff(fact_5486_Diff__insert2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,B2)) = minus_1356011639430497352at_nat(minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)),B2) ).

% Diff_insert2
tff(fact_5487_Diff__insert2,axiom,
    ! [A2: set_real,A: real,B2: set_real] : minus_minus_set_real(A2,insert_real(A,B2)) = minus_minus_set_real(minus_minus_set_real(A2,insert_real(A,bot_bot_set_real)),B2) ).

% Diff_insert2
tff(fact_5488_Diff__insert2,axiom,
    ! [A2: set_int,A: int,B2: set_int] : minus_minus_set_int(A2,insert_int(A,B2)) = minus_minus_set_int(minus_minus_set_int(A2,insert_int(A,bot_bot_set_int)),B2) ).

% Diff_insert2
tff(fact_5489_Diff__insert2,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] : minus_minus_set_nat(A2,insert_nat(A,B2)) = minus_minus_set_nat(minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat)),B2) ).

% Diff_insert2
tff(fact_5490_insert__Diff,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A2)
     => ( insert9069300056098147895at_nat(A,minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_5491_insert__Diff,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A2)
     => ( insert_set_nat(A,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_5492_insert__Diff,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A2)
     => ( insert8211810215607154385at_nat(A,minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_5493_insert__Diff,axiom,
    ! [A: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),A2)
     => ( insert_real(A,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = A2 ) ) ).

% insert_Diff
tff(fact_5494_insert__Diff,axiom,
    ! [A: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => ( insert_int(A,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = A2 ) ) ).

% insert_Diff
tff(fact_5495_insert__Diff,axiom,
    ! [A: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),A2)
     => ( insert_nat(A,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_5496_Diff__insert,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(A,B2)) = minus_3314409938677909166at_nat(minus_3314409938677909166at_nat(A2,B2),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)) ).

% Diff_insert
tff(fact_5497_Diff__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,B2)) = minus_1356011639430497352at_nat(minus_1356011639430497352at_nat(A2,B2),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)) ).

% Diff_insert
tff(fact_5498_Diff__insert,axiom,
    ! [A2: set_real,A: real,B2: set_real] : minus_minus_set_real(A2,insert_real(A,B2)) = minus_minus_set_real(minus_minus_set_real(A2,B2),insert_real(A,bot_bot_set_real)) ).

% Diff_insert
tff(fact_5499_Diff__insert,axiom,
    ! [A2: set_int,A: int,B2: set_int] : minus_minus_set_int(A2,insert_int(A,B2)) = minus_minus_set_int(minus_minus_set_int(A2,B2),insert_int(A,bot_bot_set_int)) ).

% Diff_insert
tff(fact_5500_Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] : minus_minus_set_nat(A2,insert_nat(A,B2)) = minus_minus_set_nat(minus_minus_set_nat(A2,B2),insert_nat(A,bot_bot_set_nat)) ).

% Diff_insert
tff(fact_5501_subset__Diff__insert,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,C4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),minus_3314409938677909166at_nat(B2,insert9069300056098147895at_nat(X,C4)))
    <=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),minus_3314409938677909166at_nat(B2,C4))
        & ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A2) ) ) ).

% subset_Diff_insert
tff(fact_5502_subset__Diff__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,C4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),minus_1356011639430497352at_nat(B2,insert8211810215607154385at_nat(X,C4)))
    <=> ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),minus_1356011639430497352at_nat(B2,C4))
        & ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A2) ) ) ).

% subset_Diff_insert
tff(fact_5503_subset__Diff__insert,axiom,
    ! [A2: set_real,B2: set_real,X: real,C4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),minus_minus_set_real(B2,insert_real(X,C4)))
    <=> ( aa_set_real_o(ord_less_eq_set_real(A2),minus_minus_set_real(B2,C4))
        & ~ aa_set_real_o(member_real(X),A2) ) ) ).

% subset_Diff_insert
tff(fact_5504_subset__Diff__insert,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,X: set_nat,C4: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),minus_2163939370556025621et_nat(B2,insert_set_nat(X,C4)))
    <=> ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),minus_2163939370556025621et_nat(B2,C4))
        & ~ aa_set_set_nat_o(member_set_nat(X),A2) ) ) ).

% subset_Diff_insert
tff(fact_5505_subset__Diff__insert,axiom,
    ! [A2: set_nat,B2: set_nat,X: nat,C4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),minus_minus_set_nat(B2,insert_nat(X,C4)))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),minus_minus_set_nat(B2,C4))
        & ~ aa_set_nat_o(member_nat(X),A2) ) ) ).

% subset_Diff_insert
tff(fact_5506_subset__Diff__insert,axiom,
    ! [A2: set_int,B2: set_int,X: int,C4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),minus_minus_set_int(B2,insert_int(X,C4)))
    <=> ( aa_set_int_o(ord_less_eq_set_int(A2),minus_minus_set_int(B2,C4))
        & ~ aa_set_int_o(member_int(X),A2) ) ) ).

% subset_Diff_insert
tff(fact_5507_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,X)),Y) ) ).

% real_0_less_add_iff
tff(fact_5508_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(uminus_uminus_real,X)) ) ).

% real_add_less_0_iff
tff(fact_5509_zminus1__lemma,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),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,Q4),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q4)),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_5510_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K)
     => ( aa_int_o(P,K)
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),K)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_le_induct
tff(fact_5511_int__less__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),K)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I3),K)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_less_induct
tff(fact_5512_pos__zmult__eq__1__iff__lemma,axiom,
    ! [Mb: int,Na: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Na) = one_one_int )
     => ( ( Mb = one_one_int )
        | ( Mb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_5513_zmult__eq__1__iff,axiom,
    ! [Mb: int,Na: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Na) = one_one_int )
    <=> ( ( ( Mb = one_one_int )
          & ( Na = one_one_int ) )
        | ( ( Mb = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( Na = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_5514_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Na: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Na),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_5515_take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_minus_bit0
tff(fact_5516_take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Na),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) ) ) ).

% take_bit_int_less_eq
tff(fact_5517_atLeastatMost__psubset__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ord_less_set_set_int(set_or370866239135849197et_int(A,B),set_or370866239135849197et_int(C,D))
    <=> ( ( ~ aa_set_int_o(ord_less_eq_set_int(A),B)
          | ( aa_set_int_o(ord_less_eq_set_int(C),A)
            & aa_set_int_o(ord_less_eq_set_int(B),D)
            & ( aa_set_int_o(ord_less_set_int(C),A)
              | aa_set_int_o(ord_less_set_int(B),D) ) ) )
        & aa_set_int_o(ord_less_eq_set_int(C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5518_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_5519_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_5520_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,set_or1269000886237332187st_nat(A,B)),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_5521_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_set_int_o(ord_less_set_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B)),aa_int_set_int(aa_int8639817059373041269et_int(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_5522_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_5523_take__bit__signed__take__bit,axiom,
    ! [Mb: nat,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Na))
     => ( aa_int_int(bit_se2923211474154528505it_int(Mb),aa_int_int(bit_ri631733984087533419it_int(Na),A)) = aa_int_int(bit_se2923211474154528505it_int(Mb),A) ) ) ).

% take_bit_signed_take_bit
tff(fact_5524_not__zero__le__neg__numeral,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) ).

% not_zero_le_neg_numeral
tff(fact_5525_not__zero__le__neg__numeral,axiom,
    ! [Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) ).

% not_zero_le_neg_numeral
tff(fact_5526_not__zero__le__neg__numeral,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) ).

% not_zero_le_neg_numeral
tff(fact_5527_not__zero__le__neg__numeral,axiom,
    ! [Na: 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,Na))) ).

% not_zero_le_neg_numeral
tff(fact_5528_neg__numeral__le__zero,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))),zero_zero_real) ).

% neg_numeral_le_zero
tff(fact_5529_neg__numeral__le__zero,axiom,
    ! [Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))),zero_z3403309356797280102nteger) ).

% neg_numeral_le_zero
tff(fact_5530_neg__numeral__le__zero,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))),zero_zero_rat) ).

% neg_numeral_le_zero
tff(fact_5531_neg__numeral__le__zero,axiom,
    ! [Na: 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,Na))),zero_zero_int) ).

% neg_numeral_le_zero
tff(fact_5532_neg__numeral__less__zero,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))),zero_zero_int) ).

% neg_numeral_less_zero
tff(fact_5533_neg__numeral__less__zero,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))),zero_zero_real) ).

% neg_numeral_less_zero
tff(fact_5534_neg__numeral__less__zero,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))),zero_zero_rat) ).

% neg_numeral_less_zero
tff(fact_5535_neg__numeral__less__zero,axiom,
    ! [Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))),zero_z3403309356797280102nteger) ).

% neg_numeral_less_zero
tff(fact_5536_not__zero__less__neg__numeral,axiom,
    ! [Na: 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,Na))) ).

% not_zero_less_neg_numeral
tff(fact_5537_not__zero__less__neg__numeral,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) ).

% not_zero_less_neg_numeral
tff(fact_5538_not__zero__less__neg__numeral,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) ).

% not_zero_less_neg_numeral
tff(fact_5539_not__zero__less__neg__numeral,axiom,
    ! [Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) ).

% not_zero_less_neg_numeral
tff(fact_5540_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_5541_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_5542_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_5543_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_5544_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_5545_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_5546_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_5547_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_5548_less__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(3)
tff(fact_5549_less__minus__one__simps_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% less_minus_one_simps(3)
tff(fact_5550_less__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% less_minus_one_simps(3)
tff(fact_5551_less__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% less_minus_one_simps(3)
tff(fact_5552_less__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% less_minus_one_simps(1)
tff(fact_5553_less__minus__one__simps_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).

% less_minus_one_simps(1)
tff(fact_5554_less__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).

% less_minus_one_simps(1)
tff(fact_5555_less__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% less_minus_one_simps(1)
tff(fact_5556_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_5557_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),one_one_Code_integer) ).

% neg_numeral_le_one
tff(fact_5558_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_5559_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_5560_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_5561_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(Mb)) ).

% neg_one_le_numeral
tff(fact_5562_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_5563_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_5564_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_5565_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% neg_numeral_le_neg_one
tff(fact_5566_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_5567_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_5568_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_5569_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_le_neg_one
tff(fact_5570_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_5571_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_5572_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_5573_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))) ).

% not_one_le_neg_numeral
tff(fact_5574_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_5575_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_5576_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_5577_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_5578_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_5579_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,numera6620942414471956472nteger(Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_5580_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_5581_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_5582_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_5583_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))) ).

% not_one_less_neg_numeral
tff(fact_5584_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_5585_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_5586_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_5587_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_less_neg_one
tff(fact_5588_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_5589_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_5590_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_5591_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(Mb)) ).

% neg_one_less_numeral
tff(fact_5592_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_5593_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_5594_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_5595_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),one_one_Code_integer) ).

% neg_numeral_less_one
tff(fact_5596_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(uminus1482373934393186551omplex,B),A = zero_zero_complex) ) ).

% eq_minus_divide_eq
tff(fact_5597_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(uminus_uminus_real,B),A = zero_zero_real) ) ).

% eq_minus_divide_eq
tff(fact_5598_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(uminus_uminus_rat,B),A = zero_zero_rat) ) ).

% eq_minus_divide_eq
tff(fact_5599_minus__divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) = A )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B) = aa_complex_complex(times_times_complex(A),C),A = zero_zero_complex) ) ).

% minus_divide_eq_eq
tff(fact_5600_minus__divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = A )
    <=> $ite(C != zero_zero_real,aa_real_real(uminus_uminus_real,B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C),A = zero_zero_real) ) ).

% minus_divide_eq_eq
tff(fact_5601_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) = A )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B) = aa_rat_rat(times_times_rat(A),C),A = zero_zero_rat) ) ).

% minus_divide_eq_eq
tff(fact_5602_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = C )
      <=> ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(times_times_complex(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5603_nonzero__neg__divide__eq__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = C )
      <=> ( aa_real_real(uminus_uminus_real,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5604_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = C )
      <=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(times_times_rat(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5605_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) )
      <=> ( aa_complex_complex(times_times_complex(C),B) = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5606_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: real,C: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( C = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) = aa_real_real(uminus_uminus_real,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5607_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) )
      <=> ( aa_rat_rat(times_times_rat(C),B) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5608_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( B != zero_zero_complex )
        & ( A = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_5609_divide__eq__minus__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( ( B != zero_zero_real )
        & ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_5610_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( ( B != zero_zero_rat )
        & ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_5611_finite__empty__induct,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A2)
     => ( aa_set5844364881195198285_nat_o(P,A2)
       => ( ! [A3: produc3843707927480180839at_nat,A5: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A5)
             => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),A5)
               => ( aa_set5844364881195198285_nat_o(P,A5)
                 => aa_set5844364881195198285_nat_o(P,minus_3314409938677909166at_nat(A5,insert9069300056098147895at_nat(A3,bot_bo228742789529271731at_nat))) ) ) )
         => aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat) ) ) ) ).

% finite_empty_induct
tff(fact_5612_finite__empty__induct,axiom,
    ! [A2: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(P,A2)
       => ( ! [A3: set_nat,A5: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A5)
             => ( aa_set_set_nat_o(member_set_nat(A3),A5)
               => ( aa_set_set_nat_o(P,A5)
                 => aa_set_set_nat_o(P,minus_2163939370556025621et_nat(A5,insert_set_nat(A3,bot_bot_set_set_nat))) ) ) )
         => aa_set_set_nat_o(P,bot_bot_set_set_nat) ) ) ) ).

% finite_empty_induct
tff(fact_5613_finite__empty__induct,axiom,
    ! [A2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(P,A2)
       => ( ! [A3: complex,A5: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A5)
             => ( aa_set_complex_o(member_complex(A3),A5)
               => ( aa_set_complex_o(P,A5)
                 => aa_set_complex_o(P,minus_811609699411566653omplex(A5,insert_complex(A3,bot_bot_set_complex))) ) ) )
         => aa_set_complex_o(P,bot_bot_set_complex) ) ) ) ).

% finite_empty_induct
tff(fact_5614_finite__empty__induct,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( aa_set3702741264012733747_nat_o(P,A2)
       => ( ! [A3: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A5)
             => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),A5)
               => ( aa_set3702741264012733747_nat_o(P,A5)
                 => aa_set3702741264012733747_nat_o(P,minus_1356011639430497352at_nat(A5,insert8211810215607154385at_nat(A3,bot_bo2099793752762293965at_nat))) ) ) )
         => aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat) ) ) ) ).

% finite_empty_induct
tff(fact_5615_finite__empty__induct,axiom,
    ! [A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(P,A2)
       => ( ! [A3: real,A5: set_real] :
              ( aa_set_real_o(finite_finite_real,A5)
             => ( aa_set_real_o(member_real(A3),A5)
               => ( aa_set_real_o(P,A5)
                 => aa_set_real_o(P,minus_minus_set_real(A5,insert_real(A3,bot_bot_set_real))) ) ) )
         => aa_set_real_o(P,bot_bot_set_real) ) ) ) ).

% finite_empty_induct
tff(fact_5616_finite__empty__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(P,A2)
       => ( ! [A3: int,A5: set_int] :
              ( aa_set_int_o(finite_finite_int,A5)
             => ( aa_set_int_o(member_int(A3),A5)
               => ( aa_set_int_o(P,A5)
                 => aa_set_int_o(P,minus_minus_set_int(A5,insert_int(A3,bot_bot_set_int))) ) ) )
         => aa_set_int_o(P,bot_bot_set_int) ) ) ) ).

% finite_empty_induct
tff(fact_5617_finite__empty__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(P,A2)
       => ( ! [A3: nat,A5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A5)
             => ( aa_set_nat_o(member_nat(A3),A5)
               => ( aa_set_nat_o(P,A5)
                 => aa_set_nat_o(P,minus_minus_set_nat(A5,insert_nat(A3,bot_bot_set_nat))) ) ) )
         => aa_set_nat_o(P,bot_bot_set_nat) ) ) ) ).

% finite_empty_induct
tff(fact_5618_infinite__coinduct,axiom,
    ! [X5: fun_se7788542033685991124_nat_o,A2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(X5,A2)
     => ( ! [A5: set_Pr4329608150637261639at_nat] :
            ( aa_set5844364881195198285_nat_o(X5,A5)
           => ? [X2: produc3843707927480180839at_nat] :
                ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X2),A5)
                & ( aa_set5844364881195198285_nat_o(X5,minus_3314409938677909166at_nat(A5,insert9069300056098147895at_nat(X2,bot_bo228742789529271731at_nat)))
                  | ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,minus_3314409938677909166at_nat(A5,insert9069300056098147895at_nat(X2,bot_bo228742789529271731at_nat))) ) ) )
       => ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A2) ) ) ).

% infinite_coinduct
tff(fact_5619_infinite__coinduct,axiom,
    ! [X5: fun_set_complex_o,A2: set_complex] :
      ( aa_set_complex_o(X5,A2)
     => ( ! [A5: set_complex] :
            ( aa_set_complex_o(X5,A5)
           => ? [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),A5)
                & ( aa_set_complex_o(X5,minus_811609699411566653omplex(A5,insert_complex(X2,bot_bot_set_complex)))
                  | ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A5,insert_complex(X2,bot_bot_set_complex))) ) ) )
       => ~ aa_set_complex_o(finite3207457112153483333omplex,A2) ) ) ).

% infinite_coinduct
tff(fact_5620_infinite__coinduct,axiom,
    ! [X5: fun_se4782206882984618170_nat_o,A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(X5,A2)
     => ( ! [A5: set_Pr1261947904930325089at_nat] :
            ( aa_set3702741264012733747_nat_o(X5,A5)
           => ? [X2: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),A5)
                & ( aa_set3702741264012733747_nat_o(X5,minus_1356011639430497352at_nat(A5,insert8211810215607154385at_nat(X2,bot_bo2099793752762293965at_nat)))
                  | ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A5,insert8211810215607154385at_nat(X2,bot_bo2099793752762293965at_nat))) ) ) )
       => ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2) ) ) ).

% infinite_coinduct
tff(fact_5621_infinite__coinduct,axiom,
    ! [X5: fun_set_real_o,A2: set_real] :
      ( aa_set_real_o(X5,A2)
     => ( ! [A5: set_real] :
            ( aa_set_real_o(X5,A5)
           => ? [X2: real] :
                ( aa_set_real_o(member_real(X2),A5)
                & ( aa_set_real_o(X5,minus_minus_set_real(A5,insert_real(X2,bot_bot_set_real)))
                  | ~ aa_set_real_o(finite_finite_real,minus_minus_set_real(A5,insert_real(X2,bot_bot_set_real))) ) ) )
       => ~ aa_set_real_o(finite_finite_real,A2) ) ) ).

% infinite_coinduct
tff(fact_5622_infinite__coinduct,axiom,
    ! [X5: fun_set_int_o,A2: set_int] :
      ( aa_set_int_o(X5,A2)
     => ( ! [A5: set_int] :
            ( aa_set_int_o(X5,A5)
           => ? [X2: int] :
                ( aa_set_int_o(member_int(X2),A5)
                & ( aa_set_int_o(X5,minus_minus_set_int(A5,insert_int(X2,bot_bot_set_int)))
                  | ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(A5,insert_int(X2,bot_bot_set_int))) ) ) )
       => ~ aa_set_int_o(finite_finite_int,A2) ) ) ).

% infinite_coinduct
tff(fact_5623_infinite__coinduct,axiom,
    ! [X5: fun_set_nat_o,A2: set_nat] :
      ( aa_set_nat_o(X5,A2)
     => ( ! [A5: set_nat] :
            ( aa_set_nat_o(X5,A5)
           => ? [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),A5)
                & ( aa_set_nat_o(X5,minus_minus_set_nat(A5,insert_nat(X2,bot_bot_set_nat)))
                  | ~ aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A5,insert_nat(X2,bot_bot_set_nat))) ) ) )
       => ~ aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% infinite_coinduct
tff(fact_5624_infinite__remove,axiom,
    ! [S3: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,S3)
     => ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,minus_3314409938677909166at_nat(S3,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) ) ).

% infinite_remove
tff(fact_5625_infinite__remove,axiom,
    ! [S3: set_complex,A: complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex))) ) ).

% infinite_remove
tff(fact_5626_infinite__remove,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S3)
     => ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(S3,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) ) ).

% infinite_remove
tff(fact_5627_infinite__remove,axiom,
    ! [S3: set_real,A: real] :
      ( ~ aa_set_real_o(finite_finite_real,S3)
     => ~ aa_set_real_o(finite_finite_real,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real))) ) ).

% infinite_remove
tff(fact_5628_infinite__remove,axiom,
    ! [S3: set_int,A: int] :
      ( ~ aa_set_int_o(finite_finite_int,S3)
     => ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int))) ) ).

% infinite_remove
tff(fact_5629_infinite__remove,axiom,
    ! [S3: set_nat,A: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
     => ~ aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(S3,insert_nat(A,bot_bot_set_nat))) ) ).

% infinite_remove
tff(fact_5630_Diff__single__insert,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))),B2)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),insert9069300056098147895at_nat(X,B2)) ) ).

% Diff_single_insert
tff(fact_5631_Diff__single__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B2)
     => aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),insert8211810215607154385at_nat(X,B2)) ) ).

% Diff_single_insert
tff(fact_5632_Diff__single__insert,axiom,
    ! [A2: set_real,X: real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(minus_minus_set_real(A2,insert_real(X,bot_bot_set_real))),B2)
     => aa_set_real_o(ord_less_eq_set_real(A2),insert_real(X,B2)) ) ).

% Diff_single_insert
tff(fact_5633_Diff__single__insert,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,minus_minus_set_nat(A2,insert_nat(X,bot_bot_set_nat))),B2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat(X,B2)) ) ).

% Diff_single_insert
tff(fact_5634_Diff__single__insert,axiom,
    ! [A2: set_int,X: int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A2,insert_int(X,bot_bot_set_int))),B2)
     => aa_set_int_o(ord_less_eq_set_int(A2),insert_int(X,B2)) ) ).

% Diff_single_insert
tff(fact_5635_subset__insert__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),insert9069300056098147895at_nat(X,B2))
    <=> $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A2),aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))),B2),aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),B2)) ) ).

% subset_insert_iff
tff(fact_5636_subset__insert__iff,axiom,
    ! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),insert_set_nat(X,B2))
    <=> $ite(aa_set_set_nat_o(member_set_nat(X),A2),aa_set_set_nat_o(ord_le6893508408891458716et_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(X,bot_bot_set_set_nat))),B2),aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)) ) ).

% subset_insert_iff
tff(fact_5637_subset__insert__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),insert8211810215607154385at_nat(X,B2))
    <=> $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A2),aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B2),aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),B2)) ) ).

% subset_insert_iff
tff(fact_5638_subset__insert__iff,axiom,
    ! [A2: set_real,X: real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),insert_real(X,B2))
    <=> $ite(aa_set_real_o(member_real(X),A2),aa_set_real_o(ord_less_eq_set_real(minus_minus_set_real(A2,insert_real(X,bot_bot_set_real))),B2),aa_set_real_o(ord_less_eq_set_real(A2),B2)) ) ).

% subset_insert_iff
tff(fact_5639_subset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat(X,B2))
    <=> $ite(aa_set_nat_o(member_nat(X),A2),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,minus_minus_set_nat(A2,insert_nat(X,bot_bot_set_nat))),B2),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)) ) ).

% subset_insert_iff
tff(fact_5640_subset__insert__iff,axiom,
    ! [A2: set_int,X: int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),insert_int(X,B2))
    <=> $ite(aa_set_int_o(member_int(X),A2),aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A2,insert_int(X,bot_bot_set_int))),B2),aa_set_int_o(ord_less_eq_set_int(A2),B2)) ) ).

% subset_insert_iff
tff(fact_5641_take__bit__Suc__minus__1__eq,axiom,
    ! [Na: nat] : bit_se1745604003318907178nteger(aa_nat_nat(suc,Na),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(suc,Na)),one_one_Code_integer) ).

% take_bit_Suc_minus_1_eq
tff(fact_5642_take__bit__Suc__minus__1__eq,axiom,
    ! [Na: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),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,Na))),one_one_int) ).

% take_bit_Suc_minus_1_eq
tff(fact_5643_atLeast0__atMost__Suc,axiom,
    ! [Na: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Na)) = insert_nat(aa_nat_nat(suc,Na),set_or1269000886237332187st_nat(zero_zero_nat,Na)) ).

% atLeast0_atMost_Suc
tff(fact_5644_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_5645_Icc__eq__insert__lb__nat,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( set_or1269000886237332187st_nat(Mb,Na) = insert_nat(Mb,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Na)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_5646_atLeastAtMostSuc__conv,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Na))
     => ( set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Na)) = insert_nat(aa_nat_nat(suc,Na),set_or1269000886237332187st_nat(Mb,Na)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_5647_atLeastAtMost__insertL,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( insert_nat(Mb,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Na)) = set_or1269000886237332187st_nat(Mb,Na) ) ) ).

% atLeastAtMost_insertL
tff(fact_5648_plusinfinity,axiom,
    ! [D: int,P3: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(P3,X3)
          <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
             => ( aa_int_o(P,X3)
              <=> aa_int_o(P3,X3) ) )
         => ( ? [X_12: int] : aa_int_o(P3,X_12)
           => ? [X_1: int] : aa_int_o(P,X_1) ) ) ) ) ).

% plusinfinity
tff(fact_5649_minusinfinity,axiom,
    ! [D: int,P1: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(P1,X3)
          <=> aa_int_o(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
             => ( aa_int_o(P,X3)
              <=> aa_int_o(P1,X3) ) )
         => ( ? [X_12: int] : aa_int_o(P1,X_12)
           => ? [X_1: int] : aa_int_o(P,X_1) ) ) ) ) ).

% minusinfinity
tff(fact_5650_int__induct,axiom,
    ! [P: fun_int_o,K: int,I: int] :
      ( aa_int_o(P,K)
     => ( ! [I3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I3)
           => ( aa_int_o(P,I3)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),K)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_induct
tff(fact_5651_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N4: set_nat,Na: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N4),set_or1269000886237332187st_nat(zero_zero_nat,Na))
     => aa_set_nat_o(finite_finite_nat,N4) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_5652_take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Na),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% take_bit_int_less_exp
tff(fact_5653_pos__minus__divide__less__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_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_5654_pos__minus__divide__less__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_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_5655_pos__less__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_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_5656_pos__less__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_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_5657_neg__minus__divide__less__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_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_5658_neg__minus__divide__less__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_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_5659_neg__less__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_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_5660_neg__less__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_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_5661_minus__divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,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_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_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_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% minus_divide_less_eq
tff(fact_5662_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_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_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_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_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_less_eq
tff(fact_5663_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,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(aa_rea7340605660300375871l_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_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_minus_divide_eq
tff(fact_5664_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_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(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_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_minus_divide_eq
tff(fact_5665_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W2: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),C),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5666_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5667_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5668_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: complex,C: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),C) = B,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5669_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C) = B,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5670_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C) = B,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5671_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5672_minus__divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z))),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5673_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(X),Z))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5674_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_5675_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = $ite(Z = zero_zero_real,B,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_5676_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_5677_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_5678_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_5679_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_5680_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),aa_complex_complex(times_times_complex(B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_5681_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_5682_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_5683_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5684_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z))),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5685_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(X),Z))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5686_remove__induct,axiom,
    ! [P: fun_se7788542033685991124_nat_o,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
     => ( ( ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,B2)
         => aa_set5844364881195198285_nat_o(P,B2) )
       => ( ! [A5: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A5)
             => ( ( A5 != bot_bo228742789529271731at_nat )
               => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A5),B2)
                 => ( ! [X2: produc3843707927480180839at_nat] :
                        ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X2),A5)
                       => aa_set5844364881195198285_nat_o(P,minus_3314409938677909166at_nat(A5,insert9069300056098147895at_nat(X2,bot_bo228742789529271731at_nat))) )
                   => aa_set5844364881195198285_nat_o(P,A5) ) ) ) )
         => aa_set5844364881195198285_nat_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_5687_remove__induct,axiom,
    ! [P: fun_set_set_nat_o,B2: set_set_nat] :
      ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
     => ( ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,B2)
         => aa_set_set_nat_o(P,B2) )
       => ( ! [A5: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A5)
             => ( ( A5 != bot_bot_set_set_nat )
               => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A5),B2)
                 => ( ! [X2: set_nat] :
                        ( aa_set_set_nat_o(member_set_nat(X2),A5)
                       => aa_set_set_nat_o(P,minus_2163939370556025621et_nat(A5,insert_set_nat(X2,bot_bot_set_set_nat))) )
                   => aa_set_set_nat_o(P,A5) ) ) ) )
         => aa_set_set_nat_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_5688_remove__induct,axiom,
    ! [P: fun_set_complex_o,B2: set_complex] :
      ( aa_set_complex_o(P,bot_bot_set_complex)
     => ( ( ~ aa_set_complex_o(finite3207457112153483333omplex,B2)
         => aa_set_complex_o(P,B2) )
       => ( ! [A5: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A5)
             => ( ( A5 != bot_bot_set_complex )
               => ( aa_set_complex_o(ord_le211207098394363844omplex(A5),B2)
                 => ( ! [X2: complex] :
                        ( aa_set_complex_o(member_complex(X2),A5)
                       => aa_set_complex_o(P,minus_811609699411566653omplex(A5,insert_complex(X2,bot_bot_set_complex))) )
                   => aa_set_complex_o(P,A5) ) ) ) )
         => aa_set_complex_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_5689_remove__induct,axiom,
    ! [P: fun_se4782206882984618170_nat_o,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
     => ( ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
         => aa_set3702741264012733747_nat_o(P,B2) )
       => ( ! [A5: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A5)
             => ( ( A5 != bot_bo2099793752762293965at_nat )
               => ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A5),B2)
                 => ( ! [X2: product_prod_nat_nat] :
                        ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),A5)
                       => aa_set3702741264012733747_nat_o(P,minus_1356011639430497352at_nat(A5,insert8211810215607154385at_nat(X2,bot_bo2099793752762293965at_nat))) )
                   => aa_set3702741264012733747_nat_o(P,A5) ) ) ) )
         => aa_set3702741264012733747_nat_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_5690_remove__induct,axiom,
    ! [P: fun_set_real_o,B2: set_real] :
      ( aa_set_real_o(P,bot_bot_set_real)
     => ( ( ~ aa_set_real_o(finite_finite_real,B2)
         => aa_set_real_o(P,B2) )
       => ( ! [A5: set_real] :
              ( aa_set_real_o(finite_finite_real,A5)
             => ( ( A5 != bot_bot_set_real )
               => ( aa_set_real_o(ord_less_eq_set_real(A5),B2)
                 => ( ! [X2: real] :
                        ( aa_set_real_o(member_real(X2),A5)
                       => aa_set_real_o(P,minus_minus_set_real(A5,insert_real(X2,bot_bot_set_real))) )
                   => aa_set_real_o(P,A5) ) ) ) )
         => aa_set_real_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_5691_remove__induct,axiom,
    ! [P: fun_set_nat_o,B2: set_nat] :
      ( aa_set_nat_o(P,bot_bot_set_nat)
     => ( ( ~ aa_set_nat_o(finite_finite_nat,B2)
         => aa_set_nat_o(P,B2) )
       => ( ! [A5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A5)
             => ( ( A5 != bot_bot_set_nat )
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A5),B2)
                 => ( ! [X2: nat] :
                        ( aa_set_nat_o(member_nat(X2),A5)
                       => aa_set_nat_o(P,minus_minus_set_nat(A5,insert_nat(X2,bot_bot_set_nat))) )
                   => aa_set_nat_o(P,A5) ) ) ) )
         => aa_set_nat_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_5692_remove__induct,axiom,
    ! [P: fun_set_int_o,B2: set_int] :
      ( aa_set_int_o(P,bot_bot_set_int)
     => ( ( ~ aa_set_int_o(finite_finite_int,B2)
         => aa_set_int_o(P,B2) )
       => ( ! [A5: set_int] :
              ( aa_set_int_o(finite_finite_int,A5)
             => ( ( A5 != bot_bot_set_int )
               => ( aa_set_int_o(ord_less_eq_set_int(A5),B2)
                 => ( ! [X2: int] :
                        ( aa_set_int_o(member_int(X2),A5)
                       => aa_set_int_o(P,minus_minus_set_int(A5,insert_int(X2,bot_bot_set_int))) )
                   => aa_set_int_o(P,A5) ) ) ) )
         => aa_set_int_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_5693_finite__remove__induct,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,B2)
     => ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
       => ( ! [A5: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A5)
             => ( ( A5 != bot_bo228742789529271731at_nat )
               => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A5),B2)
                 => ( ! [X2: produc3843707927480180839at_nat] :
                        ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X2),A5)
                       => aa_set5844364881195198285_nat_o(P,minus_3314409938677909166at_nat(A5,insert9069300056098147895at_nat(X2,bot_bo228742789529271731at_nat))) )
                   => aa_set5844364881195198285_nat_o(P,A5) ) ) ) )
         => aa_set5844364881195198285_nat_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5694_finite__remove__induct,axiom,
    ! [B2: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B2)
     => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
       => ( ! [A5: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A5)
             => ( ( A5 != bot_bot_set_set_nat )
               => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A5),B2)
                 => ( ! [X2: set_nat] :
                        ( aa_set_set_nat_o(member_set_nat(X2),A5)
                       => aa_set_set_nat_o(P,minus_2163939370556025621et_nat(A5,insert_set_nat(X2,bot_bot_set_set_nat))) )
                   => aa_set_set_nat_o(P,A5) ) ) ) )
         => aa_set_set_nat_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5695_finite__remove__induct,axiom,
    ! [B2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [A5: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A5)
             => ( ( A5 != bot_bot_set_complex )
               => ( aa_set_complex_o(ord_le211207098394363844omplex(A5),B2)
                 => ( ! [X2: complex] :
                        ( aa_set_complex_o(member_complex(X2),A5)
                       => aa_set_complex_o(P,minus_811609699411566653omplex(A5,insert_complex(X2,bot_bot_set_complex))) )
                   => aa_set_complex_o(P,A5) ) ) ) )
         => aa_set_complex_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5696_finite__remove__induct,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
     => ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
       => ( ! [A5: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A5)
             => ( ( A5 != bot_bo2099793752762293965at_nat )
               => ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A5),B2)
                 => ( ! [X2: product_prod_nat_nat] :
                        ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),A5)
                       => aa_set3702741264012733747_nat_o(P,minus_1356011639430497352at_nat(A5,insert8211810215607154385at_nat(X2,bot_bo2099793752762293965at_nat))) )
                   => aa_set3702741264012733747_nat_o(P,A5) ) ) ) )
         => aa_set3702741264012733747_nat_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5697_finite__remove__induct,axiom,
    ! [B2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [A5: set_real] :
              ( aa_set_real_o(finite_finite_real,A5)
             => ( ( A5 != bot_bot_set_real )
               => ( aa_set_real_o(ord_less_eq_set_real(A5),B2)
                 => ( ! [X2: real] :
                        ( aa_set_real_o(member_real(X2),A5)
                       => aa_set_real_o(P,minus_minus_set_real(A5,insert_real(X2,bot_bot_set_real))) )
                   => aa_set_real_o(P,A5) ) ) ) )
         => aa_set_real_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5698_finite__remove__induct,axiom,
    ! [B2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [A5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A5)
             => ( ( A5 != bot_bot_set_nat )
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A5),B2)
                 => ( ! [X2: nat] :
                        ( aa_set_nat_o(member_nat(X2),A5)
                       => aa_set_nat_o(P,minus_minus_set_nat(A5,insert_nat(X2,bot_bot_set_nat))) )
                   => aa_set_nat_o(P,A5) ) ) ) )
         => aa_set_nat_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5699_finite__remove__induct,axiom,
    ! [B2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [A5: set_int] :
              ( aa_set_int_o(finite_finite_int,A5)
             => ( ( A5 != bot_bot_set_int )
               => ( aa_set_int_o(ord_less_eq_set_int(A5),B2)
                 => ( ! [X2: int] :
                        ( aa_set_int_o(member_int(X2),A5)
                       => aa_set_int_o(P,minus_minus_set_int(A5,insert_int(X2,bot_bot_set_int))) )
                   => aa_set_int_o(P,A5) ) ) ) )
         => aa_set_int_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5700_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(bit_ri631733984087533419it_int(Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_5701_finite__induct__select,axiom,
    ! [S3: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,S3)
     => ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
       => ( ! [T4: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(T4),S3)
             => ( aa_set5844364881195198285_nat_o(P,T4)
               => ? [X2: produc3843707927480180839at_nat] :
                    ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X2),minus_3314409938677909166at_nat(S3,T4))
                    & aa_set5844364881195198285_nat_o(P,insert9069300056098147895at_nat(X2,T4)) ) ) )
         => aa_set5844364881195198285_nat_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_5702_finite__induct__select,axiom,
    ! [S3: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [T4: set_complex] :
              ( aa_set_complex_o(ord_less_set_complex(T4),S3)
             => ( aa_set_complex_o(P,T4)
               => ? [X2: complex] :
                    ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(S3,T4))
                    & aa_set_complex_o(P,insert_complex(X2,T4)) ) ) )
         => aa_set_complex_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_5703_finite__induct__select,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S3)
     => ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
       => ( ! [T4: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(T4),S3)
             => ( aa_set3702741264012733747_nat_o(P,T4)
               => ? [X2: product_prod_nat_nat] :
                    ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),minus_1356011639430497352at_nat(S3,T4))
                    & aa_set3702741264012733747_nat_o(P,insert8211810215607154385at_nat(X2,T4)) ) ) )
         => aa_set3702741264012733747_nat_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_5704_finite__induct__select,axiom,
    ! [S3: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [T4: set_real] :
              ( aa_set_real_o(ord_less_set_real(T4),S3)
             => ( aa_set_real_o(P,T4)
               => ? [X2: real] :
                    ( aa_set_real_o(member_real(X2),minus_minus_set_real(S3,T4))
                    & aa_set_real_o(P,insert_real(X2,T4)) ) ) )
         => aa_set_real_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_5705_finite__induct__select,axiom,
    ! [S3: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [T4: set_int] :
              ( aa_set_int_o(ord_less_set_int(T4),S3)
             => ( aa_set_int_o(P,T4)
               => ? [X2: int] :
                    ( aa_set_int_o(member_int(X2),minus_minus_set_int(S3,T4))
                    & aa_set_int_o(P,insert_int(X2,T4)) ) ) )
         => aa_set_int_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_5706_finite__induct__select,axiom,
    ! [S3: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [T4: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,T4),S3)
             => ( aa_set_nat_o(P,T4)
               => ? [X2: nat] :
                    ( aa_set_nat_o(member_nat(X2),minus_minus_set_nat(S3,T4))
                    & aa_set_nat_o(P,insert_nat(X2,T4)) ) ) )
         => aa_set_nat_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_5707_psubset__insert__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(A2),insert9069300056098147895at_nat(X,B2))
    <=> $ite(
          aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),B2),
          aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(A2),B2),
          $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A2),aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))),B2),aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_5708_psubset__insert__iff,axiom,
    ! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A2),insert_set_nat(X,B2))
    <=> $ite(
          aa_set_set_nat_o(member_set_nat(X),B2),
          aa_set_set_nat_o(ord_less_set_set_nat(A2),B2),
          $ite(aa_set_set_nat_o(member_set_nat(X),A2),aa_set_set_nat_o(ord_less_set_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(X,bot_bot_set_set_nat))),B2),aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_5709_psubset__insert__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(A2),insert8211810215607154385at_nat(X,B2))
    <=> $ite(
          aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),B2),
          aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(A2),B2),
          $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A2),aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B2),aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_5710_psubset__insert__iff,axiom,
    ! [A2: set_real,X: real,B2: set_real] :
      ( aa_set_real_o(ord_less_set_real(A2),insert_real(X,B2))
    <=> $ite(
          aa_set_real_o(member_real(X),B2),
          aa_set_real_o(ord_less_set_real(A2),B2),
          $ite(aa_set_real_o(member_real(X),A2),aa_set_real_o(ord_less_set_real(minus_minus_set_real(A2,insert_real(X,bot_bot_set_real))),B2),aa_set_real_o(ord_less_eq_set_real(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_5711_psubset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),insert_nat(X,B2))
    <=> $ite(
          aa_set_nat_o(member_nat(X),B2),
          aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2),
          $ite(aa_set_nat_o(member_nat(X),A2),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,minus_minus_set_nat(A2,insert_nat(X,bot_bot_set_nat))),B2),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_5712_psubset__insert__iff,axiom,
    ! [A2: set_int,X: int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),insert_int(X,B2))
    <=> $ite(
          aa_set_int_o(member_int(X),B2),
          aa_set_int_o(ord_less_set_int(A2),B2),
          $ite(aa_set_int_o(member_int(X),A2),aa_set_int_o(ord_less_set_int(minus_minus_set_int(A2,insert_int(X,bot_bot_set_int))),B2),aa_set_int_o(ord_less_eq_set_int(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_5713_decr__mult__lemma,axiom,
    ! [D: int,P: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X3: int] :
            ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X2: int] :
              ( aa_int_o(P,X2)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_5714_verit__less__mono__div__int2,axiom,
    ! [A2: int,B2: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,Na))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(B2),Na)),aa_int_int(divide_divide_int(A2),Na)) ) ) ).

% verit_less_mono_div_int2
tff(fact_5715_div__eq__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_5716_take__bit__Suc__bit0,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_5717_take__bit__Suc__bit0,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Na),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_5718_take__bit__nat__eq__self,axiom,
    ! [Mb: nat,Na: 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))),Na))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),Mb) = Mb ) ) ).

% take_bit_nat_eq_self
tff(fact_5719_take__bit__nat__less__exp,axiom,
    ! [Na: 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,Na),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% take_bit_nat_less_exp
tff(fact_5720_take__bit__nat__eq__self__iff,axiom,
    ! [Na: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),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))),Na)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_5721_take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Na),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_5722_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_se2923211474154528505it_int(Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_5723_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_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_5724_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(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(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(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_5725_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_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_5726_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(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(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(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_5727_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_5728_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(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(times_times_rat(A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_5729_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(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5730_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(divide_divide_rat(B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5731_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(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5732_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(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5733_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5734_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(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(times_times_rat(A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5735_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),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(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),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,W2)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5736_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),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(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),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,W2)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5737_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: 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,W2))),aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),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(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5738_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: 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,W2))),aa_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(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),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(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5739_signed__take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Na),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Na)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_5740_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5741_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5742_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5743_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5744_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5745_signed__take__bit__int__eq__self,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_int(bit_ri631733984087533419it_int(Na),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_5746_signed__take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(Na),K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_5747_take__bit__eq__0__iff,axiom,
    ! [Na: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger(Na,A) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)),A) ) ).

% take_bit_eq_0_iff
tff(fact_5748_take__bit__eq__0__iff,axiom,
    ! [Na: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),A) ) ).

% take_bit_eq_0_iff
tff(fact_5749_take__bit__eq__0__iff,axiom,
    ! [Na: nat,A: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Na),A) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),A) ) ).

% take_bit_eq_0_iff
tff(fact_5750_take__bit__nat__less__self__iff,axiom,
    ! [Na: 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,Na),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))),Na)),Mb) ) ).

% take_bit_nat_less_self_iff
tff(fact_5751_take__bit__int__eq__self,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_int(bit_se2923211474154528505it_int(Na),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_5752_take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Na),K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_5753_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),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(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),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,W2)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5754_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),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(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),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,W2)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5755_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: 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,W2))),aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),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(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),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,W2))),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5756_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: 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,W2))),aa_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(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),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(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),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,W2))),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5757_square__le__1,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ) ) ).

% square_le_1
tff(fact_5758_square__le__1,axiom,
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),X)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% square_le_1
tff(fact_5759_square__le__1,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat) ) ) ).

% square_le_1
tff(fact_5760_square__le__1,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% square_le_1
tff(fact_5761_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_5762_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int)
       => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_5763_take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),aa_int_int(bit_se2923211474154528505it_int(Na),K)) ) ).

% take_bit_int_greater_eq
tff(fact_5764_power__minus1__odd,axiom,
    ! [Na: 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))),Na))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_5765_power__minus1__odd,axiom,
    ! [Na: 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))),Na))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_5766_power__minus1__odd,axiom,
    ! [Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_5767_power__minus1__odd,axiom,
    ! [Na: 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))),Na))) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% power_minus1_odd
tff(fact_5768_power__minus1__odd,axiom,
    ! [Na: 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))),Na))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% power_minus1_odd
tff(fact_5769_take__bit__Suc,axiom,
    ! [Na: nat,A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,Na),A) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(Na,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))),numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_5770_take__bit__Suc,axiom,
    ! [Na: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Na)),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,Na),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_5771_take__bit__Suc,axiom,
    ! [Na: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),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(Na),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_5772_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Na)))),aa_int_int(bit_ri631733984087533419it_int(Na),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_5773_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,Na: nat] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( bit_se1745604003318907178nteger(Na,A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na),one_one_Code_integer)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_5774_stable__imp__take__bit__eq,axiom,
    ! [A: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),one_one_nat)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_5775_stable__imp__take__bit__eq,axiom,
    ! [A: int,Na: nat] :
      ( ( aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(bit_se2923211474154528505it_int(Na),A) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),one_one_int)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_5776_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Na),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_5777_concat__bit__Suc,axiom,
    ! [Na: nat,K: int,L: int] : aa_int_int(bit_concat_bit(aa_nat_nat(suc,Na),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_concat_bit(Na,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_5778_Bolzano,axiom,
    ! [A: real,B: real,P: fun_real_fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [A3: real,B3: real,C2: real] :
            ( aa_real_o(aa_real_fun_real_o(P,A3),B3)
           => ( aa_real_o(aa_real_fun_real_o(P,B3),C2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A3),B3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B3),C2)
                 => aa_real_o(aa_real_fun_real_o(P,A3),C2) ) ) ) )
       => ( ! [X3: real] :
              ( 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)
               => ? [D3: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
                    & ! [A3: real,B3: real] :
                        ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A3),X3)
                          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B3)
                          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),A3)),D3) )
                       => aa_real_o(aa_real_fun_real_o(P,A3),B3) ) ) ) )
         => aa_real_o(aa_real_fun_real_o(P,A),B) ) ) ) ).

% Bolzano
tff(fact_5779_signed__take__bit__Suc__bit1,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Na)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Na),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_5780_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X),Xa2) = Y )
     => ( Y = $ite(
            ( aa_set_int_o(member_int(X),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))),X)
                & ~ 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))),X)
                  & ~ 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(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(Xa2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.elims
tff(fact_5781_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K),L) = $ite(
        ( aa_set_int_o(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
        & aa_set_int_o(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
        aa_int_int(uminus_uminus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
        aa_int_int(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))),K)
              & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ).

% and_int.simps
tff(fact_5782_Diff__idemp,axiom,
    ! [A2: set_nat,B2: set_nat] : minus_minus_set_nat(minus_minus_set_nat(A2,B2),B2) = minus_minus_set_nat(A2,B2) ).

% Diff_idemp
tff(fact_5783_Diff__iff,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),minus_1356011639430497352at_nat(A2,B2))
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),A2)
        & ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),B2) ) ) ).

% Diff_iff
tff(fact_5784_Diff__iff,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),minus_minus_set_real(A2,B2))
    <=> ( aa_set_real_o(member_real(C),A2)
        & ~ aa_set_real_o(member_real(C),B2) ) ) ).

% Diff_iff
tff(fact_5785_Diff__iff,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),minus_2163939370556025621et_nat(A2,B2))
    <=> ( aa_set_set_nat_o(member_set_nat(C),A2)
        & ~ aa_set_set_nat_o(member_set_nat(C),B2) ) ) ).

% Diff_iff
tff(fact_5786_Diff__iff,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A2,B2))
    <=> ( aa_set_int_o(member_int(C),A2)
        & ~ aa_set_int_o(member_int(C),B2) ) ) ).

% Diff_iff
tff(fact_5787_Diff__iff,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),minus_minus_set_nat(A2,B2))
    <=> ( aa_set_nat_o(member_nat(C),A2)
        & ~ aa_set_nat_o(member_nat(C),B2) ) ) ).

% Diff_iff
tff(fact_5788_DiffI,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),A2)
     => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),B2)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),minus_1356011639430497352at_nat(A2,B2)) ) ) ).

% DiffI
tff(fact_5789_DiffI,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),A2)
     => ( ~ aa_set_real_o(member_real(C),B2)
       => aa_set_real_o(member_real(C),minus_minus_set_real(A2,B2)) ) ) ).

% DiffI
tff(fact_5790_DiffI,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),A2)
     => ( ~ aa_set_set_nat_o(member_set_nat(C),B2)
       => aa_set_set_nat_o(member_set_nat(C),minus_2163939370556025621et_nat(A2,B2)) ) ) ).

% DiffI
tff(fact_5791_DiffI,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),A2)
     => ( ~ aa_set_int_o(member_int(C),B2)
       => aa_set_int_o(member_int(C),minus_minus_set_int(A2,B2)) ) ) ).

% DiffI
tff(fact_5792_DiffI,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),A2)
     => ( ~ aa_set_nat_o(member_nat(C),B2)
       => aa_set_nat_o(member_nat(C),minus_minus_set_nat(A2,B2)) ) ) ).

% DiffI
tff(fact_5793_ComplI,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),A2)
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),uminus6524753893492686040at_nat(A2)) ) ).

% ComplI
tff(fact_5794_ComplI,axiom,
    ! [C: real,A2: set_real] :
      ( ~ aa_set_real_o(member_real(C),A2)
     => aa_set_real_o(member_real(C),uminus612125837232591019t_real(A2)) ) ).

% ComplI
tff(fact_5795_ComplI,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(C),A2)
     => aa_set_set_nat_o(member_set_nat(C),uminus613421341184616069et_nat(A2)) ) ).

% ComplI
tff(fact_5796_ComplI,axiom,
    ! [C: nat,A2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(C),A2)
     => aa_set_nat_o(member_nat(C),uminus5710092332889474511et_nat(A2)) ) ).

% ComplI
tff(fact_5797_ComplI,axiom,
    ! [C: int,A2: set_int] :
      ( ~ aa_set_int_o(member_int(C),A2)
     => aa_set_int_o(member_int(C),uminus1532241313380277803et_int(A2)) ) ).

% ComplI
tff(fact_5798_Compl__iff,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),uminus6524753893492686040at_nat(A2))
    <=> ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),A2) ) ).

% Compl_iff
tff(fact_5799_Compl__iff,axiom,
    ! [C: real,A2: set_real] :
      ( aa_set_real_o(member_real(C),uminus612125837232591019t_real(A2))
    <=> ~ aa_set_real_o(member_real(C),A2) ) ).

% Compl_iff
tff(fact_5800_Compl__iff,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),uminus613421341184616069et_nat(A2))
    <=> ~ aa_set_set_nat_o(member_set_nat(C),A2) ) ).

% Compl_iff
tff(fact_5801_Compl__iff,axiom,
    ! [C: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(C),uminus5710092332889474511et_nat(A2))
    <=> ~ aa_set_nat_o(member_nat(C),A2) ) ).

% Compl_iff
tff(fact_5802_Compl__iff,axiom,
    ! [C: int,A2: set_int] :
      ( aa_set_int_o(member_int(C),uminus1532241313380277803et_int(A2))
    <=> ~ aa_set_int_o(member_int(C),A2) ) ).

% Compl_iff
tff(fact_5803_and__zero__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),zero_zero_int) = zero_zero_int ).

% and_zero_eq
tff(fact_5804_and__zero__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),zero_zero_nat) = zero_zero_nat ).

% and_zero_eq
tff(fact_5805_zero__and__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),A) = zero_zero_int ).

% zero_and_eq
tff(fact_5806_zero__and__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,zero_zero_nat),A) = zero_zero_nat ).

% zero_and_eq
tff(fact_5807_bit_Oconj__zero__left,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),X) = zero_zero_int ).

% bit.conj_zero_left
tff(fact_5808_bit_Oconj__zero__right,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(X),zero_zero_int) = zero_zero_int ).

% bit.conj_zero_right
tff(fact_5809_concat__bit__0,axiom,
    ! [K: int,L: int] : aa_int_int(bit_concat_bit(zero_zero_nat,K),L) = L ).

% concat_bit_0
tff(fact_5810_semiring__norm_I80_J,axiom,
    ! [Mb: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Na) ) ).

% semiring_norm(80)
tff(fact_5811_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% and_negative_int_iff
tff(fact_5812_semiring__norm_I81_J,axiom,
    ! [Mb: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Na) ) ).

% semiring_norm(81)
tff(fact_5813_semiring__norm_I77_J,axiom,
    ! [Na: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit1,Na)) ).

% semiring_norm(77)
tff(fact_5814_concat__bit__negative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_concat_bit(Na,K),L)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ).

% concat_bit_negative_iff
tff(fact_5815_semiring__norm_I79_J,axiom,
    ! [Mb: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Na) ) ).

% semiring_norm(79)
tff(fact_5816_semiring__norm_I74_J,axiom,
    ! [Mb: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Na) ) ).

% semiring_norm(74)
tff(fact_5817_and__numerals_I5_J,axiom,
    ! [X: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),one_one_int) = zero_zero_int ).

% and_numerals(5)
tff(fact_5818_and__numerals_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),one_one_nat) = zero_zero_nat ).

% and_numerals(5)
tff(fact_5819_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = zero_zero_int ).

% and_numerals(1)
tff(fact_5820_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_5821_div__Suc__eq__div__add3,axiom,
    ! [Mb: nat,Na: 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,Na)))) = 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))),Na)) ).

% div_Suc_eq_div_add3
tff(fact_5822_Suc__div__eq__add3__div__numeral,axiom,
    ! [Mb: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_5823_mod__Suc__eq__mod__add3,axiom,
    ! [Mb: nat,Na: 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,Na)))) = 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))),Na)) ).

% mod_Suc_eq_mod_add3
tff(fact_5824_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [Mb: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_5825_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% and_numerals(7)
tff(fact_5826_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% and_numerals(7)
tff(fact_5827_DiffD2,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),minus_1356011639430497352at_nat(A2,B2))
     => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),B2) ) ).

% DiffD2
tff(fact_5828_DiffD2,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),minus_minus_set_real(A2,B2))
     => ~ aa_set_real_o(member_real(C),B2) ) ).

% DiffD2
tff(fact_5829_DiffD2,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),minus_2163939370556025621et_nat(A2,B2))
     => ~ aa_set_set_nat_o(member_set_nat(C),B2) ) ).

% DiffD2
tff(fact_5830_DiffD2,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A2,B2))
     => ~ aa_set_int_o(member_int(C),B2) ) ).

% DiffD2
tff(fact_5831_DiffD2,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),minus_minus_set_nat(A2,B2))
     => ~ aa_set_nat_o(member_nat(C),B2) ) ).

% DiffD2
tff(fact_5832_DiffD1,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),minus_1356011639430497352at_nat(A2,B2))
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),A2) ) ).

% DiffD1
tff(fact_5833_DiffD1,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),minus_minus_set_real(A2,B2))
     => aa_set_real_o(member_real(C),A2) ) ).

% DiffD1
tff(fact_5834_DiffD1,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),minus_2163939370556025621et_nat(A2,B2))
     => aa_set_set_nat_o(member_set_nat(C),A2) ) ).

% DiffD1
tff(fact_5835_DiffD1,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A2,B2))
     => aa_set_int_o(member_int(C),A2) ) ).

% DiffD1
tff(fact_5836_DiffD1,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),minus_minus_set_nat(A2,B2))
     => aa_set_nat_o(member_nat(C),A2) ) ).

% DiffD1
tff(fact_5837_DiffE,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),minus_1356011639430497352at_nat(A2,B2))
     => ~ ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),A2)
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),B2) ) ) ).

% DiffE
tff(fact_5838_DiffE,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),minus_minus_set_real(A2,B2))
     => ~ ( aa_set_real_o(member_real(C),A2)
         => aa_set_real_o(member_real(C),B2) ) ) ).

% DiffE
tff(fact_5839_DiffE,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),minus_2163939370556025621et_nat(A2,B2))
     => ~ ( aa_set_set_nat_o(member_set_nat(C),A2)
         => aa_set_set_nat_o(member_set_nat(C),B2) ) ) ).

% DiffE
tff(fact_5840_DiffE,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A2,B2))
     => ~ ( aa_set_int_o(member_int(C),A2)
         => aa_set_int_o(member_int(C),B2) ) ) ).

% DiffE
tff(fact_5841_DiffE,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),minus_minus_set_nat(A2,B2))
     => ~ ( aa_set_nat_o(member_nat(C),A2)
         => aa_set_nat_o(member_nat(C),B2) ) ) ).

% DiffE
tff(fact_5842_ComplD,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),uminus6524753893492686040at_nat(A2))
     => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),A2) ) ).

% ComplD
tff(fact_5843_ComplD,axiom,
    ! [C: real,A2: set_real] :
      ( aa_set_real_o(member_real(C),uminus612125837232591019t_real(A2))
     => ~ aa_set_real_o(member_real(C),A2) ) ).

% ComplD
tff(fact_5844_ComplD,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),uminus613421341184616069et_nat(A2))
     => ~ aa_set_set_nat_o(member_set_nat(C),A2) ) ).

% ComplD
tff(fact_5845_ComplD,axiom,
    ! [C: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(C),uminus5710092332889474511et_nat(A2))
     => ~ aa_set_nat_o(member_nat(C),A2) ) ).

% ComplD
tff(fact_5846_ComplD,axiom,
    ! [C: int,A2: set_int] :
      ( aa_set_int_o(member_int(C),uminus1532241313380277803et_int(A2))
     => ~ aa_set_int_o(member_int(C),A2) ) ).

% ComplD
tff(fact_5847_concat__bit__assoc,axiom,
    ! [Na: nat,K: int,Mb: nat,L: int,R2: int] : aa_int_int(bit_concat_bit(Na,K),aa_int_int(bit_concat_bit(Mb,L),R2)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na),aa_int_int(bit_concat_bit(Na,K),L)),R2) ).

% concat_bit_assoc
tff(fact_5848_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X != product_Pair_num_num(one,one) )
     => ( ! [N: num] : X != product_Pair_num_num(one,aa_num_num(bit0,N))
       => ( ! [N: num] : X != product_Pair_num_num(one,aa_num_num(bit1,N))
         => ( ! [M: num] : X != product_Pair_num_num(aa_num_num(bit0,M),one)
           => ( ! [M: num,N: num] : X != product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit0,N))
             => ( ! [M: num,N: num] : X != product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit1,N))
               => ( ! [M: num] : X != product_Pair_num_num(aa_num_num(bit1,M),one)
                 => ( ! [M: num,N: num] : X != product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit0,N))
                   => ~ ! [M: num,N: num] : X != product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit1,N)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_5849_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),K) ) ).

% and_less_eq
tff(fact_5850_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: 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_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_5851_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: 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_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_5852_numeral__Bit1,axiom,
    ! [Na: num] : numera6690914467698888265omplex(aa_num_num(bit1,Na)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Na)),numera6690914467698888265omplex(Na))),one_one_complex) ).

% numeral_Bit1
tff(fact_5853_numeral__Bit1,axiom,
    ! [Na: num] : numera1916890842035813515d_enat(aa_num_num(bit1,Na)) = plus_p3455044024723400733d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Na),numera1916890842035813515d_enat(Na)),one_on7984719198319812577d_enat) ).

% numeral_Bit1
tff(fact_5854_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Na)),aa_num_real(numeral_numeral_real,Na))),one_one_real) ).

% numeral_Bit1
tff(fact_5855_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Na)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Na)),aa_num_rat(numeral_numeral_rat,Na))),one_one_rat) ).

% numeral_Bit1
tff(fact_5856_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,Na))),one_one_nat) ).

% numeral_Bit1
tff(fact_5857_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,Na))),one_one_int) ).

% numeral_Bit1
tff(fact_5858_simp__from__to,axiom,
    ! [I: int,J: int] :
      aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,I),J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),bot_bot_set_int,insert_int(I,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int)),J))) ).

% simp_from_to
tff(fact_5859_eval__nat__numeral_I3_J,axiom,
    ! [Na: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na))) ).

% eval_nat_numeral(3)
tff(fact_5860_cong__exp__iff__simps_I3_J,axiom,
    ! [Na: num,Q4: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_5861_cong__exp__iff__simps_I3_J,axiom,
    ! [Na: num,Q4: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_5862_cong__exp__iff__simps_I3_J,axiom,
    ! [Na: num,Q4: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Na)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_5863_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_5864_Suc3__eq__add__3,axiom,
    ! [Na: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Na) ).

% Suc3_eq_add_3
tff(fact_5865_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_5866_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_5867_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q4: 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,Q4))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Q4)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5868_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q4: num] :
      ( ( aa_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,Q4))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Q4)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5869_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q4: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Mb)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) = modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Q4)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5870_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,Na: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,Q4)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5871_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,Na: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,Q4)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5872_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,Na: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Na)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Na),numera6620942414471956472nteger(Q4)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5873_periodic__finite__ex,axiom,
    ! [D: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(P,X3)
          <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [X_13: int] : aa_int_o(P,X_13)
        <=> ? [X4: int] :
              ( aa_set_int_o(member_int(X4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D))
              & aa_int_o(P,X4) ) ) ) ) ).

% periodic_finite_ex
tff(fact_5874_bset_I3_J,axiom,
    ! [D5: int,T: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
           => ( ( X2 = T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D5) = T ) ) ) ) ) ).

% bset(3)
tff(fact_5875_bset_I4_J,axiom,
    ! [D5: int,T: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( aa_set_int_o(member_int(T),B2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
           => ( ( X2 != T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D5) != T ) ) ) ) ) ).

% bset(4)
tff(fact_5876_bset_I5_J,axiom,
    ! [D5: int,B2: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),T)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D5)),T) ) ) ) ).

% bset(5)
tff(fact_5877_bset_I7_J,axiom,
    ! [D5: int,T: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( aa_set_int_o(member_int(T),B2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,T),X2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D5)) ) ) ) ) ).

% bset(7)
tff(fact_5878_aset_I3_J,axiom,
    ! [D5: int,T: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
           => ( ( X2 = T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D5) = T ) ) ) ) ) ).

% aset(3)
tff(fact_5879_aset_I4_J,axiom,
    ! [D5: int,T: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( aa_set_int_o(member_int(T),A2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
           => ( ( X2 != T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D5) != T ) ) ) ) ) ).

% aset(4)
tff(fact_5880_aset_I5_J,axiom,
    ! [D5: int,T: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( aa_set_int_o(member_int(T),A2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),T)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D5)),T) ) ) ) ) ).

% aset(5)
tff(fact_5881_aset_I7_J,axiom,
    ! [D5: int,A2: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,T),X2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D5)) ) ) ) ).

% aset(7)
tff(fact_5882_Suc__div__eq__add3__div,axiom,
    ! [Mb: nat,Na: 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)))),Na) = 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)),Na) ).

% Suc_div_eq_add3_div
tff(fact_5883_Suc__mod__eq__add3__mod,axiom,
    ! [Mb: nat,Na: 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)))),Na) = 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)),Na) ).

% Suc_mod_eq_add3_mod
tff(fact_5884_aset_I8_J,axiom,
    ! [D5: int,A2: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D5)) ) ) ) ).

% aset(8)
tff(fact_5885_aset_I6_J,axiom,
    ! [D5: int,T: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),T)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D5)),T) ) ) ) ) ).

% aset(6)
tff(fact_5886_bset_I8_J,axiom,
    ! [D5: int,T: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D5)) ) ) ) ) ).

% bset(8)
tff(fact_5887_bset_I6_J,axiom,
    ! [D5: int,B2: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),T)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D5)),T) ) ) ) ).

% bset(6)
tff(fact_5888_cppi,axiom,
    ! [D5: int,P: fun_int_o,P3: fun_int_o,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
           => ( aa_int_o(P,X3)
            <=> aa_int_o(P3,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5))
                 => ! [Xb3: int] :
                      ( aa_set_int_o(member_int(Xb3),A2)
                     => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa) ) ) )
             => ( aa_int_o(P,X3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5)) ) )
         => ( ! [X3: int,K2: int] :
                ( aa_int_o(P3,X3)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D5))) )
           => ( ? [X_13: int] : aa_int_o(P,X_13)
            <=> ( ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5))
                    & aa_int_o(P3,X4) )
                | ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5))
                    & ? [Xa4: int] :
                        ( aa_set_int_o(member_int(Xa4),A2)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa4),X4)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_5889_cpmi,axiom,
    ! [D5: int,P: fun_int_o,P3: fun_int_o,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
           => ( aa_int_o(P,X3)
            <=> aa_int_o(P3,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5))
                 => ! [Xb3: int] :
                      ( aa_set_int_o(member_int(Xb3),B2)
                     => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa) ) ) )
             => ( aa_int_o(P,X3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5)) ) )
         => ( ! [X3: int,K2: int] :
                ( aa_int_o(P3,X3)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D5))) )
           => ( ? [X_13: int] : aa_int_o(P,X_13)
            <=> ( ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5))
                    & aa_int_o(P3,X4) )
                | ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5))
                    & ? [Xa4: int] :
                        ( aa_set_int_o(member_int(Xa4),B2)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa4),X4)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_5890_take__bit__Suc__bit1,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_Suc_bit1
tff(fact_5891_take__bit__Suc__bit1,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Na),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_bit1
tff(fact_5892_odd__mod__4__div__2,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% odd_mod_4_div_2
tff(fact_5893_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_5894_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Na: num] :
      unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit1,Na)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Na),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,Na),unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(7)
tff(fact_5895_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Na: num] :
      unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit1,Na)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Na),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,Na)),unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(7)
tff(fact_5896_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Na: num] :
      unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit1,Na)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Na),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit0,Mb))),unique4921790084139445826nteger(aa_num_num(bit1,Na),unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(7)
tff(fact_5897_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Na: num] :
      unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit1,Na)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Na),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,Na),unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(8)
tff(fact_5898_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Na: num] :
      unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit1,Na)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Na),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,Na)),unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(8)
tff(fact_5899_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Na: num] :
      unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit1,Na)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Na),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit1,Mb))),unique4921790084139445826nteger(aa_num_num(bit1,Na),unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(8)
tff(fact_5900_take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_5901_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K),L))
     => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = $ite(
            ( aa_set_int_o(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
            aa_int_int(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))),K)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.psimps
tff(fact_5902_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X),Xa2) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa2))
       => ~ ( ( Y = $ite(
                  ( aa_set_int_o(member_int(X),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))),X)
                      & ~ 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))),X)
                        & ~ 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(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(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,X),Xa2)) ) ) ) ).

% and_int.pelims
tff(fact_5903_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : aa_set_int_o(finite_finite_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),U)) ).

% finite_atLeastAtMost_int
tff(fact_5904_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_5905_and__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_5906_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_5907_and__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_5908_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_5909_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_5910_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique3479559517661332726nteger(Mb,one) = produc1086072967326762835nteger(numera6620942414471956472nteger(Mb),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_5911_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_5912_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_5913_add__neg__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_real_real(aa_rea7340605660300375871l_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_5914_add__neg__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_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_5915_add__neg__numeral__special_I6_J,axiom,
    ! [Mb: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,Mb))) ).

% add_neg_numeral_special(6)
tff(fact_5916_add__neg__numeral__special_I5_J,axiom,
    ! [Na: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,Na))) ).

% add_neg_numeral_special(5)
tff(fact_5917_add__neg__numeral__special_I5_J,axiom,
    ! [Na: 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,Na))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Na))) ).

% add_neg_numeral_special(5)
tff(fact_5918_add__neg__numeral__special_I5_J,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,Na))) ).

% add_neg_numeral_special(5)
tff(fact_5919_add__neg__numeral__special_I5_J,axiom,
    ! [Na: num] : aa_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,Na))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Na))) ).

% add_neg_numeral_special(5)
tff(fact_5920_add__neg__numeral__special_I5_J,axiom,
    ! [Na: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,Na))) ).

% add_neg_numeral_special(5)
tff(fact_5921_and__Suc__0__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_5922_Suc__0__and__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_5923_divmod__algorithm__code_I3_J,axiom,
    ! [Na: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,Na)) = 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_5924_divmod__algorithm__code_I3_J,axiom,
    ! [Na: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,Na)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_5925_divmod__algorithm__code_I3_J,axiom,
    ! [Na: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,Na)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(3)
tff(fact_5926_divmod__algorithm__code_I4_J,axiom,
    ! [Na: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,Na)) = 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_5927_divmod__algorithm__code_I4_J,axiom,
    ! [Na: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,Na)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_5928_divmod__algorithm__code_I4_J,axiom,
    ! [Na: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,Na)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(4)
tff(fact_5929_numeral__inc,axiom,
    ! [X: num] : numera6690914467698888265omplex(aa_num_num(inc,X)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(X)),one_one_complex) ).

% numeral_inc
tff(fact_5930_numeral__inc,axiom,
    ! [X: num] : numera1916890842035813515d_enat(aa_num_num(inc,X)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(X),one_on7984719198319812577d_enat) ).

% numeral_inc
tff(fact_5931_numeral__inc,axiom,
    ! [X: num] : aa_num_real(numeral_numeral_real,aa_num_num(inc,X)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,X)),one_one_real) ).

% numeral_inc
tff(fact_5932_numeral__inc,axiom,
    ! [X: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(inc,X)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,X)),one_one_rat) ).

% numeral_inc
tff(fact_5933_numeral__inc,axiom,
    ! [X: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(inc,X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) ).

% numeral_inc
tff(fact_5934_numeral__inc,axiom,
    ! [X: num] : aa_num_int(numeral_numeral_int,aa_num_num(inc,X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X)),one_one_int) ).

% numeral_inc
tff(fact_5935_divmod__int__def,axiom,
    ! [Mb: num,Na: num] : unique5052692396658037445od_int(Mb,Na) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Na))),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Na))) ).

% divmod_int_def
tff(fact_5936_divmod__def,axiom,
    ! [Mb: num,Na: num] : unique5052692396658037445od_int(Mb,Na) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Na))),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Na))) ).

% divmod_def
tff(fact_5937_divmod__def,axiom,
    ! [Mb: num,Na: num] : unique5055182867167087721od_nat(Mb,Na) = 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,Na))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Na))) ).

% divmod_def
tff(fact_5938_divmod__def,axiom,
    ! [Mb: num,Na: num] : unique3479559517661332726nteger(Mb,Na) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Na)),modulo364778990260209775nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Na))) ).

% divmod_def
tff(fact_5939_divmod_H__nat__def,axiom,
    ! [Mb: num,Na: num] : unique5055182867167087721od_nat(Mb,Na) = 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,Na))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Na))) ).

% divmod'_nat_def
tff(fact_5940_and__nat__unfold,axiom,
    ! [Mb: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Na) = $ite(
        ( ( Mb = zero_zero_nat )
        | ( Na = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% and_nat_unfold
tff(fact_5941_and__nat__rec,axiom,
    ! [Mb: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Na) = 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))),Na) ))),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_5942_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))
     => ( ! [K2: int,L3: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K2),L3))
           => ( ( ~ ( aa_set_int_o(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & aa_set_int_o(member_int(L3),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) )
             => aa_int_o(aa_int_fun_int_o(P,K2),L3) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_5943_divmod__divmod__step,axiom,
    ! [Mb: num,Na: num] :
      unique5055182867167087721od_nat(Mb,Na) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Na),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,Mb)),unique5026877609467782581ep_nat(Na,unique5055182867167087721od_nat(Mb,aa_num_num(bit0,Na)))) ).

% divmod_divmod_step
tff(fact_5944_divmod__divmod__step,axiom,
    ! [Mb: num,Na: num] :
      unique5052692396658037445od_int(Mb,Na) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Na),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(Na),unique5052692396658037445od_int(Mb,aa_num_num(bit0,Na)))) ).

% divmod_divmod_step
tff(fact_5945_divmod__divmod__step,axiom,
    ! [Mb: num,Na: num] :
      unique3479559517661332726nteger(Mb,Na) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Na),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(Mb)),unique4921790084139445826nteger(Na,unique3479559517661332726nteger(Mb,aa_num_num(bit0,Na)))) ).

% divmod_divmod_step
tff(fact_5946_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_5947_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Na: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Na),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),K)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Na))),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_5948_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_numeral_bit1
tff(fact_5949_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_bit1
tff(fact_5950_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: 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,X))),Na))
    <=> 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,X))),Na)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5951_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),Na))
    <=> 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,X))),Na)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5952_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),Na))
    <=> 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,X))),Na)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5953_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: 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,numera6620942414471956472nteger(X))),Na))
    <=> 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,X))),Na)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5954_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Na: 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,X))),Na)),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,X))),Na)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5955_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),Na)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5956_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),Na)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5957_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),Na)),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,X))),Na)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5958_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),Na))
    <=> 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,X))),Na)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5959_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: 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,numera6620942414471956472nteger(X))),Na))
    <=> 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,X))),Na)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5960_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),Na))
    <=> 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,X))),Na)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5961_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: 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,X))),Na))
    <=> 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,X))),Na)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5962_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),Na)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5963_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),Na)),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,X))),Na)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5964_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),Na)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5965_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Na: 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,X))),Na)),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,X))),Na)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5966_of__int__eq__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( aa_int_real(ring_1_of_int_real,W2) = aa_int_real(ring_1_of_int_real,Z) )
    <=> ( W2 = Z ) ) ).

% of_int_eq_iff
tff(fact_5967_of__int__eq__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,W2) = aa_int_rat(ring_1_of_int_rat,Z) )
    <=> ( W2 = Z ) ) ).

% of_int_eq_iff
tff(fact_5968_bit__0__eq,axiom,
    aa_int_fun_nat_o(bit_se1146084159140164899it_int,zero_zero_int) = bot_bot_fun_nat_o ).

% bit_0_eq
tff(fact_5969_bit__0__eq,axiom,
    aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,zero_zero_nat) = bot_bot_fun_nat_o ).

% bit_0_eq
tff(fact_5970_of__int__0,axiom,
    aa_int_complex(ring_17405671764205052669omplex,zero_zero_int) = zero_zero_complex ).

% of_int_0
tff(fact_5971_of__int__0,axiom,
    aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_5972_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_5973_of__int__0,axiom,
    aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).

% of_int_0
tff(fact_5974_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_complex = aa_int_complex(ring_17405671764205052669omplex,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_5975_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_int = aa_int_int(ring_1_of_int_int,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_5976_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_real = aa_int_real(ring_1_of_int_real,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_5977_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_rat = aa_int_rat(ring_1_of_int_rat,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_5978_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = zero_zero_complex )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_5979_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = zero_zero_int )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_5980_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = zero_zero_real )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_5981_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = zero_zero_rat )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_5982_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ).

% of_int_le_iff
tff(fact_5983_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ).

% of_int_le_iff
tff(fact_5984_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ).

% of_int_le_iff
tff(fact_5985_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = aa_num_real(numeral_numeral_real,Na) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Na) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5986_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = aa_num_rat(numeral_numeral_rat,Na) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Na) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5987_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = aa_num_int(numeral_numeral_int,Na) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Na) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5988_of__int__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K)) = aa_num_real(numeral_numeral_real,K) ).

% of_int_numeral
tff(fact_5989_of__int__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K)) = aa_num_rat(numeral_numeral_rat,K) ).

% of_int_numeral
tff(fact_5990_of__int__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,K) ).

% of_int_numeral
tff(fact_5991_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% of_int_less_iff
tff(fact_5992_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% of_int_less_iff
tff(fact_5993_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% of_int_less_iff
tff(fact_5994_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = one_one_complex )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_5995_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = one_one_int )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_5996_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = one_one_real )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_5997_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = one_one_rat )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_5998_of__int__1,axiom,
    aa_int_complex(ring_17405671764205052669omplex,one_one_int) = one_one_complex ).

% of_int_1
tff(fact_5999_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_6000_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_6001_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_6002_of__int__mult,axiom,
    ! [W2: int,Z: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z)) = aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,W2)),aa_int_complex(ring_17405671764205052669omplex,Z)) ).

% of_int_mult
tff(fact_6003_of__int__mult,axiom,
    ! [W2: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_mult
tff(fact_6004_of__int__mult,axiom,
    ! [W2: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z)) = aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_mult
tff(fact_6005_of__int__mult,axiom,
    ! [W2: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_mult
tff(fact_6006_of__int__add,axiom,
    ! [W2: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_add
tff(fact_6007_of__int__add,axiom,
    ! [W2: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_add
tff(fact_6008_of__int__add,axiom,
    ! [W2: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),Z)) = aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_add
tff(fact_6009_of__int__minus,axiom,
    ! [Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,Z)) = aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_minus
tff(fact_6010_of__int__minus,axiom,
    ! [Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,Z)) = aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_minus
tff(fact_6011_of__int__minus,axiom,
    ! [Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,Z)) = aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_minus
tff(fact_6012_of__int__minus,axiom,
    ! [Z: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,Z)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ).

% of_int_minus
tff(fact_6013_of__int__diff,axiom,
    ! [W2: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(minus_minus_int,W2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_diff
tff(fact_6014_of__int__diff,axiom,
    ! [W2: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(minus_minus_int,W2),Z)) = aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_diff
tff(fact_6015_of__int__diff,axiom,
    ! [W2: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,W2),Z)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_diff
tff(fact_6016_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_6017_eq__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,Na) )
    <=> ( aa_num_nat(pred_numeral,K) = Na ) ) ).

% eq_numeral_Suc
tff(fact_6018_Suc__eq__numeral,axiom,
    ! [Na: nat,K: num] :
      ( ( aa_nat_nat(suc,Na) = aa_num_nat(numeral_numeral_nat,K) )
    <=> ( Na = aa_num_nat(pred_numeral,K) ) ) ).

% Suc_eq_numeral
tff(fact_6019_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,X) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2) )
    <=> ( X = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_6020_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( aa_int_real(ring_1_of_int_real,X) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W2) )
    <=> ( X = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_6021_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( aa_int_int(ring_1_of_int_int,X) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2) )
    <=> ( X = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_6022_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,X) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W2) )
    <=> ( X = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_6023_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2) = aa_int_rat(ring_1_of_int_rat,X) )
    <=> ( aa_nat_int(power_power_int(B),W2) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_6024_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W2) = aa_int_real(ring_1_of_int_real,X) )
    <=> ( aa_nat_int(power_power_int(B),W2) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_6025_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2) = aa_int_int(ring_1_of_int_int,X) )
    <=> ( aa_nat_int(power_power_int(B),W2) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_6026_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W2) = aa_int_complex(ring_17405671764205052669omplex,X) )
    <=> ( aa_nat_int(power_power_int(B),W2) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_6027_of__int__power,axiom,
    ! [Z: int,Na: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(power_power_int(Z),Na)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z)),Na) ).

% of_int_power
tff(fact_6028_of__int__power,axiom,
    ! [Z: int,Na: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Z),Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,Z)),Na) ).

% of_int_power
tff(fact_6029_of__int__power,axiom,
    ! [Z: int,Na: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Z),Na)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Z)),Na) ).

% of_int_power
tff(fact_6030_of__int__power,axiom,
    ! [Z: int,Na: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(power_power_int(Z),Na)) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,Z)),Na) ).

% of_int_power
tff(fact_6031_of__int__of__bool,axiom,
    ! [P: $o] : aa_int_real(ring_1_of_int_real,aa_o_int(zero_n2684676970156552555ol_int,(P))) = zero_n3304061248610475627l_real((P)) ).

% of_int_of_bool
tff(fact_6032_of__int__of__bool,axiom,
    ! [P: $o] : aa_int_rat(ring_1_of_int_rat,aa_o_int(zero_n2684676970156552555ol_int,(P))) = zero_n2052037380579107095ol_rat((P)) ).

% of_int_of_bool
tff(fact_6033_of__int__of__bool,axiom,
    ! [P: $o] : aa_int_int(ring_1_of_int_int,aa_o_int(zero_n2684676970156552555ol_int,(P))) = aa_o_int(zero_n2684676970156552555ol_int,(P)) ).

% of_int_of_bool
tff(fact_6034_bit__numeral__Bit0__Suc__iff,axiom,
    ! [Mb: num,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,Mb)),Na) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_6035_bit__numeral__Bit0__Suc__iff,axiom,
    ! [Mb: num,Na: 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,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Mb)),Na) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_6036_bit__numeral__Bit1__Suc__iff,axiom,
    ! [Mb: num,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,Mb)),Na) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_6037_bit__numeral__Bit1__Suc__iff,axiom,
    ! [Mb: num,Na: 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,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Mb)),Na) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_6038_less__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(pred_numeral,K)),Na) ) ).

% less_numeral_Suc
tff(fact_6039_less__Suc__numeral,axiom,
    ! [Na: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(pred_numeral,K)) ) ).

% less_Suc_numeral
tff(fact_6040_le__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),Na) ) ).

% le_numeral_Suc
tff(fact_6041_le__Suc__numeral,axiom,
    ! [Na: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_num_nat(pred_numeral,K)) ) ).

% le_Suc_numeral
tff(fact_6042_diff__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_num_nat(pred_numeral,K)) ).

% diff_Suc_numeral
tff(fact_6043_diff__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),Na) ).

% diff_numeral_Suc
tff(fact_6044_signed__take__bit__negative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Na),K)),zero_zero_int)
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na) ) ).

% signed_take_bit_negative_iff
tff(fact_6045_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_6046_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_6047_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_6048_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_6049_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_6050_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_6051_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_6052_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_6053_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_6054_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_6055_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_6056_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_6057_of__int__le__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_le_numeral_iff
tff(fact_6058_of__int__le__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_le_numeral_iff
tff(fact_6059_of__int__le__numeral__iff,axiom,
    ! [Z: int,Na: 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,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_le_numeral_iff
tff(fact_6060_of__int__numeral__le__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Na)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6061_of__int__numeral__le__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Na)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6062_of__int__numeral__le__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),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,Na)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6063_of__int__less__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_less_numeral_iff
tff(fact_6064_of__int__less__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_less_numeral_iff
tff(fact_6065_of__int__less__numeral__iff,axiom,
    ! [Z: int,Na: 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,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_less_numeral_iff
tff(fact_6066_of__int__numeral__less__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Na)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_6067_of__int__numeral__less__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Na)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_6068_of__int__numeral__less__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),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,Na)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_6069_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_6070_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_6071_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_6072_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_6073_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_6074_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_6075_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_6076_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_6077_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_6078_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_6079_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_6080_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_6081_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W2)),aa_int_real(ring_1_of_int_real,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W2)),X) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6082_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: 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)),W2)),aa_int_rat(ring_1_of_int_rat,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W2)),X) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6083_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: 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)),W2)),aa_int_int(ring_1_of_int_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W2)),X) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6084_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W2)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_6085_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W2)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_6086_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W2)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_6087_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_6088_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_6089_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_6090_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_6091_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),Na) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_6092_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),Na) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_6093_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),Na) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_6094_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_6095_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W2)),aa_int_real(ring_1_of_int_real,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W2)),X) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_6096_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: 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)),W2)),aa_int_rat(ring_1_of_int_rat,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W2)),X) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_6097_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: 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)),W2)),aa_int_int(ring_1_of_int_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W2)),X) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_6098_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(power_power_int(B),W2)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_6099_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(power_power_int(B),W2)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_6100_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(power_power_int(B),W2)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_6101_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W2: num,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W2)))),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W2))),Na) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_6102_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W2: num,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W2)))),aa_nat_nat(suc,Na))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W2)),Na) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_6103_bit__0,axiom,
    ! [A: code_integer] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),zero_zero_nat)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% bit_0
tff(fact_6104_bit__0,axiom,
    ! [A: int] :
      ( aa_nat_o(aa_int_fun_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_6105_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_6106_bit__mod__2__iff,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),Na)
    <=> ( ( Na = zero_zero_nat )
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ) ).

% bit_mod_2_iff
tff(fact_6107_bit__mod__2__iff,axiom,
    ! [A: int,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),Na)
    <=> ( ( Na = zero_zero_nat )
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ) ).

% bit_mod_2_iff
tff(fact_6108_bit__mod__2__iff,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Na)
    <=> ( ( Na = zero_zero_nat )
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ) ).

% bit_mod_2_iff
tff(fact_6109_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),Na)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_6110_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),Na)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_6111_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)),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,X)),Na)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_6112_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_6113_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_6114_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: 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,X)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_6115_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),Na)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_6116_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),Na)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_6117_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)),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,X)),Na)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_6118_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_6119_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_6120_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: 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,X)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_6121_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X))),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6122_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6123_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6124_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6125_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6126_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X))),Na) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6127_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6128_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),Na) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6129_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),Na) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6130_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),Na) = aa_int_Code_integer(ring_18347121197199848620nteger,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6131_bit__disjunctive__add__iff,axiom,
    ! [A: int,B: int,Na: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),N)
          | ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,B),N) )
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Na)
      <=> ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),Na)
          | aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,B),Na) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_6132_bit__disjunctive__add__iff,axiom,
    ! [A: nat,B: nat,Na: 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)),Na)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na)
          | aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),Na) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_6133_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z2)) ).

% ex_le_of_int
tff(fact_6134_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% ex_le_of_int
tff(fact_6135_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z2)) ).

% ex_less_of_int
tff(fact_6136_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% ex_less_of_int
tff(fact_6137_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),X) ).

% ex_of_int_less
tff(fact_6138_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X) ).

% ex_of_int_less
tff(fact_6139_mult__of__int__commute,axiom,
    ! [X: int,Y: complex] : aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,X)),Y) = aa_complex_complex(times_times_complex(Y),aa_int_complex(ring_17405671764205052669omplex,X)) ).

% mult_of_int_commute
tff(fact_6140_mult__of__int__commute,axiom,
    ! [X: int,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_int_real(ring_1_of_int_real,X)) ).

% mult_of_int_commute
tff(fact_6141_mult__of__int__commute,axiom,
    ! [X: int,Y: rat] : aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,X)),Y) = aa_rat_rat(times_times_rat(Y),aa_int_rat(ring_1_of_int_rat,X)) ).

% mult_of_int_commute
tff(fact_6142_mult__of__int__commute,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_int_int(ring_1_of_int_int,X)) ).

% mult_of_int_commute
tff(fact_6143_not__bit__1__Suc,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,one_one_int),aa_nat_nat(suc,Na)) ).

% not_bit_1_Suc
tff(fact_6144_not__bit__1__Suc,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),aa_nat_nat(suc,Na)) ).

% not_bit_1_Suc
tff(fact_6145_bit__1__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,one_one_int),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_6146_bit__1__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_6147_bit__take__bit__iff,axiom,
    ! [Mb: nat,A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),A)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na) ) ) ).

% bit_take_bit_iff
tff(fact_6148_bit__take__bit__iff,axiom,
    ! [Mb: nat,A: int,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_se2923211474154528505it_int(Mb),A)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),Na) ) ) ).

% bit_take_bit_iff
tff(fact_6149_bit__of__bool__iff,axiom,
    ! [B: $o,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_o_int(zero_n2684676970156552555ol_int,(B))),Na)
    <=> ( (B)
        & ( Na = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_6150_bit__of__bool__iff,axiom,
    ! [B: $o,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B))),Na)
    <=> ( (B)
        & ( Na = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_6151_numeral__eq__Suc,axiom,
    ! [K: num] : aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,aa_num_nat(pred_numeral,K)) ).

% numeral_eq_Suc
tff(fact_6152_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_6153_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_6154_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_6155_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)) ) ).

% of_int_pos
tff(fact_6156_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% of_int_pos
tff(fact_6157_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)) ) ).

% of_int_pos
tff(fact_6158_floor__exists,axiom,
    ! [X: real] :
    ? [Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int))) ) ).

% floor_exists
tff(fact_6159_floor__exists,axiom,
    ! [X: rat] :
    ? [Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int))) ) ).

% floor_exists
tff(fact_6160_floor__exists1,axiom,
    ! [X: real] :
    ? [X3: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X3)),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),one_one_int)))
      & ! [Y4: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y4)),X)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int))) )
         => ( Y4 = X3 ) ) ) ).

% floor_exists1
tff(fact_6161_floor__exists1,axiom,
    ! [X: rat] :
    ? [X3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X3)),X)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),one_one_int)))
      & ! [Y4: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y4)),X)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int))) )
         => ( Y4 = X3 ) ) ) ).

% floor_exists1
tff(fact_6162_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)) ).

% of_int_neg_numeral
tff(fact_6163_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K)) ).

% of_int_neg_numeral
tff(fact_6164_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K)) ).

% of_int_neg_numeral
tff(fact_6165_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K)) ).

% of_int_neg_numeral
tff(fact_6166_bit__imp__take__bit__positive,axiom,
    ! [Na: nat,Mb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(Mb),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_6167_int__le__real__less,axiom,
    ! [Na: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Mb)),one_one_real)) ) ).

% int_le_real_less
tff(fact_6168_int__less__real__le,axiom,
    ! [Na: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Na),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real)),aa_int_real(ring_1_of_int_real,Mb)) ) ).

% int_less_real_le
tff(fact_6169_bit__concat__bit__iff,axiom,
    ! [Mb: nat,K: int,L: int,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_concat_bit(Mb,K),L)),Na)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Mb)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_6170_exp__eq__0__imp__not__bit,axiom,
    ! [Na: nat,A: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) = zero_zero_int )
     => ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),Na) ) ).

% exp_eq_0_imp_not_bit
tff(fact_6171_exp__eq__0__imp__not__bit,axiom,
    ! [Na: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) = zero_zero_nat )
     => ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na) ) ).

% exp_eq_0_imp_not_bit
tff(fact_6172_bit__Suc,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),aa_nat_nat(suc,Na))
    <=> aa_nat_o(bit_se9216721137139052372nteger(divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),Na) ) ).

% bit_Suc
tff(fact_6173_bit__Suc,axiom,
    ! [A: int,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),Na) ) ).

% bit_Suc
tff(fact_6174_bit__Suc,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,Na))
    <=> 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)))),Na) ) ).

% bit_Suc
tff(fact_6175_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M4)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),M4)
              <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
              <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N) ) ) ) ).

% int_bit_bound
tff(fact_6176_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: int,Na: nat] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = zero_zero_int )
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),Na) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_6177_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = zero_zero_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_6178_even__bit__succ__iff,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se9216721137139052372nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A)),Na)
      <=> ( aa_nat_o(bit_se9216721137139052372nteger(A),Na)
          | ( Na = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_6179_even__bit__succ__iff,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),Na)
      <=> ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),Na)
          | ( Na = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_6180_even__bit__succ__iff,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),Na)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na)
          | ( Na = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_6181_odd__bit__iff__bit__pred,axiom,
    ! [A: code_integer,Na: nat] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se9216721137139052372nteger(A),Na)
      <=> ( aa_nat_o(bit_se9216721137139052372nteger(minus_8373710615458151222nteger(A,one_one_Code_integer)),Na)
          | ( Na = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_6182_odd__bit__iff__bit__pred,axiom,
    ! [A: int,Na: nat] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),Na)
      <=> ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),one_one_int)),Na)
          | ( Na = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_6183_odd__bit__iff__bit__pred,axiom,
    ! [A: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),one_one_nat)),Na)
          | ( Na = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_6184_bit__sum__mult__2__cases,axiom,
    ! [A: code_integer,B: code_integer,Na: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(bit_se9216721137139052372nteger(A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(bit_se9216721137139052372nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),Na)
      <=> $ite(Na = zero_zero_nat,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),aa_nat_o(bit_se9216721137139052372nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)),Na)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_6185_bit__sum__mult__2__cases,axiom,
    ! [A: int,B: int,Na: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(aa_int_fun_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))),Na)
      <=> $ite(Na = zero_zero_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),aa_nat_o(aa_int_fun_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)),Na)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_6186_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B: nat,Na: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),Na)
      <=> $ite(Na = zero_zero_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)),Na)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_6187_bit__rec,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),Na)
    <=> $ite(Na = zero_zero_nat,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),aa_nat_o(bit_se9216721137139052372nteger(divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ).

% bit_rec
tff(fact_6188_bit__rec,axiom,
    ! [A: int,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),Na)
    <=> $ite(Na = zero_zero_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ).

% bit_rec
tff(fact_6189_bit__rec,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na)
    <=> $ite(Na = zero_zero_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ).

% bit_rec
tff(fact_6190_take__bit__Suc__from__most,axiom,
    ! [Na: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na)))),aa_int_int(bit_se2923211474154528505it_int(Na),K)) ).

% take_bit_Suc_from_most
tff(fact_6191_round__unique,axiom,
    ! [X: real,Y: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,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(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( aa_real_int(archim8280529875227126926d_real,X) = Y ) ) ) ).

% round_unique
tff(fact_6192_round__unique,axiom,
    ! [X: rat,Y: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y)),aa_rat_rat(plus_plus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X) = Y ) ) ) ).

% round_unique
tff(fact_6193_of__int__round__gt,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X))) ).

% of_int_round_gt
tff(fact_6194_of__int__round__gt,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))) ).

% of_int_round_gt
tff(fact_6195_of__int__round__ge,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X))) ).

% of_int_round_ge
tff(fact_6196_of__int__round__ge,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))) ).

% of_int_round_ge
tff(fact_6197_of__int__round__le,axiom,
    ! [X: 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,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_6198_of__int__round__le,axiom,
    ! [X: 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,X))),aa_rat_rat(plus_plus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_6199_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_6200_mask__numeral,axiom,
    ! [Na: num] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(numeral_numeral_nat,Na)) = 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,Na)))) ).

% mask_numeral
tff(fact_6201_mask__numeral,axiom,
    ! [Na: num] : aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(numeral_numeral_nat,Na)) = 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,Na)))) ).

% mask_numeral
tff(fact_6202_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)),X) ) ).

% neg_numeral_le_ceiling
tff(fact_6203_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),X) ) ).

% neg_numeral_le_ceiling
tff(fact_6204_mask__nat__positive__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% mask_nat_positive_iff
tff(fact_6205_ceiling__zero,axiom,
    aa_rat_int(archim2889992004027027881ng_rat,zero_zero_rat) = zero_zero_int ).

% ceiling_zero
tff(fact_6206_ceiling__zero,axiom,
    aa_real_int(archim7802044766580827645g_real,zero_zero_real) = zero_zero_int ).

% ceiling_zero
tff(fact_6207_mask__eq__0__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(bit_se2002935070580805687sk_nat,Na) = zero_zero_nat )
    <=> ( Na = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_6208_mask__eq__0__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_int(bit_se2000444600071755411sk_int,Na) = zero_zero_int )
    <=> ( Na = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_6209_mask__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,zero_zero_nat) = zero_zero_nat ).

% mask_0
tff(fact_6210_mask__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,zero_zero_nat) = zero_zero_int ).

% mask_0
tff(fact_6211_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_6212_round__0,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,zero_zero_rat) = zero_zero_int ).

% round_0
tff(fact_6213_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_6214_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_6215_ceiling__add__of__int,axiom,
    ! [X: rat,Z: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(X),aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z) ).

% ceiling_add_of_int
tff(fact_6216_ceiling__add__of__int,axiom,
    ! [X: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),Z) ).

% ceiling_add_of_int
tff(fact_6217_ceiling__le__zero,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ).

% ceiling_le_zero
tff(fact_6218_ceiling__le__zero,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat) ) ).

% ceiling_le_zero
tff(fact_6219_zero__less__ceiling,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X) ) ).

% zero_less_ceiling
tff(fact_6220_zero__less__ceiling,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X) ) ).

% zero_less_ceiling
tff(fact_6221_ceiling__le__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_num_real(numeral_numeral_real,V)) ) ).

% ceiling_le_numeral
tff(fact_6222_ceiling__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_num_rat(numeral_numeral_rat,V)) ) ).

% ceiling_le_numeral
tff(fact_6223_ceiling__less__one,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ).

% ceiling_less_one
tff(fact_6224_ceiling__less__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat) ) ).

% ceiling_less_one
tff(fact_6225_one__le__ceiling,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X) ) ).

% one_le_ceiling
tff(fact_6226_one__le__ceiling,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X) ) ).

% one_le_ceiling
tff(fact_6227_numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,V)),X) ) ).

% numeral_less_ceiling
tff(fact_6228_numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,V)),X) ) ).

% numeral_less_ceiling
tff(fact_6229_ceiling__le__one,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ).

% ceiling_le_one
tff(fact_6230_ceiling__le__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),one_one_rat) ) ).

% ceiling_le_one
tff(fact_6231_one__less__ceiling,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),X) ) ).

% one_less_ceiling
tff(fact_6232_one__less__ceiling,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X) ) ).

% one_less_ceiling
tff(fact_6233_ceiling__add__numeral,axiom,
    ! [X: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_6234_ceiling__add__numeral,axiom,
    ! [X: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(X),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_6235_ceiling__add__one,axiom,
    ! [X: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(X),one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int) ).

% ceiling_add_one
tff(fact_6236_ceiling__add__one,axiom,
    ! [X: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int) ).

% ceiling_add_one
tff(fact_6237_ceiling__less__zero,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% ceiling_less_zero
tff(fact_6238_ceiling__less__zero,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% ceiling_less_zero
tff(fact_6239_zero__le__ceiling,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X) ) ).

% zero_le_ceiling
tff(fact_6240_zero__le__ceiling,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X) ) ).

% zero_le_ceiling
tff(fact_6241_ceiling__less__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)) ) ).

% ceiling_less_numeral
tff(fact_6242_ceiling__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)) ) ).

% ceiling_less_numeral
tff(fact_6243_numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)),X) ) ).

% numeral_le_ceiling
tff(fact_6244_numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),X) ) ).

% numeral_le_ceiling
tff(fact_6245_ceiling__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) ) ).

% ceiling_le_neg_numeral
tff(fact_6246_ceiling__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) ) ).

% ceiling_le_neg_numeral
tff(fact_6247_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),X) ) ).

% neg_numeral_less_ceiling
tff(fact_6248_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),X) ) ).

% neg_numeral_less_ceiling
tff(fact_6249_ceiling__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)) ) ).

% ceiling_less_neg_numeral
tff(fact_6250_ceiling__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)) ) ).

% ceiling_less_neg_numeral
tff(fact_6251_less__eq__mask,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) ).

% less_eq_mask
tff(fact_6252_bit__Suc__0__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_6253_not__bit__Suc__0__Suc,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,Na)) ).

% not_bit_Suc_0_Suc
tff(fact_6254_ceiling__mono,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Y)),aa_real_int(archim7802044766580827645g_real,X)) ) ).

% ceiling_mono
tff(fact_6255_ceiling__mono,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Y)),aa_rat_int(archim2889992004027027881ng_rat,X)) ) ).

% ceiling_mono
tff(fact_6256_le__of__int__ceiling,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X))) ).

% le_of_int_ceiling
tff(fact_6257_le__of__int__ceiling,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X))) ).

% le_of_int_ceiling
tff(fact_6258_ceiling__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,Y))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ).

% ceiling_less_cancel
tff(fact_6259_ceiling__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim7802044766580827645g_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ).

% ceiling_less_cancel
tff(fact_6260_not__mask__negative__int,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,Na)),zero_zero_int) ).

% not_mask_negative_int
tff(fact_6261_not__bit__Suc__0__numeral,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,Na)) ).

% not_bit_Suc_0_numeral
tff(fact_6262_ceiling__le__iff,axiom,
    ! [X: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z)) ) ).

% ceiling_le_iff
tff(fact_6263_ceiling__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% ceiling_le_iff
tff(fact_6264_ceiling__le,axiom,
    ! [X: real,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),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,X)),A) ) ).

% ceiling_le
tff(fact_6265_ceiling__le,axiom,
    ! [X: rat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),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,X)),A) ) ).

% ceiling_le
tff(fact_6266_less__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),X) ) ).

% less_ceiling_iff
tff(fact_6267_less__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),X) ) ).

% less_ceiling_iff
tff(fact_6268_ceiling__add__le,axiom,
    ! [X: rat,Y: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(X),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,Y))) ).

% ceiling_add_le
tff(fact_6269_ceiling__add__le,axiom,
    ! [X: real,Y: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim7802044766580827645g_real,Y))) ).

% ceiling_add_le
tff(fact_6270_round__mono,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,X)),aa_rat_int(archim7778729529865785530nd_rat,Y)) ) ).

% round_mono
tff(fact_6271_less__mask,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) ) ).

% less_mask
tff(fact_6272_eval__nat__numeral_I2_J,axiom,
    ! [Na: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,Na))) ).

% eval_nat_numeral(2)
tff(fact_6273_of__int__ceiling__le__add__one,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),one_one_real)) ).

% of_int_ceiling_le_add_one
tff(fact_6274_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(plus_plus_rat(R2),one_one_rat)) ).

% of_int_ceiling_le_add_one
tff(fact_6275_of__int__ceiling__diff__one__le,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),one_one_real)),R2) ).

% of_int_ceiling_diff_one_le
tff(fact_6276_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R2))),one_one_rat)),R2) ).

% of_int_ceiling_diff_one_le
tff(fact_6277_ceiling__split,axiom,
    ! [P: fun_int_o,T: real] :
      ( aa_int_o(P,aa_real_int(archim7802044766580827645g_real,T))
    <=> ! [I2: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,I2)),one_one_real)),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),aa_int_real(ring_1_of_int_real,I2)) )
         => aa_int_o(P,I2) ) ) ).

% ceiling_split
tff(fact_6278_ceiling__split,axiom,
    ! [P: fun_int_o,T: rat] :
      ( aa_int_o(P,aa_rat_int(archim2889992004027027881ng_rat,T))
    <=> ! [I2: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I2)),one_one_rat)),T)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,T),aa_int_rat(ring_1_of_int_rat,I2)) )
         => aa_int_o(P,I2) ) ) ).

% ceiling_split
tff(fact_6279_ceiling__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,X) = A )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,A)),one_one_real)),X)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,A)) ) ) ).

% ceiling_eq_iff
tff(fact_6280_ceiling__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( aa_rat_int(archim2889992004027027881ng_rat,X) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,A)) ) ) ).

% ceiling_eq_iff
tff(fact_6281_ceiling__unique,axiom,
    ! [Z: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z))
       => ( aa_real_int(archim7802044766580827645g_real,X) = Z ) ) ) ).

% ceiling_unique
tff(fact_6282_ceiling__unique,axiom,
    ! [Z: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z))
       => ( aa_rat_int(archim2889992004027027881ng_rat,X) = Z ) ) ) ).

% ceiling_unique
tff(fact_6283_ceiling__correct,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X))),one_one_real)),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X))) ) ).

% ceiling_correct
tff(fact_6284_ceiling__correct,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X))),one_one_rat)),X)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X))) ) ).

% ceiling_correct
tff(fact_6285_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(aa_rea7340605660300375871l_real(times_times_real,A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B))) ) ) ).

% mult_ceiling_le
tff(fact_6286_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(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_6287_ceiling__less__iff,axiom,
    ! [X: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)) ) ).

% ceiling_less_iff
tff(fact_6288_ceiling__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)) ) ).

% ceiling_less_iff
tff(fact_6289_le__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X) ) ).

% le_ceiling_iff
tff(fact_6290_le__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X) ) ).

% le_ceiling_iff
tff(fact_6291_ceiling__divide__upper,axiom,
    ! [Q4: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q4)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,P2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q4)))),Q4)) ) ).

% ceiling_divide_upper
tff(fact_6292_ceiling__divide__upper,axiom,
    ! [Q4: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q4)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P2),Q4)))),Q4)) ) ).

% ceiling_divide_upper
tff(fact_6293_Suc__mask__eq__exp,axiom,
    ! [Na: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ).

% Suc_mask_eq_exp
tff(fact_6294_mask__nat__less__exp,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% mask_nat_less_exp
tff(fact_6295_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(bit_se2119862282449309892nteger,Na))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_6296_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,Na))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_6297_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,Na))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_6298_ceiling__divide__lower,axiom,
    ! [Q4: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q4)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q4)))),one_one_real)),Q4)),P2) ) ).

% ceiling_divide_lower
tff(fact_6299_ceiling__divide__lower,axiom,
    ! [Q4: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q4)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P2),Q4)))),one_one_rat)),Q4)),P2) ) ).

% ceiling_divide_lower
tff(fact_6300_ceiling__eq,axiom,
    ! [Na: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Na)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real))
       => ( aa_real_int(archim7802044766580827645g_real,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_6301_ceiling__eq,axiom,
    ! [Na: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Na)),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Na)),one_one_rat))
       => ( aa_rat_int(archim2889992004027027881ng_rat,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_6302_Suc__0__xor__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% Suc_0_xor_eq
tff(fact_6303_xor__Suc__0__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% xor_Suc_0_eq
tff(fact_6304_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))),aa_list_o_nat(size_size_list_o,Bs))) ).

% horner_sum_of_bool_2_less
tff(fact_6305_dbl__dec__simps_I2_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,zero_zero_complex) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% dbl_dec_simps(2)
tff(fact_6306_dbl__dec__simps_I2_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_dec_simps(2)
tff(fact_6307_dbl__dec__simps_I2_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,zero_zero_real) = aa_real_real(uminus_uminus_real,one_one_real) ).

% dbl_dec_simps(2)
tff(fact_6308_dbl__dec__simps_I2_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,zero_zero_rat) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_dec_simps(2)
tff(fact_6309_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_6310_the__elem__eq,axiom,
    ! [X: produc3843707927480180839at_nat] : the_el221668144340439132at_nat(insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) = X ).

% the_elem_eq
tff(fact_6311_the__elem__eq,axiom,
    ! [X: product_prod_nat_nat] : the_el2281957884133575798at_nat(insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) = X ).

% the_elem_eq
tff(fact_6312_the__elem__eq,axiom,
    ! [X: real] : the_elem_real(insert_real(X,bot_bot_set_real)) = X ).

% the_elem_eq
tff(fact_6313_the__elem__eq,axiom,
    ! [X: nat] : aa_set_nat_nat(the_elem_nat,insert_nat(X,bot_bot_set_nat)) = X ).

% the_elem_eq
tff(fact_6314_the__elem__eq,axiom,
    ! [X: int] : the_elem_int(insert_int(X,bot_bot_set_int)) = X ).

% the_elem_eq
tff(fact_6315_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(X),Xa2)
      <=> (Y) )
     => ( ! [A3: $o,B3: $o] :
            ( ( X = vEBT_Leaf((A3),(B3)) )
           => ( (Y)
            <=> ~ $ite(
                    Xa2 = zero_zero_nat,
                    (A3),
                    $ite(Xa2 = one_one_nat,(B3),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => (Y) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
             => (Y) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => (Y) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)
                   => ( (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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_6316_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_set_vebt(T) = collect_nat(vEBT_vebt_member(T)) ).

% set_vebt'_def
tff(fact_6317_finite__Collect__conjI,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( ( aa_set_real_o(finite_finite_real,collect_real(P))
        | aa_set_real_o(finite_finite_real,collect_real(Q)) )
     => aa_set_real_o(finite_finite_real,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_af(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_6318_finite__Collect__conjI,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ( finite8100373058378681591st_nat(collect_list_nat(P))
        | finite8100373058378681591st_nat(collect_list_nat(Q)) )
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_ag(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_6319_finite__Collect__conjI,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(P))
        | aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(Q)) )
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ah(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_6320_finite__Collect__conjI,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
        | aa_set_nat_o(finite_finite_nat,collect_nat(Q)) )
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ai(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_6321_finite__Collect__conjI,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ( aa_set_int_o(finite_finite_int,collect_int(P))
        | aa_set_int_o(finite_finite_int,collect_int(Q)) )
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_aj(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_6322_finite__Collect__conjI,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P))
        | aa_set_complex_o(finite3207457112153483333omplex,collect_complex(Q)) )
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_ak(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_6323_finite__Collect__conjI,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] :
      ( ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(P))
        | aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Q)) )
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_al(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_6324_finite__Collect__disjI,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_am(P),Q)))
    <=> ( aa_set_real_o(finite_finite_real,collect_real(P))
        & aa_set_real_o(finite_finite_real,collect_real(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_6325_finite__Collect__disjI,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( finite8100373058378681591st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_an(P),Q)))
    <=> ( finite8100373058378681591st_nat(collect_list_nat(P))
        & finite8100373058378681591st_nat(collect_list_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_6326_finite__Collect__disjI,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ao(P),Q)))
    <=> ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(P))
        & aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_6327_finite__Collect__disjI,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ap(P),Q)))
    <=> ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
        & aa_set_nat_o(finite_finite_nat,collect_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_6328_finite__Collect__disjI,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_aq(P),Q)))
    <=> ( aa_set_int_o(finite_finite_int,collect_int(P))
        & aa_set_int_o(finite_finite_int,collect_int(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_6329_finite__Collect__disjI,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_ar(P),Q)))
    <=> ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P))
        & aa_set_complex_o(finite3207457112153483333omplex,collect_complex(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_6330_finite__Collect__disjI,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_as(P),Q)))
    <=> ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(P))
        & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_6331_bit_Oxor__self,axiom,
    ! [X: int] : aa_int_int(bit_se6526347334894502574or_int(X),X) = zero_zero_int ).

% bit.xor_self
tff(fact_6332_xor__self__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),A) = zero_zero_nat ).

% xor_self_eq
tff(fact_6333_xor__self__eq,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(A),A) = zero_zero_int ).

% xor_self_eq
tff(fact_6334_xor_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,zero_zero_nat),A) = A ).

% xor.left_neutral
tff(fact_6335_xor_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(zero_zero_int),A) = A ).

% xor.left_neutral
tff(fact_6336_xor_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),zero_zero_nat) = A ).

% xor.right_neutral
tff(fact_6337_xor_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(A),zero_zero_int) = A ).

% xor.right_neutral
tff(fact_6338_finite__Collect__bounded__ex,axiom,
    ! [P: fun_real_o,Q: fun_real_fun_real_o] :
      ( aa_set_real_o(finite_finite_real,collect_real(P))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun8588574755773516687real_o(aTP_Lamm_at(P),Q)))
      <=> ! [Y5: real] :
            ( aa_real_o(P,Y5)
           => aa_set_real_o(finite_finite_real,collect_real(aa_real_fun_real_o(aTP_Lamm_au(Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_6339_finite__Collect__bounded__ex,axiom,
    ! [P: fun_real_o,Q: fun_nat_fun_real_o] :
      ( aa_set_real_o(finite_finite_real,collect_real(P))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1522457866930155023_nat_o(aTP_Lamm_av(P),Q)))
      <=> ! [Y5: real] :
            ( aa_real_o(P,Y5)
           => aa_set_nat_o(finite_finite_nat,collect_nat(aa_real_fun_nat_o(aTP_Lamm_aw(Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_6340_finite__Collect__bounded__ex,axiom,
    ! [P: fun_real_o,Q: fun_int_fun_real_o] :
      ( aa_set_real_o(finite_finite_real,collect_real(P))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun8063507495001755407_int_o(aTP_Lamm_ax(P),Q)))
      <=> ! [Y5: real] :
            ( aa_real_o(P,Y5)
           => aa_set_int_o(finite_finite_int,collect_int(aa_real_fun_int_o(aTP_Lamm_ay(Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_6341_finite__Collect__bounded__ex,axiom,
    ! [P: fun_real_o,Q: fun_co4998825143550818160real_o] :
      ( aa_set_real_o(finite_finite_real,collect_real(P))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun4501470310227681423plex_o(aTP_Lamm_az(P),Q)))
      <=> ! [Y5: real] :
            ( aa_real_o(P,Y5)
           => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_rea2382570592068293209plex_o(aTP_Lamm_ba(Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_6342_finite__Collect__bounded__ex,axiom,
    ! [P: fun_nat_o,Q: fun_real_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun8982401453414893235real_o(aTP_Lamm_bb(P),Q)))
      <=> ! [Y5: nat] :
            ( aa_nat_o(P,Y5)
           => aa_set_real_o(finite_finite_real,collect_real(aa_nat_fun_real_o(aTP_Lamm_bc(Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_6343_finite__Collect__bounded__ex,axiom,
    ! [P: fun_nat_o,Q: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6854469565811772723_nat_o(aTP_Lamm_bd(P),Q)))
      <=> ! [Y5: nat] :
            ( aa_nat_o(P,Y5)
           => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_be(Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_6344_finite__Collect__bounded__ex,axiom,
    ! [P: fun_nat_o,Q: fun_int_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7458983532533439539_int_o(aTP_Lamm_bf(P),Q)))
      <=> ! [Y5: nat] :
            ( aa_nat_o(P,Y5)
           => aa_set_int_o(finite_finite_int,collect_int(aa_nat_fun_int_o(aTP_Lamm_bg(Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_6345_finite__Collect__bounded__ex,axiom,
    ! [P: fun_nat_o,Q: fun_co5676208753217392844_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun3010802591474140851plex_o(aTP_Lamm_bh(P),Q)))
      <=> ! [Y5: nat] :
            ( aa_nat_o(P,Y5)
           => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_bi(Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_6346_finite__Collect__bounded__ex,axiom,
    ! [P: fun_int_o,Q: fun_real_fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(P))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3710474044975826703real_o(aTP_Lamm_bj(P),Q)))
      <=> ! [Y5: int] :
            ( aa_int_o(P,Y5)
           => aa_set_real_o(finite_finite_real,collect_real(aa_int_fun_real_o(aTP_Lamm_bk(Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_6347_finite__Collect__bounded__ex,axiom,
    ! [P: fun_int_o,Q: fun_nat_fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(P))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3042179710660459407_nat_o(aTP_Lamm_bl(P),Q)))
      <=> ! [Y5: int] :
            ( aa_int_o(P,Y5)
           => aa_set_nat_o(finite_finite_nat,collect_nat(aa_int_fun_nat_o(aTP_Lamm_bm(Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_6348_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_bn,A2))) ) ).

% finite_Collect_subsets
tff(fact_6349_finite__Collect__subsets,axiom,
    ! [A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => finite6551019134538273531omplex(collect_set_complex(aTP_Lamm_bo(A2))) ) ).

% finite_Collect_subsets
tff(fact_6350_finite__Collect__subsets,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => finite9047747110432174090at_nat(collec5514110066124741708at_nat(aTP_Lamm_bp(A2))) ) ).

% finite_Collect_subsets
tff(fact_6351_finite__Collect__subsets,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => finite6197958912794628473et_int(collect_set_int(aTP_Lamm_bq(A2))) ) ).

% finite_Collect_subsets
tff(fact_6352_singleton__conv2,axiom,
    ! [A: produc3843707927480180839at_nat] : collec6321179662152712658at_nat(fequal7949547063978738364at_nat(A)) = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ).

% singleton_conv2
tff(fact_6353_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_6354_singleton__conv2,axiom,
    ! [A: set_nat] : collect_set_nat(aa_set7861427066675100071_nat_o(fequal_set_nat,A)) = insert_set_nat(A,bot_bot_set_set_nat) ).

% singleton_conv2
tff(fact_6355_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_6356_singleton__conv2,axiom,
    ! [A: real] : collect_real(aa_real_fun_real_o(fequal_real,A)) = insert_real(A,bot_bot_set_real) ).

% singleton_conv2
tff(fact_6357_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_6358_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_6359_singleton__conv,axiom,
    ! [A: produc3843707927480180839at_nat] : collec6321179662152712658at_nat(aTP_Lamm_br(A)) = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ).

% singleton_conv
tff(fact_6360_singleton__conv,axiom,
    ! [A: list_nat] : collect_list_nat(aTP_Lamm_bs(A)) = insert_list_nat(A,bot_bot_set_list_nat) ).

% singleton_conv
tff(fact_6361_singleton__conv,axiom,
    ! [A: set_nat] : collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_bt,A)) = insert_set_nat(A,bot_bot_set_set_nat) ).

% singleton_conv
tff(fact_6362_singleton__conv,axiom,
    ! [A: product_prod_nat_nat] : collec3392354462482085612at_nat(aa_Pro989595756709598047_nat_o(aTP_Lamm_bu,A)) = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ).

% singleton_conv
tff(fact_6363_singleton__conv,axiom,
    ! [A: real] : collect_real(aa_real_fun_real_o(aTP_Lamm_bv,A)) = insert_real(A,bot_bot_set_real) ).

% singleton_conv
tff(fact_6364_singleton__conv,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bw,A)) = insert_nat(A,bot_bot_set_nat) ).

% singleton_conv
tff(fact_6365_singleton__conv,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_o(aTP_Lamm_bx,A)) = insert_int(A,bot_bot_set_int) ).

% singleton_conv
tff(fact_6366_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_by,K))) ).

% finite_Collect_less_nat
tff(fact_6367_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bz,K))) ).

% finite_Collect_le_nat
tff(fact_6368_finite__interval__int1,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_ca(A),B))) ).

% finite_interval_int1
tff(fact_6369_finite__interval__int4,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_cb(A),B))) ).

% finite_interval_int4
tff(fact_6370_finite__interval__int2,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_cc(A),B))) ).

% finite_interval_int2
tff(fact_6371_finite__interval__int3,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_cd(A),B))) ).

% finite_interval_int3
tff(fact_6372_xor__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X)) ).

% xor_nat_numerals(4)
tff(fact_6373_xor__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% xor_nat_numerals(3)
tff(fact_6374_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_6375_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_6376_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% xor_numerals(6)
tff(fact_6377_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% xor_numerals(6)
tff(fact_6378_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% xor_numerals(4)
tff(fact_6379_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% xor_numerals(4)
tff(fact_6380_empty__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(aTP_Lamm_ce) ).

% empty_def
tff(fact_6381_empty__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(aTP_Lamm_cf) ).

% empty_def
tff(fact_6382_empty__def,axiom,
    bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(aTP_Lamm_cg) ).

% empty_def
tff(fact_6383_empty__def,axiom,
    bot_bot_set_real = collect_real(aTP_Lamm_ch) ).

% empty_def
tff(fact_6384_empty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_ci) ).

% empty_def
tff(fact_6385_empty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_cj) ).

% empty_def
tff(fact_6386_Collect__conv__if2,axiom,
    ! [A: produc3843707927480180839at_nat,P: fun_Pr3059359313727237556_nat_o] :
      collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_ck(A),P)) = $ite(aa_Pro5947980084102686253_nat_o(P,A),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat),bot_bo228742789529271731at_nat) ).

% Collect_conv_if2
tff(fact_6387_Collect__conv__if2,axiom,
    ! [A: list_nat,P: fun_list_nat_o] :
      collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_cl(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_6388_Collect__conv__if2,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_cm(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_6389_Collect__conv__if2,axiom,
    ! [A: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o] :
      collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_cn(A),P)) = $ite(aa_Pro6150637604445964947_nat_o(P,A),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat),bot_bo2099793752762293965at_nat) ).

% Collect_conv_if2
tff(fact_6390_Collect__conv__if2,axiom,
    ! [A: real,P: fun_real_o] :
      collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_co(A),P)) = $ite(aa_real_o(P,A),insert_real(A,bot_bot_set_real),bot_bot_set_real) ).

% Collect_conv_if2
tff(fact_6391_Collect__conv__if2,axiom,
    ! [A: nat,P: fun_nat_o] :
      collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cp(A),P)) = $ite(aa_nat_o(P,A),insert_nat(A,bot_bot_set_nat),bot_bot_set_nat) ).

% Collect_conv_if2
tff(fact_6392_Collect__conv__if2,axiom,
    ! [A: int,P: fun_int_o] :
      collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cq(A),P)) = $ite(aa_int_o(P,A),insert_int(A,bot_bot_set_int),bot_bot_set_int) ).

% Collect_conv_if2
tff(fact_6393_Collect__conv__if,axiom,
    ! [A: produc3843707927480180839at_nat,P: fun_Pr3059359313727237556_nat_o] :
      collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_cr(A),P)) = $ite(aa_Pro5947980084102686253_nat_o(P,A),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat),bot_bo228742789529271731at_nat) ).

% Collect_conv_if
tff(fact_6394_Collect__conv__if,axiom,
    ! [A: list_nat,P: fun_list_nat_o] :
      collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_cs(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_6395_Collect__conv__if,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ct(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_6396_Collect__conv__if,axiom,
    ! [A: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o] :
      collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_cu(A),P)) = $ite(aa_Pro6150637604445964947_nat_o(P,A),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat),bot_bo2099793752762293965at_nat) ).

% Collect_conv_if
tff(fact_6397_Collect__conv__if,axiom,
    ! [A: real,P: fun_real_o] :
      collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_cv(A),P)) = $ite(aa_real_o(P,A),insert_real(A,bot_bot_set_real),bot_bot_set_real) ).

% Collect_conv_if
tff(fact_6398_Collect__conv__if,axiom,
    ! [A: nat,P: fun_nat_o] :
      collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cw(A),P)) = $ite(aa_nat_o(P,A),insert_nat(A,bot_bot_set_nat),bot_bot_set_nat) ).

% Collect_conv_if
tff(fact_6399_Collect__conv__if,axiom,
    ! [A: int,P: fun_int_o] :
      collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cx(A),P)) = $ite(aa_int_o(P,A),insert_int(A,bot_bot_set_int),bot_bot_set_int) ).

% Collect_conv_if
tff(fact_6400_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ord_le768339454706049738_nat_o(aTP_Lamm_cy(R),aTP_Lamm_cy(S3))
    <=> aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(R),S3) ) ).

% pred_subset_eq2
tff(fact_6401_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ord_le77980682512674322_int_o(aTP_Lamm_cz(R),aTP_Lamm_cz(S3))
    <=> ord_le2843351958646193337nt_int(R,S3) ) ).

% pred_subset_eq2
tff(fact_6402_pred__subset__eq2,axiom,
    ! [R: set_Pr1200682188532975830nteger,S3: set_Pr1200682188532975830nteger] :
      ( ord_le7899862361585801933eger_o(aTP_Lamm_da(R),aTP_Lamm_da(S3))
    <=> ord_le2645283422117218614nteger(R,S3) ) ).

% pred_subset_eq2
tff(fact_6403_pred__subset__eq2,axiom,
    ! [R: set_Pr3486312109254157132nteger,S3: set_Pr3486312109254157132nteger] :
      ( ord_le4825817444062779351eger_o(aTP_Lamm_db(R),aTP_Lamm_db(S3))
    <=> ord_le2932375828722672556nteger(R,S3) ) ).

% pred_subset_eq2
tff(fact_6404_pred__subset__eq2,axiom,
    ! [R: set_Pr6700895561065273913nt_int,S3: set_Pr6700895561065273913nt_int] :
      ( ord_le4806730579060245290_int_o(aTP_Lamm_dc(R),aTP_Lamm_dc(S3))
    <=> ord_le3990897393281686169nt_int(R,S3) ) ).

% pred_subset_eq2
tff(fact_6405_pred__subset__eq2,axiom,
    ! [R: set_Pr4166680992137330533nt_int,S3: set_Pr4166680992137330533nt_int] :
      ( ord_le2876595303158420734_int_o(aTP_Lamm_dd(R),aTP_Lamm_dd(S3))
    <=> ord_le3197784583289274309nt_int(R,S3) ) ).

% pred_subset_eq2
tff(fact_6406_strict__subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( aa_set_real_o(ord_less_set_real(collect_real(aa_real_fun_real_o(aTP_Lamm_de,A))),collect_real(aa_real_fun_real_o(aTP_Lamm_de,B)))
    <=> ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
        & ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_6407_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_df,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_df,B)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_6408_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( aa_set_int_o(ord_less_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_dg,A))),collect_int(aa_int_fun_int_o(aTP_Lamm_dg,B)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_6409_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ord_le1307284697595431911nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_dh,A)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_dh,B)))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_6410_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Xa4: nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Xa4)),R)
        <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Xa4)),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_6411_pred__equals__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ! [X4: int,Xa4: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Xa4),R)
        <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Xa4),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_6412_pred__equals__eq2,axiom,
    ! [R: set_Pr1200682188532975830nteger,S3: set_Pr1200682188532975830nteger] :
      ( ! [X4: fun_Co3460408610320954864e_term,Xa4: produc8923325533196201883nteger] :
          ( member6901806533512420791nteger(aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X4),Xa4),R)
        <=> member6901806533512420791nteger(aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X4),Xa4),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_6413_pred__equals__eq2,axiom,
    ! [R: set_Pr3486312109254157132nteger,S3: set_Pr3486312109254157132nteger] :
      ( ! [X4: fun_Pr5098669352780829690e_term,Xa4: produc8923325533196201883nteger] :
          ( member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X4),Xa4),R)
        <=> member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X4),Xa4),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_6414_pred__equals__eq2,axiom,
    ! [R: set_Pr6700895561065273913nt_int,S3: set_Pr6700895561065273913nt_int] :
      ( ! [X4: fun_Pr7763680713840462443e_term,Xa4: product_prod_int_int] :
          ( member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X4),Xa4),R)
        <=> member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X4),Xa4),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_6415_pred__equals__eq2,axiom,
    ! [R: set_Pr4166680992137330533nt_int,S3: set_Pr4166680992137330533nt_int] :
      ( ! [X4: fun_in5730379987311621695e_term,Xa4: product_prod_int_int] :
          ( member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Xa4),R)
        <=> member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X4),Xa4),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_6416_subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( aa_set_real_o(ord_less_eq_set_real(collect_real(aa_real_fun_real_o(aTP_Lamm_de,A))),collect_real(aa_real_fun_real_o(aTP_Lamm_de,B)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% subset_divisors_dvd
tff(fact_6417_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_df,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_df,B)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% subset_divisors_dvd
tff(fact_6418_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ord_le7084787975880047091nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_dh,A)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_dh,B)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% subset_divisors_dvd
tff(fact_6419_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_dg,A))),collect_int(aa_int_fun_int_o(aTP_Lamm_dg,B)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% subset_divisors_dvd
tff(fact_6420_less__set__def,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(A2),B2)
    <=> aa_fun8408151826566693824at_o_o(ord_le9073604597836486254_nat_o(aTP_Lamm_a(A2)),aTP_Lamm_a(B2)) ) ).

% less_set_def
tff(fact_6421_less__set__def,axiom,
    ! [A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_set_real(A2),B2)
    <=> ord_less_fun_real_o(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2)) ) ).

% less_set_def
tff(fact_6422_less__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A2),B2)
    <=> ord_le921202756137246487_nat_o(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2)) ) ).

% less_set_def
tff(fact_6423_less__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2)
    <=> ord_less_fun_nat_o(aTP_Lamm_ad(A2),aTP_Lamm_ad(B2)) ) ).

% less_set_def
tff(fact_6424_less__set__def,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B2)
    <=> ord_less_fun_int_o(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2)) ) ).

% less_set_def
tff(fact_6425_Collect__subset,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: fun_Pr4796943543331071514_nat_o] : aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_di(A2),P))),A2) ).

% Collect_subset
tff(fact_6426_Collect__subset,axiom,
    ! [A2: set_real,P: fun_real_o] : aa_set_real_o(ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_dj(A2),P))),A2) ).

% Collect_subset
tff(fact_6427_Collect__subset,axiom,
    ! [A2: set_list_nat,P: fun_list_nat_o] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_dk(A2),P)),A2) ).

% Collect_subset
tff(fact_6428_Collect__subset,axiom,
    ! [A2: set_set_nat,P: fun_set_nat_o] : aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_dl(A2),P))),A2) ).

% Collect_subset
tff(fact_6429_Collect__subset,axiom,
    ! [A2: set_nat,P: fun_nat_o] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_dm(A2),P))),A2) ).

% Collect_subset
tff(fact_6430_Collect__subset,axiom,
    ! [A2: set_int,P: fun_int_o] : aa_set_int_o(ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_dn(A2),P))),A2) ).

% Collect_subset
tff(fact_6431_less__eq__set__def,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A2),B2)
    <=> aa_fun8408151826566693824at_o_o(ord_le1572679713340831354_nat_o(aTP_Lamm_a(A2)),aTP_Lamm_a(B2)) ) ).

% less_eq_set_def
tff(fact_6432_less__eq__set__def,axiom,
    ! [A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
    <=> ord_le3605727316730466277real_o(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2)) ) ).

% less_eq_set_def
tff(fact_6433_less__eq__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
    <=> ord_le6468474429079055883_nat_o(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2)) ) ).

% less_eq_set_def
tff(fact_6434_less__eq__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
    <=> ord_le7240017405068453697_nat_o(aTP_Lamm_ad(A2),aTP_Lamm_ad(B2)) ) ).

% less_eq_set_def
tff(fact_6435_less__eq__set__def,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
    <=> ord_le5418975059635765733_int_o(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2)) ) ).

% less_eq_set_def
tff(fact_6436_prop__restrict,axiom,
    ! [X: product_prod_nat_nat,Z4: set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat,P: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),Z4)
     => ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Z4),collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_di(X5),P)))
       => aa_Pro6150637604445964947_nat_o(P,X) ) ) ).

% prop_restrict
tff(fact_6437_prop__restrict,axiom,
    ! [X: real,Z4: set_real,X5: set_real,P: fun_real_o] :
      ( aa_set_real_o(member_real(X),Z4)
     => ( aa_set_real_o(ord_less_eq_set_real(Z4),collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_dj(X5),P)))
       => aa_real_o(P,X) ) ) ).

% prop_restrict
tff(fact_6438_prop__restrict,axiom,
    ! [X: list_nat,Z4: set_list_nat,X5: set_list_nat,P: fun_list_nat_o] :
      ( member_list_nat(X,Z4)
     => ( ord_le6045566169113846134st_nat(Z4,collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_dk(X5),P)))
       => aa_list_nat_o(P,X) ) ) ).

% prop_restrict
tff(fact_6439_prop__restrict,axiom,
    ! [X: set_nat,Z4: set_set_nat,X5: set_set_nat,P: fun_set_nat_o] :
      ( aa_set_set_nat_o(member_set_nat(X),Z4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(Z4),collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_dl(X5),P)))
       => aa_set_nat_o(P,X) ) ) ).

% prop_restrict
tff(fact_6440_prop__restrict,axiom,
    ! [X: nat,Z4: set_nat,X5: set_nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat(X),Z4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z4),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_dm(X5),P)))
       => aa_nat_o(P,X) ) ) ).

% prop_restrict
tff(fact_6441_prop__restrict,axiom,
    ! [X: int,Z4: set_int,X5: set_int,P: fun_int_o] :
      ( aa_set_int_o(member_int(X),Z4)
     => ( aa_set_int_o(ord_less_eq_set_int(Z4),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_dn(X5),P)))
       => aa_int_o(P,X) ) ) ).

% prop_restrict
tff(fact_6442_Collect__restrict,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,P: fun_Pr4796943543331071514_nat_o] : aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_di(X5),P))),X5) ).

% Collect_restrict
tff(fact_6443_Collect__restrict,axiom,
    ! [X5: set_real,P: fun_real_o] : aa_set_real_o(ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_dj(X5),P))),X5) ).

% Collect_restrict
tff(fact_6444_Collect__restrict,axiom,
    ! [X5: set_list_nat,P: fun_list_nat_o] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_dk(X5),P)),X5) ).

% Collect_restrict
tff(fact_6445_Collect__restrict,axiom,
    ! [X5: set_set_nat,P: fun_set_nat_o] : aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_dl(X5),P))),X5) ).

% Collect_restrict
tff(fact_6446_Collect__restrict,axiom,
    ! [X5: set_nat,P: fun_nat_o] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_dm(X5),P))),X5) ).

% Collect_restrict
tff(fact_6447_Collect__restrict,axiom,
    ! [X5: set_int,P: fun_int_o] : aa_set_int_o(ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_dn(X5),P))),X5) ).

% Collect_restrict
tff(fact_6448_pred__subset__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( aa_fun8408151826566693824at_o_o(ord_le1572679713340831354_nat_o(aTP_Lamm_a(R)),aTP_Lamm_a(S3))
    <=> aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(R),S3) ) ).

% pred_subset_eq
tff(fact_6449_pred__subset__eq,axiom,
    ! [R: set_real,S3: set_real] :
      ( ord_le3605727316730466277real_o(aTP_Lamm_aa(R),aTP_Lamm_aa(S3))
    <=> aa_set_real_o(ord_less_eq_set_real(R),S3) ) ).

% pred_subset_eq
tff(fact_6450_pred__subset__eq,axiom,
    ! [R: set_set_nat,S3: set_set_nat] :
      ( ord_le6468474429079055883_nat_o(aTP_Lamm_ac(R),aTP_Lamm_ac(S3))
    <=> aa_set_set_nat_o(ord_le6893508408891458716et_nat(R),S3) ) ).

% pred_subset_eq
tff(fact_6451_pred__subset__eq,axiom,
    ! [R: set_nat,S3: set_nat] :
      ( ord_le7240017405068453697_nat_o(aTP_Lamm_ad(R),aTP_Lamm_ad(S3))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,R),S3) ) ).

% pred_subset_eq
tff(fact_6452_pred__subset__eq,axiom,
    ! [R: set_int,S3: set_int] :
      ( ord_le5418975059635765733_int_o(aTP_Lamm_ae(R),aTP_Lamm_ae(S3))
    <=> aa_set_int_o(ord_less_eq_set_int(R),S3) ) ).

% pred_subset_eq
tff(fact_6453_lambda__one,axiom,
    aTP_Lamm_do = times_times_complex(one_one_complex) ).

% lambda_one
tff(fact_6454_lambda__one,axiom,
    aTP_Lamm_dp = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).

% lambda_one
tff(fact_6455_lambda__one,axiom,
    aTP_Lamm_dq = times_times_rat(one_one_rat) ).

% lambda_one
tff(fact_6456_lambda__one,axiom,
    aTP_Lamm_dr = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_6457_lambda__one,axiom,
    aTP_Lamm_ds = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_6458_lambda__zero,axiom,
    aTP_Lamm_dt = times_times_complex(zero_zero_complex) ).

% lambda_zero
tff(fact_6459_lambda__zero,axiom,
    aTP_Lamm_du = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_6460_lambda__zero,axiom,
    aTP_Lamm_dv = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_6461_lambda__zero,axiom,
    aTP_Lamm_dw = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_6462_lambda__zero,axiom,
    aTP_Lamm_dx = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_6463_uminus__set__def,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(A2) = collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(uminus1662154291547880931_nat_o,aTP_Lamm_a(A2))) ).

% uminus_set_def
tff(fact_6464_uminus__set__def,axiom,
    ! [A2: set_real] : uminus612125837232591019t_real(A2) = collect_real(uminus2572480474538279996real_o(aTP_Lamm_aa(A2))) ).

% uminus_set_def
tff(fact_6465_uminus__set__def,axiom,
    ! [A2: set_list_nat] : uminus3195874150345416415st_nat(A2) = collect_list_nat(uminus8676031017987022600_nat_o(aTP_Lamm_ab(A2))) ).

% uminus_set_def
tff(fact_6466_uminus__set__def,axiom,
    ! [A2: set_set_nat] : uminus613421341184616069et_nat(A2) = collect_set_nat(uminus1495274395075166690_nat_o(aTP_Lamm_ac(A2))) ).

% uminus_set_def
tff(fact_6467_uminus__set__def,axiom,
    ! [A2: set_nat] : uminus5710092332889474511et_nat(A2) = collect_nat(uminus6183504819137795864_nat_o(aTP_Lamm_ad(A2))) ).

% uminus_set_def
tff(fact_6468_uminus__set__def,axiom,
    ! [A2: set_int] : uminus1532241313380277803et_int(A2) = collect_int(uminus4362462473705107900_int_o(aTP_Lamm_ae(A2))) ).

% uminus_set_def
tff(fact_6469_Collect__neg__eq,axiom,
    ! [P: fun_real_o] : collect_real(aTP_Lamm_dy(P)) = uminus612125837232591019t_real(collect_real(P)) ).

% Collect_neg_eq
tff(fact_6470_Collect__neg__eq,axiom,
    ! [P: fun_list_nat_o] : collect_list_nat(aTP_Lamm_dz(P)) = uminus3195874150345416415st_nat(collect_list_nat(P)) ).

% Collect_neg_eq
tff(fact_6471_Collect__neg__eq,axiom,
    ! [P: fun_set_nat_o] : collect_set_nat(aTP_Lamm_ea(P)) = uminus613421341184616069et_nat(collect_set_nat(P)) ).

% Collect_neg_eq
tff(fact_6472_Collect__neg__eq,axiom,
    ! [P: fun_nat_o] : collect_nat(aTP_Lamm_eb(P)) = uminus5710092332889474511et_nat(collect_nat(P)) ).

% Collect_neg_eq
tff(fact_6473_Collect__neg__eq,axiom,
    ! [P: fun_int_o] : collect_int(aTP_Lamm_ec(P)) = uminus1532241313380277803et_int(collect_int(P)) ).

% Collect_neg_eq
tff(fact_6474_Compl__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(A2) = collec3392354462482085612at_nat(aTP_Lamm_ed(A2)) ).

% Compl_eq
tff(fact_6475_Compl__eq,axiom,
    ! [A2: set_real] : uminus612125837232591019t_real(A2) = collect_real(aTP_Lamm_ee(A2)) ).

% Compl_eq
tff(fact_6476_Compl__eq,axiom,
    ! [A2: set_list_nat] : uminus3195874150345416415st_nat(A2) = collect_list_nat(aTP_Lamm_ef(A2)) ).

% Compl_eq
tff(fact_6477_Compl__eq,axiom,
    ! [A2: set_set_nat] : uminus613421341184616069et_nat(A2) = collect_set_nat(aTP_Lamm_eg(A2)) ).

% Compl_eq
tff(fact_6478_Compl__eq,axiom,
    ! [A2: set_nat] : uminus5710092332889474511et_nat(A2) = collect_nat(aTP_Lamm_eh(A2)) ).

% Compl_eq
tff(fact_6479_Compl__eq,axiom,
    ! [A2: set_int] : uminus1532241313380277803et_int(A2) = collect_int(aTP_Lamm_ei(A2)) ).

% Compl_eq
tff(fact_6480_set__diff__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A2,B2) = collec3392354462482085612at_nat(aa_set1857767853180004543_nat_o(aTP_Lamm_ej(A2),B2)) ).

% set_diff_eq
tff(fact_6481_set__diff__eq,axiom,
    ! [A2: set_real,B2: set_real] : minus_minus_set_real(A2,B2) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_ek(A2),B2)) ).

% set_diff_eq
tff(fact_6482_set__diff__eq,axiom,
    ! [A2: set_list_nat,B2: set_list_nat] : minus_7954133019191499631st_nat(A2,B2) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_el(A2),B2)) ).

% set_diff_eq
tff(fact_6483_set__diff__eq,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] : minus_2163939370556025621et_nat(A2,B2) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_em(A2),B2)) ).

% set_diff_eq
tff(fact_6484_set__diff__eq,axiom,
    ! [A2: set_int,B2: set_int] : minus_minus_set_int(A2,B2) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_en(A2),B2)) ).

% set_diff_eq
tff(fact_6485_set__diff__eq,axiom,
    ! [A2: set_nat,B2: set_nat] : minus_minus_set_nat(A2,B2) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_eo(A2),B2)) ).

% set_diff_eq
tff(fact_6486_insert__compr,axiom,
    ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A,B2) = collec6321179662152712658at_nat(aa_set6353399069419443519_nat_o(aTP_Lamm_ep(A),B2)) ).

% insert_compr
tff(fact_6487_insert__compr,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A,B2) = collec3392354462482085612at_nat(aa_set1857767853180004543_nat_o(aTP_Lamm_eq(A),B2)) ).

% insert_compr
tff(fact_6488_insert__compr,axiom,
    ! [A: real,B2: set_real] : insert_real(A,B2) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_er(A),B2)) ).

% insert_compr
tff(fact_6489_insert__compr,axiom,
    ! [A: list_nat,B2: set_list_nat] : insert_list_nat(A,B2) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_es(A),B2)) ).

% insert_compr
tff(fact_6490_insert__compr,axiom,
    ! [A: set_nat,B2: set_set_nat] : insert_set_nat(A,B2) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_et(A),B2)) ).

% insert_compr
tff(fact_6491_insert__compr,axiom,
    ! [A: nat,B2: set_nat] : insert_nat(A,B2) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_eu(A),B2)) ).

% insert_compr
tff(fact_6492_insert__compr,axiom,
    ! [A: int,B2: set_int] : insert_int(A,B2) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_ev(A),B2)) ).

% insert_compr
tff(fact_6493_minus__set__def,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A2,B2) = collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(minus_2386458686304230003_nat_o(aTP_Lamm_a(A2)),aTP_Lamm_a(B2))) ).

% minus_set_def
tff(fact_6494_minus__set__def,axiom,
    ! [A2: set_real,B2: set_real] : minus_minus_set_real(A2,B2) = collect_real(minus_404528632214381484real_o(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2))) ).

% minus_set_def
tff(fact_6495_minus__set__def,axiom,
    ! [A2: set_list_nat,B2: set_list_nat] : minus_7954133019191499631st_nat(A2,B2) = collect_list_nat(minus_5761417318614613624_nat_o(aTP_Lamm_ab(A2),aTP_Lamm_ab(B2))) ).

% minus_set_def
tff(fact_6496_minus__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] : minus_2163939370556025621et_nat(A2,B2) = collect_set_nat(minus_6087510627544308050_nat_o(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2))) ).

% minus_set_def
tff(fact_6497_minus__set__def,axiom,
    ! [A2: set_int,B2: set_int] : minus_minus_set_int(A2,B2) = collect_int(minus_7151557278095657260_int_o(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2))) ).

% minus_set_def
tff(fact_6498_minus__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] : minus_minus_set_nat(A2,B2) = collect_nat(minus_8972599623528345224_nat_o(aTP_Lamm_ad(A2),aTP_Lamm_ad(B2))) ).

% minus_set_def
tff(fact_6499_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_ew(A),P)) ).

% insert_Collect
tff(fact_6500_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_ex(A),P)) ).

% insert_Collect
tff(fact_6501_insert__Collect,axiom,
    ! [A: real,P: fun_real_o] : insert_real(A,collect_real(P)) = collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_ey(A),P)) ).

% insert_Collect
tff(fact_6502_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_ez(A),P)) ).

% insert_Collect
tff(fact_6503_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_fa(A),P)) ).

% insert_Collect
tff(fact_6504_insert__Collect,axiom,
    ! [A: nat,P: fun_nat_o] : insert_nat(A,collect_nat(P)) = collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_fb(A),P)) ).

% insert_Collect
tff(fact_6505_insert__Collect,axiom,
    ! [A: int,P: fun_int_o] : insert_int(A,collect_int(P)) = collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_fc(A),P)) ).

% insert_Collect
tff(fact_6506_not__finite__existsD,axiom,
    ! [P: fun_real_o] :
      ( ~ aa_set_real_o(finite_finite_real,collect_real(P))
     => ? [X_1: real] : aa_real_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_6507_not__finite__existsD,axiom,
    ! [P: fun_list_nat_o] :
      ( ~ finite8100373058378681591st_nat(collect_list_nat(P))
     => ? [X_1: list_nat] : aa_list_nat_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_6508_not__finite__existsD,axiom,
    ! [P: fun_set_nat_o] :
      ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(P))
     => ? [X_1: set_nat] : aa_set_nat_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_6509_not__finite__existsD,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => ? [X_1: nat] : aa_nat_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_6510_not__finite__existsD,axiom,
    ! [P: fun_int_o] :
      ( ~ aa_set_int_o(finite_finite_int,collect_int(P))
     => ? [X_1: int] : aa_int_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_6511_not__finite__existsD,axiom,
    ! [P: fun_complex_o] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P))
     => ? [X_1: complex] : aa_complex_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_6512_not__finite__existsD,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(P))
     => ? [X_1: product_prod_nat_nat] : aa_Pro6150637604445964947_nat_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_6513_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B2: set_nat,R: fun_real_fun_nat_o] :
      ( ~ aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),A2)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B2)
                  & aa_nat_o(aa_real_fun_nat_o(R,X3),Xa) ) )
         => ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),B2)
              & ~ aa_set_real_o(finite_finite_real,collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_fd(A2),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6514_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B2: set_int,R: fun_real_fun_int_o] :
      ( ~ aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),A2)
             => ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),B2)
                  & aa_int_o(aa_real_fun_int_o(R,X3),Xa) ) )
         => ? [X3: int] :
              ( aa_set_int_o(member_int(X3),B2)
              & ~ aa_set_real_o(finite_finite_real,collect_real(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_fe(A2),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6515_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B2: set_complex,R: fun_re7319081385389557872plex_o] :
      ( ~ aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),A2)
             => ? [Xa: complex] :
                  ( aa_set_complex_o(member_complex(Xa),B2)
                  & aa_complex_o(aa_rea2382570592068293209plex_o(R,X3),Xa) ) )
         => ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),B2)
              & ~ aa_set_real_o(finite_finite_real,collect_real(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_ff(A2),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6516_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_nat,R: fun_nat_fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A2)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B2)
                  & aa_nat_o(aa_nat_fun_nat_o(R,X3),Xa) ) )
         => ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),B2)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_fg(A2),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6517_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_int,R: fun_nat_fun_int_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A2)
             => ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),B2)
                  & aa_int_o(aa_nat_fun_int_o(R,X3),Xa) ) )
         => ? [X3: int] :
              ( aa_set_int_o(member_int(X3),B2)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_fh(A2),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6518_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_complex,R: fun_na1554462459066888140plex_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A2)
             => ? [Xa: complex] :
                  ( aa_set_complex_o(member_complex(Xa),B2)
                  & aa_complex_o(aa_nat_fun_complex_o(R,X3),Xa) ) )
         => ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),B2)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_fi(A2),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6519_pigeonhole__infinite__rel,axiom,
    ! [A2: set_int,B2: set_nat,R: fun_int_fun_nat_o] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A2)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B2)
                  & aa_nat_o(aa_int_fun_nat_o(R,X3),Xa) ) )
         => ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),B2)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_fj(A2),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6520_pigeonhole__infinite__rel,axiom,
    ! [A2: set_int,B2: set_int,R: fun_int_fun_int_o] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A2)
             => ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),B2)
                  & aa_int_o(aa_int_fun_int_o(R,X3),Xa) ) )
         => ? [X3: int] :
              ( aa_set_int_o(member_int(X3),B2)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_fk(A2),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6521_pigeonhole__infinite__rel,axiom,
    ! [A2: set_int,B2: set_complex,R: fun_in5065666251602127472plex_o] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A2)
             => ? [Xa: complex] :
                  ( aa_set_complex_o(member_complex(Xa),B2)
                  & aa_complex_o(aa_int_fun_complex_o(R,X3),Xa) ) )
         => ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),B2)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_complex_fun_int_o(aa_fun7455058932602313788_int_o(aTP_Lamm_fl(A2),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6522_pigeonhole__infinite__rel,axiom,
    ! [A2: set_complex,B2: set_nat,R: fun_co5676208753217392844_nat_o] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A2)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B2)
                  & aa_nat_o(aa_complex_fun_nat_o(R,X3),Xa) ) )
         => ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),B2)
              & ~ aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_fm(A2),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6523_finite__image__set2,axiom,
    ! [P: fun_real_o,Q: fun_real_o,F: fun_re7681423839302404990l_real] :
      ( aa_set_real_o(finite_finite_real,collect_real(P))
     => ( aa_set_real_o(finite_finite_real,collect_real(Q))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun6833002213068288777real_o(aa_fun8577581014811940695real_o(aTP_Lamm_fn(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_6524_finite__image__set2,axiom,
    ! [P: fun_real_o,Q: fun_real_o,F: fun_re7428432522468557474al_nat] :
      ( aa_set_real_o(finite_finite_real,collect_real(P))
     => ( aa_set_real_o(finite_finite_real,collect_real(Q))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6822315102276328897_nat_o(aa_fun4096426214187263247_nat_o(aTP_Lamm_fo(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_6525_finite__image__set2,axiom,
    ! [P: fun_real_o,Q: fun_real_o,F: fun_re4938349377536830718al_int] :
      ( aa_set_real_o(finite_finite_real,collect_real(P))
     => ( aa_set_real_o(finite_finite_real,collect_real(Q))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun6878201389861000969_int_o(aa_fun3728209879534177367_int_o(aTP_Lamm_fp(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_6526_finite__image__set2,axiom,
    ! [P: fun_real_o,Q: fun_real_o,F: fun_re7812779276277946752omplex] :
      ( aa_set_real_o(finite_finite_real,collect_real(P))
     => ( aa_set_real_o(finite_finite_real,collect_real(Q))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun890640625120065285plex_o(aa_fun6537733294925973843plex_o(aTP_Lamm_fq(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_6527_finite__image__set2,axiom,
    ! [P: fun_real_o,Q: fun_nat_o,F: fun_re4238438383274954274t_real] :
      ( aa_set_real_o(finite_finite_real,collect_real(P))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(Q))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun3910182461874428645real_o(aa_fun8527292551407841495real_o(aTP_Lamm_fr(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_6528_finite__image__set2,axiom,
    ! [P: fun_real_o,Q: fun_nat_o,F: fun_real_fun_nat_nat] :
      ( aa_set_real_o(finite_finite_real,collect_real(P))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(Q))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1675651493036827037_nat_o(aa_fun957765429596782223_nat_o(aTP_Lamm_fs(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_6529_finite__image__set2,axiom,
    ! [P: fun_real_o,Q: fun_nat_o,F: fun_real_fun_nat_int] :
      ( aa_set_real_o(finite_finite_real,collect_real(P))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(Q))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun1731537780621499109_int_o(aa_fun589549094943696343_int_o(aTP_Lamm_ft(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_6530_finite__image__set2,axiom,
    ! [P: fun_real_o,Q: fun_nat_o,F: fun_re2084383175013312036omplex] :
      ( aa_set_real_o(finite_finite_real,collect_real(P))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(Q))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2810168320254036705plex_o(aa_fun467192419720471251plex_o(aTP_Lamm_fu(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_6531_finite__image__set2,axiom,
    ! [P: fun_real_o,Q: fun_int_o,F: fun_re7457141977439324286t_real] :
      ( aa_set_real_o(finite_finite_real,collect_real(P))
     => ( aa_set_int_o(finite_finite_int,collect_int(Q))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun2318301334355154825real_o(aa_fun8106147379196233687real_o(aTP_Lamm_fv(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_6532_finite__image__set2,axiom,
    ! [P: fun_real_o,Q: fun_int_o,F: fun_real_fun_int_nat] :
      ( aa_set_real_o(finite_finite_real,collect_real(P))
     => ( aa_set_int_o(finite_finite_int,collect_int(Q))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1484945577804563009_nat_o(aa_fun4067268449253971343_nat_o(aTP_Lamm_fw(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_6533_finite__image__set,axiom,
    ! [P: fun_real_o,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,collect_real(P))
     => aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_fx(P),F))) ) ).

% finite_image_set
tff(fact_6534_finite__image__set,axiom,
    ! [P: fun_real_o,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,collect_real(P))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun2333486890442116906_nat_o(aTP_Lamm_fy(P),F))) ) ).

% finite_image_set
tff(fact_6535_finite__image__set,axiom,
    ! [P: fun_real_o,F: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,collect_real(P))
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun3242049114608245362_int_o(aTP_Lamm_fz(P),F))) ) ).

% finite_image_set
tff(fact_6536_finite__image__set,axiom,
    ! [P: fun_real_o,F: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,collect_real(P))
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun1011824800444484462plex_o(aTP_Lamm_ga(P),F))) ) ).

% finite_image_set
tff(fact_6537_finite__image__set,axiom,
    ! [P: fun_nat_o,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => aa_set_real_o(finite_finite_real,collect_real(aa_fun6953832176698369998real_o(aTP_Lamm_gb(P),F))) ) ).

% finite_image_set
tff(fact_6538_finite__image__set,axiom,
    ! [P: fun_nat_o,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_gc(P),F))) ) ).

% finite_image_set
tff(fact_6539_finite__image__set,axiom,
    ! [P: fun_nat_o,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun359159518738836302_int_o(aTP_Lamm_gd(P),F))) ) ).

% finite_image_set
tff(fact_6540_finite__image__set,axiom,
    ! [P: fun_nat_o,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2681562978987535434plex_o(aTP_Lamm_ge(P),F))) ) ).

% finite_image_set
tff(fact_6541_finite__image__set,axiom,
    ! [P: fun_int_o,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,collect_int(P))
     => aa_set_real_o(finite_finite_real,collect_real(aa_fun7191409318348945010real_o(aTP_Lamm_gf(P),F))) ) ).

% finite_image_set
tff(fact_6542_finite__image__set,axiom,
    ! [P: fun_int_o,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,collect_int(P))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun7246820503061074602_nat_o(aTP_Lamm_gg(P),F))) ) ).

% finite_image_set
tff(fact_6543_bot__empty__eq2,axiom,
    ! [X2: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(bot_bo428638681919004382_int_o,X2),Xa)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Xa),bot_bo1796632182523588997nt_int) ) ).

% bot_empty_eq2
tff(fact_6544_bot__empty__eq2,axiom,
    ! [X2: fun_Co3460408610320954864e_term,Xa: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(bot_bo2238103298444869889eger_o,X2),Xa)
    <=> member6901806533512420791nteger(aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X2),Xa),bot_bo5621876822565464938nteger) ) ).

% bot_empty_eq2
tff(fact_6545_bot__empty__eq2,axiom,
    ! [X2: fun_Pr5098669352780829690e_term,Xa: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(bot_bo4567088895781559819eger_o,X2),Xa)
    <=> member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X2),Xa),bot_bo9050247954524366304nteger) ) ).

% bot_empty_eq2
tff(fact_6546_bot__empty__eq2,axiom,
    ! [X2: fun_Pr7763680713840462443e_term,Xa: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(bot_bo1117279750869432670_int_o,X2),Xa)
    <=> member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X2),Xa),bot_bo2501116328598523085nt_int) ) ).

% bot_empty_eq2
tff(fact_6547_bot__empty__eq2,axiom,
    ! [X2: fun_in5730379987311621695e_term,Xa: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(bot_bo6524246659364023090_int_o,X2),Xa)
    <=> member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X2),Xa),bot_bo5189594446080201209nt_int) ) ).

% bot_empty_eq2
tff(fact_6548_bot__empty__eq2,axiom,
    ! [X2: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bot_bo1118997454112379798_nat_o,X2),Xa)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Xa)),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq2
tff(fact_6549_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_o,I: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gh(P),I))) ).

% finite_M_bounded_by_nat
tff(fact_6550_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_gi(F),U))) ) ).

% finite_less_ub
tff(fact_6551_numeral__code_I2_J,axiom,
    ! [Na: num] :
      numera1916890842035813515d_enat(aa_num_num(bit0,Na)) = $let(
        m2: extended_enat,
        m2:= numera1916890842035813515d_enat(Na),
        plus_p3455044024723400733d_enat(m2,m2) ) ).

% numeral_code(2)
tff(fact_6552_numeral__code_I2_J,axiom,
    ! [Na: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit0,Na)) = $let(
        m2: real,
        m2:= aa_num_real(numeral_numeral_real,Na),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m2),m2) ) ).

% numeral_code(2)
tff(fact_6553_numeral__code_I2_J,axiom,
    ! [Na: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Na)) = $let(
        m2: rat,
        m2:= aa_num_rat(numeral_numeral_rat,Na),
        aa_rat_rat(plus_plus_rat(m2),m2) ) ).

% numeral_code(2)
tff(fact_6554_numeral__code_I2_J,axiom,
    ! [Na: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Na),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2) ) ).

% numeral_code(2)
tff(fact_6555_numeral__code_I2_J,axiom,
    ! [Na: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Na),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2) ) ).

% numeral_code(2)
tff(fact_6556_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_set_vebt(T) = collect_nat(vEBT_V8194947554948674370ptions(T)) ).

% set_vebt_def
tff(fact_6557_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_dg,I))) ) ).

% finite_divisors_int
tff(fact_6558_numeral__code_I3_J,axiom,
    ! [Na: num] :
      numera6690914467698888265omplex(aa_num_num(bit1,Na)) = $let(
        m2: complex,
        m2:= numera6690914467698888265omplex(Na),
        aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(m2),m2)),one_one_complex) ) ).

% numeral_code(3)
tff(fact_6559_numeral__code_I3_J,axiom,
    ! [Na: num] :
      numera1916890842035813515d_enat(aa_num_num(bit1,Na)) = $let(
        m2: extended_enat,
        m2:= numera1916890842035813515d_enat(Na),
        plus_p3455044024723400733d_enat(plus_p3455044024723400733d_enat(m2,m2),one_on7984719198319812577d_enat) ) ).

% numeral_code(3)
tff(fact_6560_numeral__code_I3_J,axiom,
    ! [Na: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit1,Na)) = $let(
        m2: real,
        m2:= aa_num_real(numeral_numeral_real,Na),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m2),m2)),one_one_real) ) ).

% numeral_code(3)
tff(fact_6561_numeral__code_I3_J,axiom,
    ! [Na: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Na)) = $let(
        m2: rat,
        m2:= aa_num_rat(numeral_numeral_rat,Na),
        aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(m2),m2)),one_one_rat) ) ).

% numeral_code(3)
tff(fact_6562_numeral__code_I3_J,axiom,
    ! [Na: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Na),
        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_6563_numeral__code_I3_J,axiom,
    ! [Na: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Na),
        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_6564_finite__divisors__nat,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_df,Mb))) ) ).

% finite_divisors_nat
tff(fact_6565_set__conv__nth,axiom,
    ! [Xs: list_real] : set_real2(Xs) = collect_real(aTP_Lamm_gj(Xs)) ).

% set_conv_nth
tff(fact_6566_set__conv__nth,axiom,
    ! [Xs: list_list_nat] : set_list_nat2(Xs) = collect_list_nat(aTP_Lamm_gk(Xs)) ).

% set_conv_nth
tff(fact_6567_set__conv__nth,axiom,
    ! [Xs: list_set_nat] : set_set_nat2(Xs) = collect_set_nat(aTP_Lamm_gl(Xs)) ).

% set_conv_nth
tff(fact_6568_set__conv__nth,axiom,
    ! [Xs: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs) = collect_VEBT_VEBT(aTP_Lamm_gm(Xs)) ).

% set_conv_nth
tff(fact_6569_set__conv__nth,axiom,
    ! [Xs: list_o] : set_o2(Xs) = collect_o(aTP_Lamm_gn(Xs)) ).

% set_conv_nth
tff(fact_6570_set__conv__nth,axiom,
    ! [Xs: list_nat] : set_nat2(Xs) = collect_nat(aTP_Lamm_go(Xs)) ).

% set_conv_nth
tff(fact_6571_set__conv__nth,axiom,
    ! [Xs: list_int] : set_int2(Xs) = collect_int(aTP_Lamm_gp(Xs)) ).

% set_conv_nth
tff(fact_6572_finite__lists__length__eq,axiom,
    ! [A2: set_complex,Na: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_gq(A2),Na))) ) ).

% finite_lists_length_eq
tff(fact_6573_finite__lists__length__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,Na: nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => finite500796754983035824at_nat(collec3343600615725829874at_nat(aa_nat1970911183850433186_nat_o(aTP_Lamm_gr(A2),Na))) ) ).

% finite_lists_length_eq
tff(fact_6574_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,Na: nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_gs(A2),Na))) ) ).

% finite_lists_length_eq
tff(fact_6575_finite__lists__length__eq,axiom,
    ! [A2: set_o,Na: nat] :
      ( finite_finite_o(A2)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_gt(A2),Na))) ) ).

% finite_lists_length_eq
tff(fact_6576_finite__lists__length__eq,axiom,
    ! [A2: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_gu(A2),Na))) ) ).

% finite_lists_length_eq
tff(fact_6577_finite__lists__length__eq,axiom,
    ! [A2: set_int,Na: nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_gv(A2),Na))) ) ).

% finite_lists_length_eq
tff(fact_6578_finite__lists__length__le,axiom,
    ! [A2: set_complex,Na: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_gw(A2),Na))) ) ).

% finite_lists_length_le
tff(fact_6579_finite__lists__length__le,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,Na: nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => finite500796754983035824at_nat(collec3343600615725829874at_nat(aa_nat1970911183850433186_nat_o(aTP_Lamm_gx(A2),Na))) ) ).

% finite_lists_length_le
tff(fact_6580_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,Na: nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_gy(A2),Na))) ) ).

% finite_lists_length_le
tff(fact_6581_finite__lists__length__le,axiom,
    ! [A2: set_o,Na: nat] :
      ( finite_finite_o(A2)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_gz(A2),Na))) ) ).

% finite_lists_length_le
tff(fact_6582_finite__lists__length__le,axiom,
    ! [A2: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_ha(A2),Na))) ) ).

% finite_lists_length_le
tff(fact_6583_finite__lists__length__le,axiom,
    ! [A2: set_int,Na: nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_hb(A2),Na))) ) ).

% finite_lists_length_le
tff(fact_6584_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_6585_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va2: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( 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,Va2))),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_6586_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeLista: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeLista,S)),X)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeLista)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_6587_dbl__dec__def,axiom,
    ! [X: complex] : aa_complex_complex(neg_nu6511756317524482435omplex,X) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(X),X)),one_one_complex) ).

% dbl_dec_def
tff(fact_6588_dbl__dec__def,axiom,
    ! [X: real] : aa_real_real(neg_nu6075765906172075777c_real,X) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),X)),one_one_real) ).

% dbl_dec_def
tff(fact_6589_dbl__dec__def,axiom,
    ! [X: rat] : aa_rat_rat(neg_nu3179335615603231917ec_rat,X) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(X),X)),one_one_rat) ).

% dbl_dec_def
tff(fact_6590_dbl__dec__def,axiom,
    ! [X: int] : aa_int_int(neg_nu3811975205180677377ec_int,X) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).

% dbl_dec_def
tff(fact_6591_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeLista,Vd2)),X)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_6592_vebt__member_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,X: 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)),X)
    <=> $ite(
          X = Mia,
          $true,
          $ite(
            X = Maa,
            $true,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Mia),
              $false,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),X),
                $false,
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeLista)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_6593_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mia: nat,Maa: nat,V: nat,TreeLista: list_VEBT_VEBT,Vc: vEBT_VEBT,X: 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,V),TreeLista,Vc)),X)
    <=> ( ( X = Mia )
        | ( X = Maa )
        | $let(
            pos: nat,
            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_6594_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(X),Xa2)
      <=> (Y) )
     => ( ! [A3: $o,B3: $o] :
            ( ( X = vEBT_Leaf((A3),(B3)) )
           => ( (Y)
            <=> ~ $ite(
                    Xa2 = zero_zero_nat,
                    (A3),
                    $ite(Xa2 = one_one_nat,(B3),$false) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
           => (Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2)
               => ( (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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_6595_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(X),Xa2)
     => ( ! [A3: $o,B3: $o] :
            ( ( X = vEBT_Leaf((A3),(B3)) )
           => ~ $ite(
                  Xa2 = zero_zero_nat,
                  (A3),
                  $ite(Xa2 = one_one_nat,(B3),$false) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [S2: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2)
             => ~ $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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_6596_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(X),Xa2)
     => ( ! [A3: $o,B3: $o] :
            ( ( X = vEBT_Leaf((A3),(B3)) )
           => $ite(
                Xa2 = zero_zero_nat,
                (A3),
                $ite(Xa2 = one_one_nat,(B3),$false) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2)
               => $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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_6597_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(X),Xa2)
     => ( ! [Mi: nat,Ma: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2)
           => ~ ( ( Xa2 = Mi )
                | ( Xa2 = Ma ) ) )
       => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
         => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd)
               => ~ $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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_6598_xor__nat__unfold,axiom,
    ! [Mb: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Na) = $ite(
        Mb = zero_zero_nat,
        Na,
        $ite(Na = 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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% xor_nat_unfold
tff(fact_6599_xor__nat__rec,axiom,
    ! [Mb: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Na) = 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))),Na))),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_6600_one__xor__eq,axiom,
    ! [A: code_integer] : bit_se3222712562003087583nteger(one_one_Code_integer,A) = minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))),zero_n356916108424825756nteger(~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) ).

% one_xor_eq
tff(fact_6601_one__xor__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_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_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_xor_eq
tff(fact_6602_one__xor__eq,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(one_one_int),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_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_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_xor_eq
tff(fact_6603_xor__one__eq,axiom,
    ! [A: code_integer] : bit_se3222712562003087583nteger(A,one_one_Code_integer) = minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))),zero_n356916108424825756nteger(~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) ).

% xor_one_eq
tff(fact_6604_xor__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_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_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))) ).

% xor_one_eq
tff(fact_6605_xor__one__eq,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(A),one_one_int) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_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_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))) ).

% xor_one_eq
tff(fact_6606_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( aa_nat_o(vEBT_vebt_member(X),Xa2)
     => ( ! [A3: $o,B3: $o] :
            ( ( X = vEBT_Leaf((A3),(B3)) )
           => ~ $ite(
                  Xa2 = zero_zero_nat,
                  (A3),
                  $ite(Xa2 = one_one_nat,(B3),$false) ) )
       => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Summary: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)
             => ~ $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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_6607_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(X),Xa2)
     => ( ! [Uu2: $o,Uv2: $o] : X != vEBT_Leaf((Uu2),(Uv2))
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2)
               => ( ( Xa2 = Mi )
                  | ( Xa2 = Ma ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd)
                   => $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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_6608_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(X),Xa2)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : X = vEBT_Leaf((Uu2),(Uv2))
         => (Y) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
           => (Y) )
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2)
               => ( (Y)
                <=> ~ ( ( Xa2 = Mi )
                      | ( Xa2 = Ma ) ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd)
                   => ( (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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_6609_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,groups9119017779487936845_o_nat(zero_n2687167440665602831ol_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)),Bs)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_o_nat(size_size_list_o,Bs))
        & aa_nat_o(nth_o(Bs),Na) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_6610_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Bs)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_o_nat(size_size_list_o,Bs))
        & aa_nat_o(nth_o(Bs),Na) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_6611_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(X),Xa2)
     => ( ! [A3: $o,B3: $o] :
            ( ( X = vEBT_Leaf((A3),(B3)) )
           => $ite(
                Xa2 = zero_zero_nat,
                (A3),
                $ite(Xa2 = one_one_nat,(B3),$false) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)
                   => $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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_6612_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_complex(ring_17405671764205052669omplex,K) = $ite(
        K = zero_zero_int,
        zero_zero_complex,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: complex,
            l:= aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_complex_complex(plus_plus_complex(l),one_one_complex)) ) ) ) ).

% of_int_code_if
tff(fact_6613_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_int(ring_1_of_int_int,K) = $ite(
        K = zero_zero_int,
        zero_zero_int,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: int,
            l:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_int_int(aa_int_fun_int_int(plus_plus_int,l),one_one_int)) ) ) ) ).

% of_int_code_if
tff(fact_6614_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_real(ring_1_of_int_real,K) = $ite(
        K = zero_zero_int,
        zero_zero_real,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: real,
            l:= aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,l),one_one_real)) ) ) ) ).

% of_int_code_if
tff(fact_6615_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_rat(ring_1_of_int_rat,K) = $ite(
        K = zero_zero_int,
        zero_zero_rat,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: rat,
            l:= aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_rat_rat(plus_plus_rat(l),one_one_rat)) ) ) ) ).

% of_int_code_if
tff(fact_6616_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_Code_integer(ring_18347121197199848620nteger,K) = $ite(
        K = zero_zero_int,
        zero_z3403309356797280102nteger,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% of_int_code_if
tff(fact_6617_finite__nth__roots,axiom,
    ! [Na: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_hc(Na),C))) ) ).

% finite_nth_roots
tff(fact_6618_finite__roots__unity,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => aa_set_real_o(finite_finite_real,collect_real(aa_nat_fun_real_o(aTP_Lamm_hd,Na))) ) ).

% finite_roots_unity
tff(fact_6619_finite__roots__unity,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_he,Na))) ) ).

% finite_roots_unity
tff(fact_6620_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_complex,Y: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_hf(I5),X)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_hf(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_hg(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6621_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_complex,Y: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_hh(I5),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_hh(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_hi(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6622_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_complex,Y: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_hj(I5),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_hj(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_hk(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6623_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_complex_complex,Y: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_hl(I5),X)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_hl(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_hm(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6624_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_real,Y: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_hn(I5),X)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_hn(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_ho(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6625_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_real,Y: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_hp(I5),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_hp(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_hq(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6626_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_real,Y: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_hr(I5),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_hr(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_hs(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6627_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_complex_real,Y: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_ht(I5),X)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_ht(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_hu(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6628_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_rat,Y: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_hv(I5),X)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_hv(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_hw(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6629_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_rat,Y: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_hx(I5),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_hx(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_hy(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6630_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(X),Xa2)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: $o,B3: $o] :
              ( ( X = 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)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xa2)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ( ~ (Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xa2)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ~ (Y)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa2)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                     => ( ( (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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$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,Va2)),TreeList,Summary),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_6631_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(X),Xa2)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: $o,B3: $o] :
              ( ( X = 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) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xa2)) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xa2)) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa2)) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                     => ( 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,Va2)),TreeList,Summary),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_6632_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(K),L)),zero_zero_int)
    <=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% xor_negative_int_iff
tff(fact_6633_XOR__upper,axiom,
    ! [X: int,Na: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(X),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ) ).

% XOR_upper
tff(fact_6634_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( aa_nat_o(vEBT_vebt_member(X),Xa2)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: $o,B3: $o] :
              ( ( X = 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,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
               => ( 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,Va2)),TreeList,Summary),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_6635_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_complex,Y: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_hz(I5),X)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_hz(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_ia(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6636_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_complex,Y: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_ib(I5),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_ib(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_ic(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6637_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_complex,Y: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_id(I5),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_id(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_ie(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6638_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_complex_complex,Y: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_if(I5),X)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_if(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_ig(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6639_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_real,Y: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_ih(I5),X)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_ih(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_ii(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6640_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_real,Y: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_ij(I5),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_ij(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_ik(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6641_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_real,Y: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_il(I5),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_il(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_im(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6642_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_complex_real,Y: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_in(I5),X)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_in(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_io(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6643_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_rat,Y: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_ip(I5),X)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_ip(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_iq(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6644_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_rat,Y: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_ir(I5),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_ir(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_is(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6645_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(X),Xa2)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: $o,B3: $o] :
              ( ( X = 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)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xa2)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2) )
                 => ( ( (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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2),Xa2)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_6646_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(X),Xa2)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: $o,B3: $o] :
              ( ( X = 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,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2),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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_6647_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(X),Xa2)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: $o,B3: $o] :
              ( ( X = 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) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xa2)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2),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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_6648_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(X),Xa2)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xa2)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xa2)) )
           => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),Xa2))
                   => ( ( Xa2 = Mi )
                      | ( Xa2 = Ma ) ) ) )
             => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),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,V2),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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd),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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_6649_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(X),Xa2)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Y)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xa2)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = 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,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,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,Va3,Vb2),Xa2)) ) )
             => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),Xa2)) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd) )
                     => ( ( (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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd),Xa2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_6650_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(X),Xa2)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),Xa2))
               => ~ ( ( Xa2 = Mi )
                    | ( Xa2 = Ma ) ) ) )
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),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,V2),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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd),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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_6651_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_6652_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_6653_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_6654_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),X)
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat) ) )
         => ( ( ( X = 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)) ) )
           => ~ ! [Va2: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( 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,Va2))),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),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,Va2))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_6655_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Na: num] : unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit0,Na)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_it),unique5052692396658037445od_int(Mb,Na)) ).

% divmod_algorithm_code(6)
tff(fact_6656_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Na: num] : unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit0,Na)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_iu),unique5055182867167087721od_nat(Mb,Na)) ).

% divmod_algorithm_code(6)
tff(fact_6657_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Na: num] : unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit0,Na)) = produc6916734918728496179nteger(aTP_Lamm_iv,unique3479559517661332726nteger(Mb,Na)) ).

% divmod_algorithm_code(6)
tff(fact_6658_artanh__def,axiom,
    ! [X: real] : aa_real_real(artanh_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_6659_ln__inj__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( ( aa_real_real(ln_ln_real,X) = aa_real_real(ln_ln_real,Y) )
        <=> ( X = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_6660_ln__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( 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,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ) ).

% ln_less_cancel_iff
tff(fact_6661_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_6662_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_6663_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_6664_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_6665_case__prod__conv,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_prod_conv
tff(fact_6666_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( 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(ln_ln_real,X)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_6667_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,X)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ) ).

% ln_less_zero_iff
tff(fact_6668_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X) ) ) ).

% ln_gt_zero_iff
tff(fact_6669_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( ( aa_real_real(ln_ln_real,X) = zero_zero_real )
      <=> ( X = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_6670_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_6671_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,X)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ) ).

% ln_le_zero_iff
tff(fact_6672_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ) ).

% ln_ge_zero_iff
tff(fact_6673_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Na: num] : unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit0,Na)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_iw),unique5052692396658037445od_int(Mb,Na)) ).

% divmod_algorithm_code(5)
tff(fact_6674_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Na: num] : unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit0,Na)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ix),unique5055182867167087721od_nat(Mb,Na)) ).

% divmod_algorithm_code(5)
tff(fact_6675_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Na: num] : unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit0,Na)) = produc6916734918728496179nteger(aTP_Lamm_iy,unique3479559517661332726nteger(Mb,Na)) ).

% divmod_algorithm_code(5)
tff(fact_6676_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_iz(H),F)),Prod) ).

% prod.case_distrib
tff(fact_6677_prod_Ocase__distrib,axiom,
    ! [H: fun_o_o,F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_ja(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_6678_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(H,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun4545072666309863515_int_o(aTP_Lamm_jb(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_6679_prod_Ocase__distrib,axiom,
    ! [H: fun_o_8988054265482332260nt_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_P3134438092330336605nt_int(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_fun4023131440832735197nt_int(aTP_Lamm_jc(H),F)),Prod) ).

% prod.case_distrib
tff(fact_6680_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_jd(H),F)),Prod) ).

% prod.case_distrib
tff(fact_6681_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_je(H),F)),Prod) ).

% prod.case_distrib
tff(fact_6682_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_jf(H),F)),Prod) ).

% prod.case_distrib
tff(fact_6683_prod_Ocase__distrib,axiom,
    ! [H: fun_na8713714802964096592at_nat,F: fun_nat_fun_nat_nat,Prod: product_prod_nat_nat] : aa_nat7997280740862404369at_nat(H,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),Prod)) = aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aa_fun2484538639213223237at_nat(aTP_Lamm_jg(H),F)),Prod) ).

% prod.case_distrib
tff(fact_6684_prod_Ocase__distrib,axiom,
    ! [H: fun_fu1413608398870722896at_nat,F: fun_na1668100129766719005at_nat,Prod: product_prod_nat_nat] : aa_fun7582795104022253201at_nat(H,aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),Prod)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aa_fun8817627472026143685at_nat(aTP_Lamm_jh(H),F)),Prod) ).

% prod.case_distrib
tff(fact_6685_prod_Ocase__distrib,axiom,
    ! [H: fun_fu3275390434952296369_nat_o,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] : aa_fun359341089595923442_nat_o(H,aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),Prod)) = aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aa_fun824319262034445606_nat_o(aTP_Lamm_ji(H),F)),Prod) ).

% prod.case_distrib
tff(fact_6686_old_Oprod_Ocase,axiom,
    ! [F: fun_nat_fun_nat_nat,X1: nat,X22: nat] : aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X22)) = aa_nat_nat(aa_nat_fun_nat_nat(F,X1),X22) ).

% old.prod.case
tff(fact_6687_old_Oprod_Ocase,axiom,
    ! [F: fun_na1668100129766719005at_nat,X1: nat,X22: nat] : aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X22)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X1),X22) ).

% old.prod.case
tff(fact_6688_old_Oprod_Ocase,axiom,
    ! [F: fun_na5914473681949881652_nat_o,X1: nat,X22: nat] : aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X22)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X1),X22) ).

% old.prod.case
tff(fact_6689_old_Oprod_Ocase,axiom,
    ! [F: fun_in1696471567231710611nt_int,X1: int,X22: int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X1),X22) ).

% old.prod.case
tff(fact_6690_old_Oprod_Ocase,axiom,
    ! [F: fun_int_fun_int_o,X1: int,X22: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22))
    <=> aa_int_o(aa_int_fun_int_o(F,X1),X22) ) ).

% old.prod.case
tff(fact_6691_split__cong,axiom,
    ! [Q4: product_prod_nat_nat,F: fun_nat_fun_nat_nat,G: fun_nat_fun_nat_nat,P2: product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3) = Q4 )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(F,X3),Y3) = aa_nat_nat(aa_nat_fun_nat_nat(G,X3),Y3) ) )
     => ( ( P2 = Q4 )
       => ( aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),P2) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(G),Q4) ) ) ) ).

% split_cong
tff(fact_6692_split__cong,axiom,
    ! [Q4: product_prod_nat_nat,F: fun_na1668100129766719005at_nat,G: fun_na1668100129766719005at_nat,P2: product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3) = Q4 )
         => ( aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X3),Y3) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(G,X3),Y3) ) )
     => ( ( P2 = Q4 )
       => ( aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),P2) = aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(G),Q4) ) ) ) ).

% split_cong
tff(fact_6693_split__cong,axiom,
    ! [Q4: product_prod_nat_nat,F: fun_na5914473681949881652_nat_o,G: fun_na5914473681949881652_nat_o,P2: product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3) = Q4 )
         => ( aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X3),Y3) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(G,X3),Y3) ) )
     => ( ( P2 = Q4 )
       => ( aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),P2) = aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(G),Q4) ) ) ) ).

% split_cong
tff(fact_6694_split__cong,axiom,
    ! [Q4: product_prod_int_int,F: fun_in1696471567231710611nt_int,G: fun_in1696471567231710611nt_int,P2: product_prod_int_int] :
      ( ! [X3: int,Y3: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) = Q4 )
         => ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X3),Y3) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(G,X3),Y3) ) )
     => ( ( P2 = Q4 )
       => ( aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),P2) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(G),Q4) ) ) ) ).

% split_cong
tff(fact_6695_split__cong,axiom,
    ! [Q4: product_prod_int_int,F: fun_int_fun_int_o,G: fun_int_fun_int_o,P2: product_prod_int_int] :
      ( ! [X3: int,Y3: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) = Q4 )
         => ( aa_int_o(aa_int_fun_int_o(F,X3),Y3)
          <=> aa_int_o(aa_int_fun_int_o(G,X3),Y3) ) )
     => ( ( P2 = Q4 )
       => ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),P2)
        <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(G),Q4) ) ) ) ).

% split_cong
tff(fact_6696_cond__case__prod__eta,axiom,
    ! [F: fun_nat_fun_nat_nat,G: fun_Pr2602117396471556196at_nat] :
      ( ! [X3: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(F,X3),Y3) = aa_Pro4243173770044397973at_nat(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3))
     => ( produc6842872674320459806at_nat(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_6697_cond__case__prod__eta,axiom,
    ! [F: fun_na1668100129766719005at_nat,G: fun_Pr2477239776880934807at_nat] :
      ( ! [X3: nat,Y3: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X3),Y3) = aa_Pro8023792847356852296at_nat(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3))
     => ( produc4528391901023682385at_nat(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_6698_cond__case__prod__eta,axiom,
    ! [F: fun_na5914473681949881652_nat_o,G: fun_Pr2028404973800837934_nat_o] :
      ( ! [X3: nat,Y3: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X3),Y3) = aa_Pro989595756709598047_nat_o(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3))
     => ( produc6395135316500328296_nat_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_6699_cond__case__prod__eta,axiom,
    ! [F: fun_in1696471567231710611nt_int,G: fun_Pr2478283820680522243nt_int] :
      ( ! [X3: int,Y3: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X3),Y3) = aa_Pro6970504127699319548nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3))
     => ( produc4245557441103728435nt_int(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_6700_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_o,G: fun_Pr3238374608701176162_int_o] :
      ( ! [X3: int,Y3: int] :
          ( aa_int_o(aa_int_fun_int_o(F,X3),Y3)
        <=> aa_Pro4592068669816069595_int_o(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3)) )
     => ( produc4947309494688390418_int_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_6701_case__prod__eta,axiom,
    ! [F: fun_Pr2602117396471556196at_nat] : produc6842872674320459806at_nat(aTP_Lamm_jj(F)) = F ).

% case_prod_eta
tff(fact_6702_case__prod__eta,axiom,
    ! [F: fun_Pr2477239776880934807at_nat] : produc4528391901023682385at_nat(aTP_Lamm_jk(F)) = F ).

% case_prod_eta
tff(fact_6703_case__prod__eta,axiom,
    ! [F: fun_Pr2028404973800837934_nat_o] : produc6395135316500328296_nat_o(aTP_Lamm_jl(F)) = F ).

% case_prod_eta
tff(fact_6704_case__prod__eta,axiom,
    ! [F: fun_Pr2478283820680522243nt_int] : produc4245557441103728435nt_int(aTP_Lamm_jm(F)) = F ).

% case_prod_eta
tff(fact_6705_case__prod__eta,axiom,
    ! [F: fun_Pr3238374608701176162_int_o] : produc4947309494688390418_int_o(aTP_Lamm_jn(F)) = F ).

% case_prod_eta
tff(fact_6706_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))
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3) )
           => ~ aa_nat_o(Q,aa_nat_nat(aa_nat_fun_nat_nat(P,X3),Y3)) ) ) ).

% case_prodE2
tff(fact_6707_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))
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3) )
           => ~ aa_fun3538214411680374423_nat_o(Q,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(P,X3),Y3)) ) ) ).

% case_prodE2
tff(fact_6708_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))
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3) )
           => ~ aa_fun8408151826566693824at_o_o(Q,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(P,X3),Y3)) ) ) ).

% case_prodE2
tff(fact_6709_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))
     => ~ ! [X3: int,Y3: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_Pro4592068669816069595_int_o(Q,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(P,X3),Y3)) ) ) ).

% case_prodE2
tff(fact_6710_case__prodE2,axiom,
    ! [Q: fun_o_o,P: fun_int_fun_int_o,Z: product_prod_int_int] :
      ( aa_o_o(Q,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(P),Z))
     => ~ ! [X3: int,Y3: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_o_o(Q,aa_int_o(aa_int_fun_int_o(P,X3),Y3)) ) ) ).

% case_prodE2
tff(fact_6711_ln__less__self,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,X)),X) ) ).

% ln_less_self
tff(fact_6712_ln__bound,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,X)),X) ) ).

% ln_bound
tff(fact_6713_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_6714_ln__less__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,X)),zero_zero_real) ) ) ).

% ln_less_zero
tff(fact_6715_ln__gt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X)) ) ).

% ln_gt_zero
tff(fact_6716_ln__div,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_div
tff(fact_6717_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_6718_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_6719_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( 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(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),Y)) ) ) ).

% ln_diff_le
tff(fact_6720_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_mult
tff(fact_6721_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( ( aa_real_real(ln_ln_real,X) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real) )
       => ( X = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_6722_ln__le__minus__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)) ) ).

% ln_le_minus_one
tff(fact_6723_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X) ) ).

% ln_add_one_self_le_self2
tff(fact_6724_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X))),aa_real_real(uminus_uminus_real,X)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_6725_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_jo(L)),Qr) ).

% divmod_step_nat_def
tff(fact_6726_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_jp(L)),Qr) ).

% divmod_step_int_def
tff(fact_6727_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_jo(L)),Qr) ).

% divmod_step_def
tff(fact_6728_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_jp(L)),Qr) ).

% divmod_step_def
tff(fact_6729_divmod__step__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_jq(L),Qr) ).

% divmod_step_def
tff(fact_6730_divmod__nat__if,axiom,
    ! [Mb: nat,Na: nat] :
      divmod_nat(Mb,Na) = $ite(
        ( ( Na = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ),
        aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),Mb),
        aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_jr),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na),Na)) ) ).

% divmod_nat_if
tff(fact_6731_tanh__ln__real,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_6732_Sum__Icc__nat,axiom,
    ! [Mb: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_dr),set_or1269000886237332187st_nat(Mb,Na)) = 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,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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_6733_arith__series__nat,axiom,
    ! [A: nat,D: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_js(A),D)),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_6734_case__prod__Pair__iden,axiom,
    ! [P2: produc7509114972854644512nteger] : produc6766542753387345183nteger(produc7264323995540650962nteger,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_6735_case__prod__Pair__iden,axiom,
    ! [P2: produc2453742332425399446nteger] : produc697581972998607883nteger(produc565448451787809096nteger,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_6736_case__prod__Pair__iden,axiom,
    ! [P2: produc3043004577601930755nt_int] : produc6823905543215200229nt_int(produc7320969046608631221nt_int,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_6737_case__prod__Pair__iden,axiom,
    ! [P2: produc4788131091073775919nt_int] : produc7102243260366339645nt_int(produc4729913460834754785nt_int,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_6738_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_int_int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(product_Pair_int_int),P2) = P2 ).

% case_prod_Pair_iden
tff(fact_6739_round__unique_H,axiom,
    ! [X: real,Na: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_int_real(ring_1_of_int_real,Na)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_int(archim8280529875227126926d_real,X) = Na ) ) ).

% round_unique'
tff(fact_6740_round__unique_H,axiom,
    ! [X: rat,Na: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),aa_int_rat(ring_1_of_int_rat,Na)))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
     => ( aa_rat_int(archim7778729529865785530nd_rat,X) = Na ) ) ).

% round_unique'
tff(fact_6741_abs__abs,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_abs
tff(fact_6742_abs__abs,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_abs
tff(fact_6743_abs__abs,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_abs
tff(fact_6744_abs__abs,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_abs
tff(fact_6745_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_6746_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_6747_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_6748_abs__idempotent,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_idempotent
tff(fact_6749_case__prodI,axiom,
    ! [F: fun_fu3422731660242565741eger_o,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(F,A),B)
     => aa_Pro4652312591602923194eger_o(produc7733317944277263833eger_o(F),aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A),B)) ) ).

% case_prodI
tff(fact_6750_case__prodI,axiom,
    ! [F: fun_fu7695273699059480439eger_o,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(F,A),B)
     => aa_Pro4148064003102580676eger_o(produc7373287791920631523eger_o(F),aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A),B)) ) ).

% case_prodI
tff(fact_6751_case__prodI,axiom,
    ! [F: fun_fu6865642474282033866_int_o,A: fun_Pr7763680713840462443e_term,B: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(F,A),B)
     => aa_Pro2601676399929874711_int_o(produc2207054017359051062_int_o(F),aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B)) ) ).

% case_prodI
tff(fact_6752_case__prodI,axiom,
    ! [F: fun_fu2978539155105838238_int_o,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(F,A),B)
     => aa_Pro5451197003417746155_int_o(produc7271678899836001034_int_o(F),aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B)) ) ).

% case_prodI
tff(fact_6753_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_6754_case__prodI2,axiom,
    ! [P2: produc7509114972854644512nteger,C: fun_fu3422731660242565741eger_o] :
      ( ! [A3: fun_Co3460408610320954864e_term,B3: produc8923325533196201883nteger] :
          ( ( P2 = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A3),B3) )
         => aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(C,A3),B3) )
     => aa_Pro4652312591602923194eger_o(produc7733317944277263833eger_o(C),P2) ) ).

% case_prodI2
tff(fact_6755_case__prodI2,axiom,
    ! [P2: produc2453742332425399446nteger,C: fun_fu7695273699059480439eger_o] :
      ( ! [A3: fun_Pr5098669352780829690e_term,B3: produc8923325533196201883nteger] :
          ( ( P2 = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A3),B3) )
         => aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(C,A3),B3) )
     => aa_Pro4148064003102580676eger_o(produc7373287791920631523eger_o(C),P2) ) ).

% case_prodI2
tff(fact_6756_case__prodI2,axiom,
    ! [P2: produc3043004577601930755nt_int,C: fun_fu6865642474282033866_int_o] :
      ( ! [A3: fun_Pr7763680713840462443e_term,B3: product_prod_int_int] :
          ( ( P2 = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A3),B3) )
         => aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(C,A3),B3) )
     => aa_Pro2601676399929874711_int_o(produc2207054017359051062_int_o(C),P2) ) ).

% case_prodI2
tff(fact_6757_case__prodI2,axiom,
    ! [P2: produc4788131091073775919nt_int,C: fun_fu2978539155105838238_int_o] :
      ( ! [A3: fun_in5730379987311621695e_term,B3: product_prod_int_int] :
          ( ( P2 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A3),B3) )
         => aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(C,A3),B3) )
     => aa_Pro5451197003417746155_int_o(produc7271678899836001034_int_o(C),P2) ) ).

% case_prodI2
tff(fact_6758_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_6759_mem__case__prodI,axiom,
    ! [Z: real,C: fun_in6553099857819052212t_real,A: int,B: int] :
      ( aa_set_real_o(member_real(Z),aa_int_set_real(aa_int4847878404602763637t_real(C,A),B))
     => aa_set_real_o(member_real(Z),produc6452406959799940328t_real(C,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% mem_case_prodI
tff(fact_6760_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_in2630332840903238488et_nat,A: int,B: int] :
      ( aa_set_nat_o(member_nat(Z),aa_int_set_nat(aa_int1906528167449992217et_nat(C,A),B))
     => aa_set_nat_o(member_nat(Z),produc4251311855443802252et_nat(C,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% mem_case_prodI
tff(fact_6761_mem__case__prodI,axiom,
    ! [Z: int,C: fun_in7040156458970095540et_int,A: int,B: int] :
      ( aa_set_int_o(member_int(Z),aa_int_set_int(aa_int8639817059373041269et_int(C,A),B))
     => aa_set_int_o(member_int(Z),produc73460835934605544et_int(C,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% mem_case_prodI
tff(fact_6762_mem__case__prodI,axiom,
    ! [Z: set_nat,C: fun_in7207726656141197838et_nat,A: int,B: int] :
      ( aa_set_set_nat_o(member_set_nat(Z),aa_int_set_set_nat(aa_int1843896668418082255et_nat(C,A),B))
     => aa_set_set_nat_o(member_set_nat(Z),produc5233655623923918146et_nat(C,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% mem_case_prodI
tff(fact_6763_mem__case__prodI,axiom,
    ! [Z: product_prod_nat_nat,C: fun_in1904380326158164027at_nat,A: int,B: int] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z),aa_int8822742982810151799at_nat(aa_int6362027936895332260at_nat(C,A),B))
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z),produc1656060378719767003at_nat(C,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% mem_case_prodI
tff(fact_6764_mem__case__prodI,axiom,
    ! [Z: real,C: fun_fu7234941744988193529t_real,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( aa_set_real_o(member_real(Z),aa_Pro6515153133580421057t_real(aa_fun1966338168372333922t_real(C,A),B))
     => aa_set_real_o(member_real(Z),produc5067746511659309921t_real(C,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A),B))) ) ).

% mem_case_prodI
tff(fact_6765_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_fu6383124537394581533et_nat,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( aa_set_nat_o(member_nat(Z),aa_Pro381016180347496421et_nat(aa_fun3554285010146031238et_nat(C,A),B))
     => aa_set_nat_o(member_nat(Z),produc7961109426542857093et_nat(C,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A),B))) ) ).

% mem_case_prodI
tff(fact_6766_mem__case__prodI,axiom,
    ! [Z: int,C: fun_fu1569576118606662777et_int,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( aa_set_int_o(member_int(Z),aa_Pro5426537197693075521et_int(aa_fun1064201865214304482et_int(C,A),B))
     => aa_set_int_o(member_int(Z),produc3783258407033660385et_int(C,aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A),B))) ) ).

% mem_case_prodI
tff(fact_6767_mem__case__prodI,axiom,
    ! [Z: real,C: fun_fu604837885242254216t_real,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_set_real_o(member_real(Z),aa_Pro7989103684195733727t_real(aa_fun5197189019902991601t_real(C,A),B))
     => aa_set_real_o(member_real(Z),produc5863038274793111792t_real(C,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B))) ) ).

% mem_case_prodI
tff(fact_6768_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_fu9079770784915425324et_nat,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_set_nat_o(member_nat(Z),aa_Pro3599192995643711491et_nat(aa_fun8409204002232819093et_nat(C,A),B))
     => aa_set_nat_o(member_nat(Z),produc5011043943518292116et_nat(C,aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B))) ) ).

% mem_case_prodI
tff(fact_6769_mem__case__prodI2,axiom,
    ! [P2: product_prod_int_int,Z: real,C: fun_in6553099857819052212t_real] :
      ( ! [A3: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_set_real_o(member_real(Z),aa_int_set_real(aa_int4847878404602763637t_real(C,A3),B3)) )
     => aa_set_real_o(member_real(Z),produc6452406959799940328t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6770_mem__case__prodI2,axiom,
    ! [P2: product_prod_int_int,Z: nat,C: fun_in2630332840903238488et_nat] :
      ( ! [A3: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_set_nat_o(member_nat(Z),aa_int_set_nat(aa_int1906528167449992217et_nat(C,A3),B3)) )
     => aa_set_nat_o(member_nat(Z),produc4251311855443802252et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6771_mem__case__prodI2,axiom,
    ! [P2: product_prod_int_int,Z: int,C: fun_in7040156458970095540et_int] :
      ( ! [A3: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_set_int_o(member_int(Z),aa_int_set_int(aa_int8639817059373041269et_int(C,A3),B3)) )
     => aa_set_int_o(member_int(Z),produc73460835934605544et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6772_mem__case__prodI2,axiom,
    ! [P2: product_prod_int_int,Z: set_nat,C: fun_in7207726656141197838et_nat] :
      ( ! [A3: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_set_set_nat_o(member_set_nat(Z),aa_int_set_set_nat(aa_int1843896668418082255et_nat(C,A3),B3)) )
     => aa_set_set_nat_o(member_set_nat(Z),produc5233655623923918146et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6773_mem__case__prodI2,axiom,
    ! [P2: product_prod_int_int,Z: product_prod_nat_nat,C: fun_in1904380326158164027at_nat] :
      ( ! [A3: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z),aa_int8822742982810151799at_nat(aa_int6362027936895332260at_nat(C,A3),B3)) )
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z),produc1656060378719767003at_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6774_mem__case__prodI2,axiom,
    ! [P2: produc7509114972854644512nteger,Z: real,C: fun_fu7234941744988193529t_real] :
      ( ! [A3: fun_Co3460408610320954864e_term,B3: produc8923325533196201883nteger] :
          ( ( P2 = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A3),B3) )
         => aa_set_real_o(member_real(Z),aa_Pro6515153133580421057t_real(aa_fun1966338168372333922t_real(C,A3),B3)) )
     => aa_set_real_o(member_real(Z),produc5067746511659309921t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6775_mem__case__prodI2,axiom,
    ! [P2: produc7509114972854644512nteger,Z: nat,C: fun_fu6383124537394581533et_nat] :
      ( ! [A3: fun_Co3460408610320954864e_term,B3: produc8923325533196201883nteger] :
          ( ( P2 = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A3),B3) )
         => aa_set_nat_o(member_nat(Z),aa_Pro381016180347496421et_nat(aa_fun3554285010146031238et_nat(C,A3),B3)) )
     => aa_set_nat_o(member_nat(Z),produc7961109426542857093et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6776_mem__case__prodI2,axiom,
    ! [P2: produc7509114972854644512nteger,Z: int,C: fun_fu1569576118606662777et_int] :
      ( ! [A3: fun_Co3460408610320954864e_term,B3: produc8923325533196201883nteger] :
          ( ( P2 = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A3),B3) )
         => aa_set_int_o(member_int(Z),aa_Pro5426537197693075521et_int(aa_fun1064201865214304482et_int(C,A3),B3)) )
     => aa_set_int_o(member_int(Z),produc3783258407033660385et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6777_mem__case__prodI2,axiom,
    ! [P2: produc4788131091073775919nt_int,Z: real,C: fun_fu604837885242254216t_real] :
      ( ! [A3: fun_in5730379987311621695e_term,B3: product_prod_int_int] :
          ( ( P2 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A3),B3) )
         => aa_set_real_o(member_real(Z),aa_Pro7989103684195733727t_real(aa_fun5197189019902991601t_real(C,A3),B3)) )
     => aa_set_real_o(member_real(Z),produc5863038274793111792t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6778_mem__case__prodI2,axiom,
    ! [P2: produc4788131091073775919nt_int,Z: nat,C: fun_fu9079770784915425324et_nat] :
      ( ! [A3: fun_in5730379987311621695e_term,B3: product_prod_int_int] :
          ( ( P2 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A3),B3) )
         => aa_set_nat_o(member_nat(Z),aa_Pro3599192995643711491et_nat(aa_fun8409204002232819093et_nat(C,A3),B3)) )
     => aa_set_nat_o(member_nat(Z),produc5011043943518292116et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6779_case__prodI2_H,axiom,
    ! [P2: product_prod_nat_nat,C: fun_na5914473681949881652_nat_o,X: 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),X) )
     => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(C),P2),X) ) ).

% case_prodI2'
tff(fact_6780_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_6781_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_6782_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_6783_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_6784_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_6785_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_6786_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_6787_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_6788_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_6789_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_6790_abs__zero,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_6791_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_6792_abs__0,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_6793_abs__0,axiom,
    aa_complex_complex(abs_abs_complex,zero_zero_complex) = zero_zero_complex ).

% abs_0
tff(fact_6794_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_6795_abs__0,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_6796_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_6797_abs__mult__self__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) ).

% abs_mult_self_eq
tff(fact_6798_abs__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) ).

% abs_mult_self_eq
tff(fact_6799_abs__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(times_times_rat(A),A) ).

% abs_mult_self_eq
tff(fact_6800_abs__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,A)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).

% abs_mult_self_eq
tff(fact_6801_abs__1,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_6802_abs__1,axiom,
    aa_complex_complex(abs_abs_complex,one_one_complex) = one_one_complex ).

% abs_1
tff(fact_6803_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_6804_abs__1,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_1
tff(fact_6805_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_6806_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_6807_abs__add__abs,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_add_abs
tff(fact_6808_abs__add__abs,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_add_abs
tff(fact_6809_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_6810_abs__divide,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_divide
tff(fact_6811_abs__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_divide
tff(fact_6812_abs__minus,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_minus
tff(fact_6813_abs__minus,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_minus
tff(fact_6814_abs__minus,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_minus
tff(fact_6815_abs__minus,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).

% abs_minus
tff(fact_6816_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_6817_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_6818_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_6819_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_6820_dvd__abs__iff,axiom,
    ! [Mb: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),aa_int_int(abs_abs_int,K))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),K) ) ).

% dvd_abs_iff
tff(fact_6821_dvd__abs__iff,axiom,
    ! [Mb: real,K: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Mb),aa_real_real(abs_abs_real,K))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Mb),K) ) ).

% dvd_abs_iff
tff(fact_6822_dvd__abs__iff,axiom,
    ! [Mb: rat,K: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Mb),aa_rat_rat(abs_abs_rat,K))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Mb),K) ) ).

% dvd_abs_iff
tff(fact_6823_dvd__abs__iff,axiom,
    ! [Mb: code_integer,K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Mb),abs_abs_Code_integer(K))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Mb),K) ) ).

% dvd_abs_iff
tff(fact_6824_abs__dvd__iff,axiom,
    ! [Mb: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(abs_abs_int,Mb)),K)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),K) ) ).

% abs_dvd_iff
tff(fact_6825_abs__dvd__iff,axiom,
    ! [Mb: real,K: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(abs_abs_real,Mb)),K)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Mb),K) ) ).

% abs_dvd_iff
tff(fact_6826_abs__dvd__iff,axiom,
    ! [Mb: rat,K: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(abs_abs_rat,Mb)),K)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Mb),K) ) ).

% abs_dvd_iff
tff(fact_6827_abs__dvd__iff,axiom,
    ! [Mb: code_integer,K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,abs_abs_Code_integer(Mb)),K)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Mb),K) ) ).

% abs_dvd_iff
tff(fact_6828_of__int__abs,axiom,
    ! [X: int] : aa_int_int(ring_1_of_int_int,aa_int_int(abs_abs_int,X)) = aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,X)) ).

% of_int_abs
tff(fact_6829_of__int__abs,axiom,
    ! [X: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(abs_abs_int,X)) = abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,X)) ).

% of_int_abs
tff(fact_6830_of__int__abs,axiom,
    ! [X: int] : aa_int_real(ring_1_of_int_real,aa_int_int(abs_abs_int,X)) = aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,X)) ).

% of_int_abs
tff(fact_6831_of__int__abs,axiom,
    ! [X: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(abs_abs_int,X)) = aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,X)) ).

% of_int_abs
tff(fact_6832_abs__bool__eq,axiom,
    ! [P: $o] : aa_real_real(abs_abs_real,zero_n3304061248610475627l_real((P))) = zero_n3304061248610475627l_real((P)) ).

% abs_bool_eq
tff(fact_6833_abs__bool__eq,axiom,
    ! [P: $o] : aa_rat_rat(abs_abs_rat,zero_n2052037380579107095ol_rat((P))) = zero_n2052037380579107095ol_rat((P)) ).

% abs_bool_eq
tff(fact_6834_abs__bool__eq,axiom,
    ! [P: $o] : abs_abs_Code_integer(zero_n356916108424825756nteger((P))) = zero_n356916108424825756nteger((P)) ).

% abs_bool_eq
tff(fact_6835_abs__bool__eq,axiom,
    ! [P: $o] : aa_int_int(abs_abs_int,aa_o_int(zero_n2684676970156552555ol_int,(P))) = aa_o_int(zero_n2684676970156552555ol_int,(P)) ).

% abs_bool_eq
tff(fact_6836_zdvd1__eq,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),one_one_int)
    <=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_6837_tanh__0,axiom,
    aa_complex_complex(tanh_complex,zero_zero_complex) = zero_zero_complex ).

% tanh_0
tff(fact_6838_tanh__0,axiom,
    aa_real_real(tanh_real,zero_zero_real) = zero_zero_real ).

% tanh_0
tff(fact_6839_tanh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ).

% tanh_real_less_iff
tff(fact_6840_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_dw),A2) = zero_zero_nat ).

% sum.neutral_const
tff(fact_6841_sum_Oneutral__const,axiom,
    ! [A2: set_complex] : groups7754918857620584856omplex(aTP_Lamm_dt,A2) = zero_zero_complex ).

% sum.neutral_const
tff(fact_6842_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : groups6591440286371151544t_real(aTP_Lamm_jt,A2) = zero_zero_real ).

% sum.neutral_const
tff(fact_6843_of__int__sum,axiom,
    ! [F: fun_complex_int,A2: set_complex] : aa_int_complex(ring_17405671764205052669omplex,groups5690904116761175830ex_int(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_ju(F),A2) ).

% of_int_sum
tff(fact_6844_of__int__sum,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_real(ring_1_of_int_real,groups3539618377306564664at_int(F,A2)) = groups6591440286371151544t_real(aTP_Lamm_jv(F),A2) ).

% of_int_sum
tff(fact_6845_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_6846_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_6847_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_6848_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_6849_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_6850_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_6851_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_6852_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_6853_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_6854_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_6855_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_6856_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_6857_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_6858_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_6859_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_6860_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_6861_sum_Oempty,axiom,
    ! [G: fun_real_complex] : groups5754745047067104278omplex(G,bot_bot_set_real) = zero_zero_complex ).

% sum.empty
tff(fact_6862_sum_Oempty,axiom,
    ! [G: fun_real_real] : groups8097168146408367636l_real(G,bot_bot_set_real) = zero_zero_real ).

% sum.empty
tff(fact_6863_sum_Oempty,axiom,
    ! [G: fun_real_rat] : groups1300246762558778688al_rat(G,bot_bot_set_real) = zero_zero_rat ).

% sum.empty
tff(fact_6864_sum_Oempty,axiom,
    ! [G: fun_real_nat] : groups1935376822645274424al_nat(G,bot_bot_set_real) = zero_zero_nat ).

% sum.empty
tff(fact_6865_sum_Oempty,axiom,
    ! [G: fun_real_int] : groups1932886352136224148al_int(G,bot_bot_set_real) = zero_zero_int ).

% sum.empty
tff(fact_6866_sum_Oempty,axiom,
    ! [G: fun_nat_complex] : groups2073611262835488442omplex(G,bot_bot_set_nat) = zero_zero_complex ).

% sum.empty
tff(fact_6867_sum_Oempty,axiom,
    ! [G: fun_nat_rat] : groups2906978787729119204at_rat(G,bot_bot_set_nat) = zero_zero_rat ).

% sum.empty
tff(fact_6868_sum_Oempty,axiom,
    ! [G: fun_nat_int] : groups3539618377306564664at_int(G,bot_bot_set_nat) = zero_zero_int ).

% sum.empty
tff(fact_6869_sum_Oempty,axiom,
    ! [G: fun_int_complex] : groups3049146728041665814omplex(G,bot_bot_set_int) = zero_zero_complex ).

% sum.empty
tff(fact_6870_sum_Oempty,axiom,
    ! [G: fun_int_real] : groups8778361861064173332t_real(G,bot_bot_set_int) = zero_zero_real ).

% sum.empty
tff(fact_6871_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_complex] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( groups2073611262835488442omplex(G,A2) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_6872_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups3049146728041665814omplex(G,A2) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_6873_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups8778361861064173332t_real(G,A2) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_6874_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(G,A2) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_6875_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_rat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( groups2906978787729119204at_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_6876_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_6877_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_6878_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(G,A2) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_6879_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_nat] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5693394587270226106ex_nat(G,A2) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_6880_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_int] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( groups3539618377306564664at_int(G,A2) = zero_zero_int ) ) ).

% sum.infinite
tff(fact_6881_sum__eq__0__iff,axiom,
    ! [F3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( ( groups4541462559716669496nt_nat(F,F3) = zero_zero_nat )
      <=> ! [X4: int] :
            ( aa_set_int_o(member_int(X4),F3)
           => ( aa_int_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_6882_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F3)
     => ( ( groups5693394587270226106ex_nat(F,F3) = zero_zero_nat )
      <=> ! [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),F3)
           => ( aa_complex_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_6883_sum__eq__0__iff,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
     => ( ( groups977919841031483927at_nat(F,F3) = zero_zero_nat )
      <=> ! [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),F3)
           => ( aa_Pro4243173770044397973at_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_6884_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),F3) = zero_zero_nat )
      <=> ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),F3)
           => ( aa_nat_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_6885_artanh__minus__real,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,X)) ) ) ).

% artanh_minus_real
tff(fact_6886_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Z)),one_one_int)
    <=> ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_6887_tanh__real__pos__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tanh_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X) ) ).

% tanh_real_pos_iff
tff(fact_6888_tanh__real__neg__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,X)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real) ) ).

% tanh_real_neg_iff
tff(fact_6889_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_jw(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta
tff(fact_6890_sum_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_jx(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta
tff(fact_6891_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_jy(A),B),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_int_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta
tff(fact_6892_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_jz(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_6893_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ka(A),B),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_int_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_6894_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_kb(A),B),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_complex_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_6895_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_kc(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_6896_sum_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_kd(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_6897_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ke(A),B),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_int_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_6898_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_kf(A),B),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_complex_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_6899_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_kg(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta'
tff(fact_6900_sum_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_kh(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta'
tff(fact_6901_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_ki(A),B),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_int_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta'
tff(fact_6902_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_kj(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_6903_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_kk(A),B),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_int_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_6904_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_kl(A),B),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_complex_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_6905_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_km(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_6906_sum_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_kn(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_6907_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ko(A),B),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_int_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_6908_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_kp(A),B),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_complex_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_6909_sum__abs,axiom,
    ! [F: fun_nat_real,A2: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,groups6591440286371151544t_real(F,A2))),groups6591440286371151544t_real(aTP_Lamm_kq(F),A2)) ).

% sum_abs
tff(fact_6910_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(aa_rea7340605660300375871l_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_6911_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(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_6912_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(aa_rea7340605660300375871l_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_6913_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(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_6914_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_6915_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_6916_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_6917_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_6918_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ~ aa_set_real_o(member_real(X),A2)
       => ( groups8097168146408367636l_real(G,insert_real(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X)),groups8097168146408367636l_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_6919_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int(X),A2)
       => ( groups8778361861064173332t_real(G,insert_int(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X)),groups8778361861064173332t_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_6920_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ~ aa_set_complex_o(member_complex(X),A2)
       => ( groups5808333547571424918x_real(G,insert_complex(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X)),groups5808333547571424918x_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_6921_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ~ aa_set_real_o(member_real(X),A2)
       => ( groups1300246762558778688al_rat(G,insert_real(X,A2)) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,X)),groups1300246762558778688al_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_6922_sum_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(X),A2)
       => ( groups2906978787729119204at_rat(G,insert_nat(X,A2)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,X)),groups2906978787729119204at_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_6923_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int(X),A2)
       => ( groups3906332499630173760nt_rat(G,insert_int(X,A2)) = aa_rat_rat(plus_plus_rat(aa_int_rat(G,X)),groups3906332499630173760nt_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_6924_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ~ aa_set_complex_o(member_complex(X),A2)
       => ( groups5058264527183730370ex_rat(G,insert_complex(X,A2)) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,X)),groups5058264527183730370ex_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_6925_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ~ aa_set_real_o(member_real(X),A2)
       => ( groups1935376822645274424al_nat(G,insert_real(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X)),groups1935376822645274424al_nat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_6926_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int(X),A2)
       => ( groups4541462559716669496nt_nat(G,insert_int(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(G,X)),groups4541462559716669496nt_nat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_6927_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ~ aa_set_complex_o(member_complex(X),A2)
       => ( groups5693394587270226106ex_nat(G,insert_complex(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X)),groups5693394587270226106ex_nat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_6928_sum__abs__ge__zero,axiom,
    ! [F: fun_nat_real,A2: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups6591440286371151544t_real(aTP_Lamm_kq(F),A2)) ).

% sum_abs_ge_zero
tff(fact_6929_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Na))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( Na = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6930_zero__less__power__abs__iff,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Na))
    <=> ( ( A != zero_zero_real )
        | ( Na = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6931_zero__less__power__abs__iff,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Na))
    <=> ( ( A != zero_zero_rat )
        | ( Na = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6932_zero__less__power__abs__iff,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Na))
    <=> ( ( A != zero_zero_int )
        | ( Na = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6933_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_complex,Mb: nat,Na: nat] :
      groups2073611262835488442omplex(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),Mb),zero_zero_complex,aa_complex_complex(plus_plus_complex(groups2073611262835488442omplex(G,set_or1269000886237332187st_nat(Mb,Na))),aa_nat_complex(G,aa_nat_nat(suc,Na)))) ).

% sum.cl_ivl_Suc
tff(fact_6934_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,Mb: nat,Na: nat] :
      groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),Mb),zero_zero_rat,aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,Na))),aa_nat_rat(G,aa_nat_nat(suc,Na)))) ).

% sum.cl_ivl_Suc
tff(fact_6935_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,Mb: nat,Na: nat] :
      groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),Mb),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,Na))),aa_nat_int(G,aa_nat_nat(suc,Na)))) ).

% sum.cl_ivl_Suc
tff(fact_6936_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,Mb: nat,Na: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),Mb),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,Na))),aa_nat_nat(G,aa_nat_nat(suc,Na)))) ).

% sum.cl_ivl_Suc
tff(fact_6937_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,Mb: nat,Na: nat] :
      groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),Mb),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(Mb,Na))),aa_nat_real(G,aa_nat_nat(suc,Na)))) ).

% sum.cl_ivl_Suc
tff(fact_6938_sum__zero__power,axiom,
    ! [C: fun_nat_complex,A2: set_nat] :
      groups2073611262835488442omplex(aTP_Lamm_kr(C),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        aa_nat_complex(C,zero_zero_nat),
        zero_zero_complex ) ).

% sum_zero_power
tff(fact_6939_sum__zero__power,axiom,
    ! [C: fun_nat_rat,A2: set_nat] :
      groups2906978787729119204at_rat(aTP_Lamm_ks(C),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        aa_nat_rat(C,zero_zero_nat),
        zero_zero_rat ) ).

% sum_zero_power
tff(fact_6940_sum__zero__power,axiom,
    ! [C: fun_nat_real,A2: set_nat] :
      groups6591440286371151544t_real(aTP_Lamm_kt(C),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        aa_nat_real(C,zero_zero_nat),
        zero_zero_real ) ).

% sum_zero_power
tff(fact_6941_sum__zero__power_H,axiom,
    ! [C: fun_nat_complex,D: fun_nat_complex,A2: set_nat] :
      groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ku(C),D),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        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_6942_sum__zero__power_H,axiom,
    ! [C: fun_nat_rat,D: fun_nat_rat,A2: set_nat] :
      groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_kv(C),D),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        aa_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_6943_sum__zero__power_H,axiom,
    ! [C: fun_nat_real,D: fun_nat_real,A2: set_nat] :
      groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_kw(C),D),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        aa_real_real(aa_rea7340605660300375871l_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_6944_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_complex,A2: set_real] :
      ( ( groups5754745047067104278omplex(G,A2) != zero_zero_complex )
     => ~ ! [A3: real] :
            ( aa_set_real_o(member_real(A3),A2)
           => ( aa_real_complex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6945_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_complex,A2: set_nat] :
      ( ( groups2073611262835488442omplex(G,A2) != zero_zero_complex )
     => ~ ! [A3: nat] :
            ( aa_set_nat_o(member_nat(A3),A2)
           => ( aa_nat_complex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6946_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_complex,A2: set_int] :
      ( ( groups3049146728041665814omplex(G,A2) != zero_zero_complex )
     => ~ ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A2)
           => ( aa_int_complex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6947_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A2: set_real] :
      ( ( groups8097168146408367636l_real(G,A2) != zero_zero_real )
     => ~ ! [A3: real] :
            ( aa_set_real_o(member_real(A3),A2)
           => ( aa_real_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6948_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A2: set_int] :
      ( ( groups8778361861064173332t_real(G,A2) != zero_zero_real )
     => ~ ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A2)
           => ( aa_int_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6949_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A2: set_real] :
      ( ( groups1300246762558778688al_rat(G,A2) != zero_zero_rat )
     => ~ ! [A3: real] :
            ( aa_set_real_o(member_real(A3),A2)
           => ( aa_real_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6950_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A2: set_nat] :
      ( ( groups2906978787729119204at_rat(G,A2) != zero_zero_rat )
     => ~ ! [A3: nat] :
            ( aa_set_nat_o(member_nat(A3),A2)
           => ( aa_nat_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6951_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_rat,A2: set_int] :
      ( ( groups3906332499630173760nt_rat(G,A2) != zero_zero_rat )
     => ~ ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A2)
           => ( aa_int_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6952_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_nat,A2: set_real] :
      ( ( groups1935376822645274424al_nat(G,A2) != zero_zero_nat )
     => ~ ! [A3: real] :
            ( aa_set_real_o(member_real(A3),A2)
           => ( aa_real_nat(G,A3) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6953_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_nat,A2: set_int] :
      ( ( groups4541462559716669496nt_nat(G,A2) != zero_zero_nat )
     => ~ ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A2)
           => ( aa_int_nat(G,A3) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6954_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A2)
         => ( aa_nat_nat(G,X3) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_6955_sum_Oneutral,axiom,
    ! [A2: set_complex,G: fun_complex_complex] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A2)
         => ( aa_complex_complex(G,X3) = zero_zero_complex ) )
     => ( groups7754918857620584856omplex(G,A2) = zero_zero_complex ) ) ).

% sum.neutral
tff(fact_6956_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_real] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A2)
         => ( aa_nat_real(G,X3) = zero_zero_real ) )
     => ( groups6591440286371151544t_real(G,A2) = zero_zero_real ) ) ).

% sum.neutral
tff(fact_6957_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_6958_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_6959_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_6960_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_6961_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_6962_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_6963_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_6964_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_6965_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_6966_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(abs_abs_complex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
tff(fact_6967_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(abs_abs_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
tff(fact_6968_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
tff(fact_6969_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
tff(fact_6970_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ).

% abs_mult
tff(fact_6971_abs__mult,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(abs_abs_complex,B)) ).

% abs_mult
tff(fact_6972_abs__mult,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_mult
tff(fact_6973_abs__mult,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_mult
tff(fact_6974_abs__mult,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_mult
tff(fact_6975_abs__one,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_6976_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_6977_abs__one,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_one
tff(fact_6978_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_6979_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_6980_abs__minus__commute,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ).

% abs_minus_commute
tff(fact_6981_abs__minus__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A)) ).

% abs_minus_commute
tff(fact_6982_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_6983_abs__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(abs_abs_int,X) = aa_int_int(abs_abs_int,Y) )
    <=> ( ( X = Y )
        | ( X = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% abs_eq_iff
tff(fact_6984_abs__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(abs_abs_real,X) = aa_real_real(abs_abs_real,Y) )
    <=> ( ( X = Y )
        | ( X = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% abs_eq_iff
tff(fact_6985_abs__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(abs_abs_rat,X) = aa_rat_rat(abs_abs_rat,Y) )
    <=> ( ( X = Y )
        | ( X = aa_rat_rat(uminus_uminus_rat,Y) ) ) ) ).

% abs_eq_iff
tff(fact_6986_abs__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( abs_abs_Code_integer(X) = abs_abs_Code_integer(Y) )
    <=> ( ( X = Y )
        | ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% abs_eq_iff
tff(fact_6987_dvd__if__abs__eq,axiom,
    ! [L: int,K: int] :
      ( ( aa_int_int(abs_abs_int,L) = aa_int_int(abs_abs_int,K) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K) ) ).

% dvd_if_abs_eq
tff(fact_6988_dvd__if__abs__eq,axiom,
    ! [L: real,K: real] :
      ( ( aa_real_real(abs_abs_real,L) = aa_real_real(abs_abs_real,K) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),K) ) ).

% dvd_if_abs_eq
tff(fact_6989_dvd__if__abs__eq,axiom,
    ! [L: rat,K: rat] :
      ( ( aa_rat_rat(abs_abs_rat,L) = aa_rat_rat(abs_abs_rat,K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),K) ) ).

% dvd_if_abs_eq
tff(fact_6990_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K: code_integer] :
      ( ( abs_abs_Code_integer(L) = abs_abs_Code_integer(K) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),K) ) ).

% dvd_if_abs_eq
tff(fact_6991_mem__case__prodE,axiom,
    ! [Z: real,C: fun_in6553099857819052212t_real,P2: product_prod_int_int] :
      ( aa_set_real_o(member_real(Z),produc6452406959799940328t_real(C,P2))
     => ~ ! [X3: int,Y3: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_set_real_o(member_real(Z),aa_int_set_real(aa_int4847878404602763637t_real(C,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6992_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_in2630332840903238488et_nat,P2: product_prod_int_int] :
      ( aa_set_nat_o(member_nat(Z),produc4251311855443802252et_nat(C,P2))
     => ~ ! [X3: int,Y3: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_set_nat_o(member_nat(Z),aa_int_set_nat(aa_int1906528167449992217et_nat(C,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6993_mem__case__prodE,axiom,
    ! [Z: int,C: fun_in7040156458970095540et_int,P2: product_prod_int_int] :
      ( aa_set_int_o(member_int(Z),produc73460835934605544et_int(C,P2))
     => ~ ! [X3: int,Y3: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_set_int_o(member_int(Z),aa_int_set_int(aa_int8639817059373041269et_int(C,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6994_mem__case__prodE,axiom,
    ! [Z: set_nat,C: fun_in7207726656141197838et_nat,P2: product_prod_int_int] :
      ( aa_set_set_nat_o(member_set_nat(Z),produc5233655623923918146et_nat(C,P2))
     => ~ ! [X3: int,Y3: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_set_set_nat_o(member_set_nat(Z),aa_int_set_set_nat(aa_int1843896668418082255et_nat(C,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6995_mem__case__prodE,axiom,
    ! [Z: product_prod_nat_nat,C: fun_in1904380326158164027at_nat,P2: product_prod_int_int] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z),produc1656060378719767003at_nat(C,P2))
     => ~ ! [X3: int,Y3: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z),aa_int8822742982810151799at_nat(aa_int6362027936895332260at_nat(C,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6996_mem__case__prodE,axiom,
    ! [Z: real,C: fun_fu7234941744988193529t_real,P2: produc7509114972854644512nteger] :
      ( aa_set_real_o(member_real(Z),produc5067746511659309921t_real(C,P2))
     => ~ ! [X3: fun_Co3460408610320954864e_term,Y3: produc8923325533196201883nteger] :
            ( ( P2 = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X3),Y3) )
           => ~ aa_set_real_o(member_real(Z),aa_Pro6515153133580421057t_real(aa_fun1966338168372333922t_real(C,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6997_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_fu6383124537394581533et_nat,P2: produc7509114972854644512nteger] :
      ( aa_set_nat_o(member_nat(Z),produc7961109426542857093et_nat(C,P2))
     => ~ ! [X3: fun_Co3460408610320954864e_term,Y3: produc8923325533196201883nteger] :
            ( ( P2 = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X3),Y3) )
           => ~ aa_set_nat_o(member_nat(Z),aa_Pro381016180347496421et_nat(aa_fun3554285010146031238et_nat(C,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6998_mem__case__prodE,axiom,
    ! [Z: int,C: fun_fu1569576118606662777et_int,P2: produc7509114972854644512nteger] :
      ( aa_set_int_o(member_int(Z),produc3783258407033660385et_int(C,P2))
     => ~ ! [X3: fun_Co3460408610320954864e_term,Y3: produc8923325533196201883nteger] :
            ( ( P2 = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X3),Y3) )
           => ~ aa_set_int_o(member_int(Z),aa_Pro5426537197693075521et_int(aa_fun1064201865214304482et_int(C,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6999_mem__case__prodE,axiom,
    ! [Z: real,C: fun_fu604837885242254216t_real,P2: produc4788131091073775919nt_int] :
      ( aa_set_real_o(member_real(Z),produc5863038274793111792t_real(C,P2))
     => ~ ! [X3: fun_in5730379987311621695e_term,Y3: product_prod_int_int] :
            ( ( P2 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X3),Y3) )
           => ~ aa_set_real_o(member_real(Z),aa_Pro7989103684195733727t_real(aa_fun5197189019902991601t_real(C,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_7000_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_fu9079770784915425324et_nat,P2: produc4788131091073775919nt_int] :
      ( aa_set_nat_o(member_nat(Z),produc5011043943518292116et_nat(C,P2))
     => ~ ! [X3: fun_in5730379987311621695e_term,Y3: product_prod_int_int] :
            ( ( P2 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X3),Y3) )
           => ~ aa_set_nat_o(member_nat(Z),aa_Pro3599192995643711491et_nat(aa_fun8409204002232819093et_nat(C,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_7001_zdvd__antisym__abs,axiom,
    ! [A: int,B: 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,B),A)
       => ( aa_int_int(abs_abs_int,A) = aa_int_int(abs_abs_int,B) ) ) ) ).

% zdvd_antisym_abs
tff(fact_7002_case__prodD,axiom,
    ! [F: fun_fu3422731660242565741eger_o,A: fun_Co3460408610320954864e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4652312591602923194eger_o(produc7733317944277263833eger_o(F),aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,A),B))
     => aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(F,A),B) ) ).

% case_prodD
tff(fact_7003_case__prodD,axiom,
    ! [F: fun_fu7695273699059480439eger_o,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger] :
      ( aa_Pro4148064003102580676eger_o(produc7373287791920631523eger_o(F),aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,A),B))
     => aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(F,A),B) ) ).

% case_prodD
tff(fact_7004_case__prodD,axiom,
    ! [F: fun_fu6865642474282033866_int_o,A: fun_Pr7763680713840462443e_term,B: product_prod_int_int] :
      ( aa_Pro2601676399929874711_int_o(produc2207054017359051062_int_o(F),aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,A),B))
     => aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(F,A),B) ) ).

% case_prodD
tff(fact_7005_case__prodD,axiom,
    ! [F: fun_fu2978539155105838238_int_o,A: fun_in5730379987311621695e_term,B: product_prod_int_int] :
      ( aa_Pro5451197003417746155_int_o(produc7271678899836001034_int_o(F),aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,A),B))
     => aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(F,A),B) ) ).

% case_prodD
tff(fact_7006_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_7007_case__prodE,axiom,
    ! [C: fun_fu3422731660242565741eger_o,P2: produc7509114972854644512nteger] :
      ( aa_Pro4652312591602923194eger_o(produc7733317944277263833eger_o(C),P2)
     => ~ ! [X3: fun_Co3460408610320954864e_term,Y3: produc8923325533196201883nteger] :
            ( ( P2 = aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,X3),Y3) )
           => ~ aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(C,X3),Y3) ) ) ).

% case_prodE
tff(fact_7008_case__prodE,axiom,
    ! [C: fun_fu7695273699059480439eger_o,P2: produc2453742332425399446nteger] :
      ( aa_Pro4148064003102580676eger_o(produc7373287791920631523eger_o(C),P2)
     => ~ ! [X3: fun_Pr5098669352780829690e_term,Y3: produc8923325533196201883nteger] :
            ( ( P2 = aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,X3),Y3) )
           => ~ aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(C,X3),Y3) ) ) ).

% case_prodE
tff(fact_7009_case__prodE,axiom,
    ! [C: fun_fu6865642474282033866_int_o,P2: produc3043004577601930755nt_int] :
      ( aa_Pro2601676399929874711_int_o(produc2207054017359051062_int_o(C),P2)
     => ~ ! [X3: fun_Pr7763680713840462443e_term,Y3: product_prod_int_int] :
            ( ( P2 = aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,X3),Y3) )
           => ~ aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(C,X3),Y3) ) ) ).

% case_prodE
tff(fact_7010_case__prodE,axiom,
    ! [C: fun_fu2978539155105838238_int_o,P2: produc4788131091073775919nt_int] :
      ( aa_Pro5451197003417746155_int_o(produc7271678899836001034_int_o(C),P2)
     => ~ ! [X3: fun_in5730379987311621695e_term,Y3: product_prod_int_int] :
            ( ( P2 = aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,X3),Y3) )
           => ~ aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(C,X3),Y3) ) ) ).

% case_prodE
tff(fact_7011_case__prodE,axiom,
    ! [C: fun_int_fun_int_o,P2: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P2)
     => ~ ! [X3: int,Y3: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_int_o(aa_int_fun_int_o(C,X3),Y3) ) ) ).

% case_prodE
tff(fact_7012_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_7013_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_7014_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_7015_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_7016_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_7017_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I3: real] :
          ( aa_set_real_o(member_real(I3),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I3)),aa_real_int(G,I3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,K5)),groups1932886352136224148al_int(G,K5)) ) ).

% sum_mono
tff(fact_7018_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I3)),aa_nat_int(G,I3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,K5)),groups3539618377306564664at_int(G,K5)) ) ).

% sum_mono
tff(fact_7019_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_int,G: fun_int_int] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,I3)),aa_int_int(G,I3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4538972089207619220nt_int(F,K5)),groups4538972089207619220nt_int(G,K5)) ) ).

% sum_mono
tff(fact_7020_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I3)),aa_nat_nat(G,I3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),K5)),aa_set_nat_nat(groups3542108847815614940at_nat(G),K5)) ) ).

% sum_mono
tff(fact_7021_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),K5)
         => 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,groups6591440286371151544t_real(F,K5)),groups6591440286371151544t_real(G,K5)) ) ).

% sum_mono
tff(fact_7022_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kx(G),H)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(H),A2)) ).

% sum.distrib
tff(fact_7023_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_ky(G),H),A2) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,A2)),groups7754918857620584856omplex(H,A2)) ).

% sum.distrib
tff(fact_7024_sum_Odistrib,axiom,
    ! [G: fun_nat_real,H: fun_nat_real,A2: set_nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_kz(G),H),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,A2)),groups6591440286371151544t_real(H,A2)) ).

% sum.distrib
tff(fact_7025_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_nat,G: fun_real_fun_nat_nat,R: fun_real_fun_nat_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups1935376822645274424al_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_lb(B2),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_ld(A2),G),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7026_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_nat,G: fun_int_fun_nat_nat,R: fun_int_fun_nat_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups4541462559716669496nt_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_lf(B2),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_lh(A2),G),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7027_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B2: set_nat,G: fun_co636942507509095880at_nat,R: fun_co5676208753217392844_nat_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups5693394587270226106ex_nat(aa_fun2265263913897893899ex_nat(aa_fun1127489917733235668ex_nat(aTP_Lamm_lj(B2),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5527826185635525933at_nat(aa_fun4894236392271416310at_nat(aTP_Lamm_ll(A2),G),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7028_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_complex,G: fun_re7208826350674946562omplex,R: fun_re7319081385389557872plex_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( groups5754745047067104278omplex(aa_fun401690246322682763omplex(aa_fun7501666206714588442omplex(aTP_Lamm_ln(B2),G),R),A2) = groups7754918857620584856omplex(aa_fun9084337531551611917omplex(aa_fun6739043078600750748omplex(aTP_Lamm_lp(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7029_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_complex,G: fun_na1591161823130732966omplex,R: fun_na1554462459066888140plex_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( groups2073611262835488442omplex(aa_fun4713049822795992587omplex(aa_fun3479084698764006518omplex(aTP_Lamm_lr(B2),G),R),A2) = groups7754918857620584856omplex(aa_fun1423577888295648489omplex(aa_fun601428455815558228omplex(aTP_Lamm_lt(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7030_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_complex,G: fun_in1307110748760623106omplex,R: fun_in5065666251602127472plex_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( groups3049146728041665814omplex(aa_fun3141780957025892107omplex(aa_fun323909820831547930omplex(aTP_Lamm_lv(B2),G),R),A2) = groups7754918857620584856omplex(aa_fun7450476157521870733omplex(aa_fun6156949752960786844omplex(aTP_Lamm_lx(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7031_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_nat,G: fun_re4238438383274954274t_real,R: fun_real_fun_nat_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups8097168146408367636l_real(aa_fun6320755270648597735l_real(aa_fun6240130701872165846l_real(aTP_Lamm_ly(B2),G),R),A2) = groups6591440286371151544t_real(aa_fun6247686994378451851t_real(aa_fun7274880274332377722t_real(aTP_Lamm_ma(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7032_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_nat,G: fun_int_fun_nat_real,R: fun_int_fun_nat_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups8778361861064173332t_real(aa_fun7382188522922153831t_real(aa_fun6619364416873904086t_real(aTP_Lamm_mb(B2),G),R),A2) = groups6591440286371151544t_real(aa_fun5195266948229132043t_real(aa_fun3400660822709534074t_real(aTP_Lamm_md(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7033_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B2: set_nat,G: fun_co8378349014452122532t_real,R: fun_co5676208753217392844_nat_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups5808333547571424918x_real(aa_fun2831877435191659751x_real(aa_fun3061107605873050452x_real(aTP_Lamm_me(B2),G),R),A2) = groups6591440286371151544t_real(aa_fun8607523022212034825t_real(aa_fun982078519811717750t_real(aTP_Lamm_mg(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7034_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_real,G: fun_nat_fun_real_nat,R: fun_nat_fun_real_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_real_o(finite_finite_real,B2)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun6477086565944264623at_nat(aa_fun888394026836741882at_nat(aTP_Lamm_mi(B2),G),R)),A2) = groups1935376822645274424al_nat(aa_fun1455215566758156043al_nat(aa_fun1180509458258223190al_nat(aTP_Lamm_ml(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_7035_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_7036_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)
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3) )
           => ~ aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,X3),Y3),Z) ) ) ).

% case_prodE'
tff(fact_7037_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A2)) ) ).

% sum_nonneg
tff(fact_7038_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A2)) ) ).

% sum_nonneg
tff(fact_7039_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A2)) ) ).

% sum_nonneg
tff(fact_7040_sum__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A2)) ) ).

% sum_nonneg
tff(fact_7041_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A2)) ) ).

% sum_nonneg
tff(fact_7042_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A2)) ) ).

% sum_nonneg
tff(fact_7043_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A2)) ) ).

% sum_nonneg
tff(fact_7044_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_int] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups1932886352136224148al_int(F,A2)) ) ).

% sum_nonneg
tff(fact_7045_sum__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups3539618377306564664at_int(F,A2)) ) ).

% sum_nonneg
tff(fact_7046_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(F,A2)) ) ).

% sum_nonneg
tff(fact_7047_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X3)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,A2)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_7048_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X3)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,A2)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_7049_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X3)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A2)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7050_sum__nonpos,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,A2)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7051_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A2)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7052_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X3)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,A2)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_7053_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X3)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A2)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_7054_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_int] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,X3)),zero_zero_int) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,A2)),zero_zero_int) ) ).

% sum_nonpos
tff(fact_7055_sum__nonpos,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X3)),zero_zero_int) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,A2)),zero_zero_int) ) ).

% sum_nonpos
tff(fact_7056_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X3)),zero_zero_int) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4538972089207619220nt_int(F,A2)),zero_zero_int) ) ).

% sum_nonpos
tff(fact_7057_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_7058_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_7059_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_7060_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_7061_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_7062_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_7063_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_7064_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_7065_sum__mono__inv,axiom,
    ! [F: fun_nat_int,I5: set_nat,G: fun_nat_int,I: nat] :
      ( ( groups3539618377306564664at_int(F,I5) = groups3539618377306564664at_int(G,I5) )
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),I5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I3)),aa_nat_int(G,I3)) )
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( aa_set_nat_o(finite_finite_nat,I5)
           => ( aa_nat_int(F,I) = aa_nat_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7066_sum__mono__inv,axiom,
    ! [F: fun_int_int,I5: set_int,G: fun_int_int,I: int] :
      ( ( groups4538972089207619220nt_int(F,I5) = groups4538972089207619220nt_int(G,I5) )
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),I5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,I3)),aa_int_int(G,I3)) )
       => ( aa_set_int_o(member_int(I),I5)
         => ( aa_set_int_o(finite_finite_int,I5)
           => ( aa_int_int(F,I) = aa_int_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7067_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_7068_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_7069_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_7070_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_7071_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_7072_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_7073_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_7074_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_7075_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_7076_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_7077_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_7078_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_7079_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_7080_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(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq
tff(fact_7081_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(plus_plus_rat(A),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq
tff(fact_7082_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_7083_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(B)),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D)) ) ) ).

% abs_mult_less
tff(fact_7084_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,B)),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D)) ) ) ).

% abs_mult_less
tff(fact_7085_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,B)),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(times_times_rat(C),D)) ) ) ).

% abs_mult_less
tff(fact_7086_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,B)),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(aa_int_fun_int_int(times_times_int,C),D)) ) ) ).

% abs_mult_less
tff(fact_7087_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_7088_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ).

% abs_triangle_ineq2
tff(fact_7089_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))) ).

% abs_triangle_ineq2
tff(fact_7090_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_7091_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_7092_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ).

% abs_triangle_ineq3
tff(fact_7093_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(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))) ).

% abs_triangle_ineq3
tff(fact_7094_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_7095_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_7096_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_7097_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_7098_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_7099_nonzero__abs__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% nonzero_abs_divide
tff(fact_7100_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% nonzero_abs_divide
tff(fact_7101_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_7102_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_7103_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_7104_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_7105_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_7106_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_7107_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_7108_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_7109_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_7110_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_7111_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_7112_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_7113_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_7114_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_7115_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_7116_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_7117_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( 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(uminus_uminus_int,A)),B) ) ) ).

% abs_less_iff
tff(fact_7118_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),B)
    <=> ( 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(uminus_uminus_real,A)),B) ) ) ).

% abs_less_iff
tff(fact_7119_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),B)
    <=> ( 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(uminus_uminus_rat,A)),B) ) ) ).

% abs_less_iff
tff(fact_7120_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ) ).

% abs_less_iff
tff(fact_7121_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A2)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X3)),A2)
           => ( aa_nat_nat(F,aa_nat_nat(suc,X3)) = aa_nat_nat(G,aa_nat_nat(suc,X3)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) ) ) ) ).

% sum_cong_Suc
tff(fact_7122_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A2)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X3)),A2)
           => ( aa_nat_real(F,aa_nat_nat(suc,X3)) = aa_nat_real(G,aa_nat_nat(suc,X3)) ) )
       => ( groups6591440286371151544t_real(F,A2) = groups6591440286371151544t_real(G,A2) ) ) ) ).

% sum_cong_Suc
tff(fact_7123_Collect__case__prod__mono,axiom,
    ! [A2: fun_int_fun_int_o,B2: fun_int_fun_int_o] :
      ( ord_le77980682512674322_int_o(A2,B2)
     => ord_le2843351958646193337nt_int(collec213857154873943460nt_int(produc4947309494688390418_int_o(A2)),collec213857154873943460nt_int(produc4947309494688390418_int_o(B2))) ) ).

% Collect_case_prod_mono
tff(fact_7124_abs__zmult__eq__1,axiom,
    ! [Mb: int,Na: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Na)) = one_one_int )
     => ( aa_int_int(abs_abs_int,Mb) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_7125_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_complex,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups5754745047067104278omplex(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_dj(A2),P))) = groups5754745047067104278omplex(aa_fun1098856492951973088omplex(aTP_Lamm_mm(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7126_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_complex,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups2073611262835488442omplex(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_dm(A2),P))) = groups2073611262835488442omplex(aa_fun1400689748890728928omplex(aTP_Lamm_mn(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7127_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_complex,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3049146728041665814omplex(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_dn(A2),P))) = groups3049146728041665814omplex(aa_fun6439487526662521568omplex(aTP_Lamm_mo(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7128_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_real,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups8097168146408367636l_real(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_dj(A2),P))) = groups8097168146408367636l_real(aa_fun4448911596306054750l_real(aTP_Lamm_mp(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7129_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_real,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups8778361861064173332t_real(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_dn(A2),P))) = groups8778361861064173332t_real(aa_fun7506669883404309598t_real(aTP_Lamm_mq(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7130_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_real,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_mr(A2),P))) = groups5808333547571424918x_real(aa_fun8261057609414086622x_real(aTP_Lamm_ms(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7131_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_rat,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1300246762558778688al_rat(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_dj(A2),P))) = groups1300246762558778688al_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_mt(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7132_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_rat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups2906978787729119204at_rat(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_dm(A2),P))) = groups2906978787729119204at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_mu(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7133_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_rat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_dn(A2),P))) = groups3906332499630173760nt_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_mv(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7134_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_rat,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_mr(A2),P))) = groups5058264527183730370ex_rat(aa_fun4010016270681809034ex_rat(aTP_Lamm_mw(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7135_sum__subtractf__nat,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,G: fun_Pr2602117396471556196at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(G,X3)),aa_Pro4243173770044397973at_nat(F,X3)) )
     => ( groups977919841031483927at_nat(aa_fun2351638177499875132at_nat(aTP_Lamm_mx(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A2)),groups977919841031483927at_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_7136_sum__subtractf__nat,axiom,
    ! [A2: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(G,X3)),aa_real_nat(F,X3)) )
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_my(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_7137_sum__subtractf__nat,axiom,
    ! [A2: set_set_nat,G: fun_set_nat_nat,F: fun_set_nat_nat] :
      ( ! [X3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X3),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(G,X3)),aa_set_nat_nat(F,X3)) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_mz(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A2)),groups8294997508430121362at_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_7138_sum__subtractf__nat,axiom,
    ! [A2: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(G,X3)),aa_int_nat(F,X3)) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_na(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_7139_sum__subtractf__nat,axiom,
    ! [A2: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(G,X3)),aa_nat_nat(F,X3)) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nb(G),F)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)) ) ) ).

% sum_subtractf_nat
tff(fact_7140_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: 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(X,I3)) )
     => ( ( groups7713935264441627589nteger(X,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_nc(X),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7141_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: 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(X,I3)) )
     => ( ( groups7501900531339628137nteger(X,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(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_nd(X),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7142_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: 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(X,I3)) )
     => ( ( groups7873554091576472773nteger(X,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_ne(X),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7143_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: 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(X,I3)) )
     => ( ( groups8097168146408367636l_real(X,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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(minus_minus_real,groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_nf(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7144_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: 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(X,I3)) )
     => ( ( groups8778361861064173332t_real(X,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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ng(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7145_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_real_rat,A: fun_real_rat,B: rat,Delta: rat] :
      ( ! [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(X,I3)) )
     => ( ( groups1300246762558778688al_rat(X,I5) = one_one_rat )
       => ( ! [I3: real] :
              ( aa_set_real_o(member_real(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_real_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(minus_minus_rat(groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_nh(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7146_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_rat,A: fun_nat_rat,B: rat,Delta: rat] :
      ( ! [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(X,I3)) )
     => ( ( groups2906978787729119204at_rat(X,I5) = one_one_rat )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_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(minus_minus_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ni(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7147_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_rat,A: fun_int_rat,B: rat,Delta: rat] :
      ( ! [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(X,I3)) )
     => ( ( groups3906332499630173760nt_rat(X,I5) = one_one_rat )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(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(minus_minus_rat(groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_nj(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7148_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_real_int,A: fun_real_int,B: int,Delta: int] :
      ( ! [I3: real] :
          ( aa_set_real_o(member_real(I3),I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(X,I3)) )
     => ( ( groups1932886352136224148al_int(X,I5) = one_one_int )
       => ( ! [I3: real] :
              ( aa_set_real_o(member_real(I3),I5)
             => 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_real_int(A,I3)),B))),Delta) )
         => 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,groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aTP_Lamm_nk(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7149_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_int,A: fun_nat_int,B: int,Delta: int] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(X,I3)) )
     => ( ( groups3539618377306564664at_int(X,I5) = one_one_int )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I5)
             => 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(A,I3)),B))),Delta) )
         => 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,groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_nl(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7150_infinite__int__iff__unbounded__le,axiom,
    ! [S3: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S3)
    <=> ! [M5: int] :
        ? [N3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M5),aa_int_int(abs_abs_int,N3))
          & aa_set_int_o(member_int(N3),S3) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_7151_infinite__int__iff__unbounded,axiom,
    ! [S3: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S3)
    <=> ! [M5: int] :
        ? [N3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,M5),aa_int_int(abs_abs_int,N3))
          & aa_set_int_o(member_int(N3),S3) ) ) ).

% infinite_int_iff_unbounded
tff(fact_7152_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Na))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_nm(G)),set_or1269000886237332187st_nat(Mb,Na)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_7153_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_real,Mb: nat,Na: nat] : groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Na))) = groups6591440286371151544t_real(aTP_Lamm_nn(G),set_or1269000886237332187st_nat(Mb,Na)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_7154_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,K: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_no(G),K)),set_or1269000886237332187st_nat(Mb,Na)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_7155_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,Mb: nat,K: nat,Na: nat] : groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K))) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_np(G),K),set_or1269000886237332187st_nat(Mb,Na)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_7156_tanh__real__lt__1,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,X)),one_one_real) ).

% tanh_real_lt_1
tff(fact_7157_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: fun_int_real,I: fun_int_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(finite_finite_int,T)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),T)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(G,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int(Xa),T)
                    & ( aa_int_int(I,Xa) = X3 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X3)),aa_int_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,S)),groups8778361861064173332t_real(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_7158_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: fun_complex_real,I: fun_complex_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),T)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),T)
                    & ( aa_complex_int(I,Xa) = X3 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X3)),aa_complex_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,S)),groups5808333547571424918x_real(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_7159_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G: fun_int_real,I: fun_int_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( aa_set_int_o(finite_finite_int,T)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),T)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(G,X3)) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int(Xa),T)
                    & ( aa_int_complex(I,Xa) = X3 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X3)),aa_int_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,S)),groups8778361861064173332t_real(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_7160_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: fun_complex_real,I: fun_complex_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),T)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X3)) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),T)
                    & ( aa_complex_complex(I,Xa) = X3 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X3)),aa_complex_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,S)),groups5808333547571424918x_real(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_7161_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: fun_nat_rat,I: fun_nat_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),T)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X3)) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat(Xa),T)
                    & ( aa_nat_nat(I,Xa) = X3 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,S)),groups2906978787729119204at_rat(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_7162_sum__le__included,axiom,
    ! [S: set_nat,T: set_int,G: fun_int_rat,I: fun_int_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_int_o(finite_finite_int,T)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),T)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X3)) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int(Xa),T)
                    & ( aa_int_nat(I,Xa) = X3 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_int_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,S)),groups3906332499630173760nt_rat(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_7163_sum__le__included,axiom,
    ! [S: set_nat,T: set_complex,G: fun_complex_rat,I: fun_complex_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),T)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X3)) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),T)
                    & ( aa_complex_nat(I,Xa) = X3 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_complex_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,S)),groups5058264527183730370ex_rat(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_7164_sum__le__included,axiom,
    ! [S: set_int,T: set_nat,G: fun_nat_rat,I: fun_nat_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),T)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat(Xa),T)
                    & ( aa_nat_int(I,Xa) = X3 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_nat_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S)),groups2906978787729119204at_rat(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_7165_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: fun_int_rat,I: fun_int_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(finite_finite_int,T)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),T)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int(Xa),T)
                    & ( aa_int_int(I,Xa) = X3 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_int_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S)),groups3906332499630173760nt_rat(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_7166_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: fun_complex_rat,I: fun_complex_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),T)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),T)
                    & ( aa_complex_int(I,Xa) = X3 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_complex_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S)),groups5058264527183730370ex_rat(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_7167_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3)) )
       => ( ( groups8097168146408367636l_real(F,A2) = zero_zero_real )
        <=> ! [X4: real] :
              ( aa_set_real_o(member_real(X4),A2)
             => ( aa_real_real(F,X4) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7168_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X3)) )
       => ( ( groups8778361861064173332t_real(F,A2) = zero_zero_real )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A2)
             => ( aa_int_real(F,X4) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7169_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X3)) )
       => ( ( groups5808333547571424918x_real(F,A2) = zero_zero_real )
        <=> ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),A2)
             => ( aa_complex_real(F,X4) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7170_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X3)) )
       => ( ( groups1300246762558778688al_rat(F,A2) = zero_zero_rat )
        <=> ! [X4: real] :
              ( aa_set_real_o(member_real(X4),A2)
             => ( aa_real_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7171_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3)) )
       => ( ( groups2906978787729119204at_rat(F,A2) = zero_zero_rat )
        <=> ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),A2)
             => ( aa_nat_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7172_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3)) )
       => ( ( groups3906332499630173760nt_rat(F,A2) = zero_zero_rat )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A2)
             => ( aa_int_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7173_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X3)) )
       => ( ( groups5058264527183730370ex_rat(F,A2) = zero_zero_rat )
        <=> ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),A2)
             => ( aa_complex_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7174_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X3)) )
       => ( ( groups1935376822645274424al_nat(F,A2) = zero_zero_nat )
        <=> ! [X4: real] :
              ( aa_set_real_o(member_real(X4),A2)
             => ( aa_real_nat(F,X4) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7175_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X3)) )
       => ( ( groups4541462559716669496nt_nat(F,A2) = zero_zero_nat )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A2)
             => ( aa_int_nat(F,X4) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7176_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X3)) )
       => ( ( groups5693394587270226106ex_nat(F,A2) = zero_zero_nat )
        <=> ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),A2)
             => ( aa_complex_nat(F,X4) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7177_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X3)),aa_int_real(G,X3)) )
       => ( ? [X2: int] :
              ( aa_set_int_o(member_int(X2),A2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X2)),aa_int_real(G,X2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7178_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X3)),aa_complex_real(G,X3)) )
       => ( ? [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),A2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X2)),aa_complex_real(G,X2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7179_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,X3)) )
       => ( ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7180_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_int_rat(G,X3)) )
       => ( ? [X2: int] :
              ( aa_set_int_o(member_int(X2),A2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7181_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X3)),aa_complex_rat(G,X3)) )
       => ( ? [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),A2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X2)),aa_complex_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7182_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X3)),aa_int_nat(G,X3)) )
       => ( ? [X2: int] :
              ( aa_set_int_o(member_int(X2),A2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X2)),aa_int_nat(G,X2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7183_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X3)),aa_complex_nat(G,X3)) )
       => ( ? [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),A2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,X2)),aa_complex_nat(G,X2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7184_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X3)),aa_nat_int(G,X3)) )
       => ( ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A2)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X2)),aa_nat_int(G,X2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,A2)),groups3539618377306564664at_int(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7185_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X3)),aa_int_int(G,X3)) )
       => ( ? [X2: int] :
              ( aa_set_int_o(member_int(X2),A2)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,X2)),aa_int_int(G,X2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7186_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,X3)),aa_complex_int(G,X3)) )
       => ( ? [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),A2)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_complex_int(F,X2)),aa_complex_int(G,X2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7187_sum_Orelated,axiom,
    ! [R: fun_co1148497455491383662plex_o,S3: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R,zero_zero_complex),zero_zero_complex)
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X23)
              & 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(X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S3)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_nat_complex(H,X3)),aa_nat_complex(G,X3)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,groups2073611262835488442omplex(H,S3)),groups2073611262835488442omplex(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_7188_sum_Orelated,axiom,
    ! [R: fun_co1148497455491383662plex_o,S3: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R,zero_zero_complex),zero_zero_complex)
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X23)
              & 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(X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S3)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S3)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_int_complex(H,X3)),aa_int_complex(G,X3)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,groups3049146728041665814omplex(H,S3)),groups3049146728041665814omplex(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_7189_sum_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S3: set_int,H: fun_int_real,G: fun_int_real] :
      ( aa_real_o(aa_real_fun_real_o(R,zero_zero_real),zero_zero_real)
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X23)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S3)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S3)
               => aa_real_o(aa_real_fun_real_o(R,aa_int_real(H,X3)),aa_int_real(G,X3)) )
           => aa_real_o(aa_real_fun_real_o(R,groups8778361861064173332t_real(H,S3)),groups8778361861064173332t_real(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_7190_sum_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S3: 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,X23: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X23)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S3)
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S3)
               => aa_real_o(aa_real_fun_real_o(R,aa_complex_real(H,X3)),aa_complex_real(G,X3)) )
           => aa_real_o(aa_real_fun_real_o(R,groups5808333547571424918x_real(H,S3)),groups5808333547571424918x_real(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_7191_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S3: 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,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S3)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_nat_rat(H,X3)),aa_nat_rat(G,X3)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups2906978787729119204at_rat(H,S3)),groups2906978787729119204at_rat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_7192_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S3: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S3)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S3)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_int_rat(H,X3)),aa_int_rat(G,X3)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups3906332499630173760nt_rat(H,S3)),groups3906332499630173760nt_rat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_7193_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S3: 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,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X23),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S3)
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S3)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_complex_rat(H,X3)),aa_complex_rat(G,X3)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups5058264527183730370ex_rat(H,S3)),groups5058264527183730370ex_rat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_7194_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S3: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,zero_zero_nat),zero_zero_nat)
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X15),X23)
              & 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,X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S3)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S3)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_int_nat(H,X3)),aa_int_nat(G,X3)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,groups4541462559716669496nt_nat(H,S3)),groups4541462559716669496nt_nat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_7195_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S3: 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,X23: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X15),X23)
              & 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,X23),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S3)
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S3)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_complex_nat(H,X3)),aa_complex_nat(G,X3)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,groups5693394587270226106ex_nat(H,S3)),groups5693394587270226106ex_nat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_7196_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S3: 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,X23: int,Y23: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R,X15),X23)
              & 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,X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S3)
               => aa_int_o(aa_int_fun_int_o(R,aa_nat_int(H,X3)),aa_nat_int(G,X3)) )
           => aa_int_o(aa_int_fun_int_o(R,groups3539618377306564664at_int(H,S3)),groups3539618377306564664at_int(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_7197_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),E) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_7198_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),E) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_7199_abs__mult__pos,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(Y)),X) = abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_7200_abs__mult__pos,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Y)),X) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_7201_abs__mult__pos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,Y)),X) = aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(Y),X)) ) ) ).

% abs_mult_pos
tff(fact_7202_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),X) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_7203_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_7204_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(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% abs_eq_mult
tff(fact_7205_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(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% abs_eq_mult
tff(fact_7206_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_7207_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X3)),aa_complex_real(G,X3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_7208_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(G,X3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_7209_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),aa_int_real(G,X3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_7210_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X3)),aa_complex_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_7211_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(F,A2)),groups1300246762558778688al_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_7212_sum__strict__mono,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_7213_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_7214_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,X3)),aa_complex_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_7215_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X3)),aa_real_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_7216_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X3)),aa_int_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_7217_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_7218_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_7219_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_7220_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_7221_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_7222_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_7223_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_7224_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_7225_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_7226_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_7227_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_7228_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_7229_zero__le__power__abs,axiom,
    ! [A: code_integer,Na: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Na)) ).

% zero_le_power_abs
tff(fact_7230_zero__le__power__abs,axiom,
    ! [A: real,Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Na)) ).

% zero_le_power_abs
tff(fact_7231_zero__le__power__abs,axiom,
    ! [A: rat,Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Na)) ).

% zero_le_power_abs
tff(fact_7232_zero__le__power__abs,axiom,
    ! [A: int,Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Na)) ).

% zero_le_power_abs
tff(fact_7233_abs__div__pos,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,X)),Y) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% abs_div_pos
tff(fact_7234_abs__div__pos,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,X)),Y) = aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(X),Y)) ) ) ).

% abs_div_pos
tff(fact_7235_abs__if__raw,axiom,
    ! [X2: int] :
      aa_int_int(abs_abs_int,X2) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X2),zero_zero_int),aa_int_int(uminus_uminus_int,X2),X2) ).

% abs_if_raw
tff(fact_7236_abs__if__raw,axiom,
    ! [X2: real] :
      aa_real_real(abs_abs_real,X2) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,X2),zero_zero_real),aa_real_real(uminus_uminus_real,X2),X2) ).

% abs_if_raw
tff(fact_7237_abs__if__raw,axiom,
    ! [X2: rat] :
      aa_rat_rat(abs_abs_rat,X2) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,X2),X2) ).

% abs_if_raw
tff(fact_7238_abs__if__raw,axiom,
    ! [X2: code_integer] :
      abs_abs_Code_integer(X2) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X2),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X2),X2) ).

% abs_if_raw
tff(fact_7239_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_7240_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_7241_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_7242_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_7243_abs__if,axiom,
    ! [A: int] :
      aa_int_int(abs_abs_int,A) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int),aa_int_int(uminus_uminus_int,A),A) ).

% abs_if
tff(fact_7244_abs__if,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,A) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real),aa_real_real(uminus_uminus_real,A),A) ).

% abs_if
tff(fact_7245_abs__if,axiom,
    ! [A: rat] :
      aa_rat_rat(abs_abs_rat,A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A),A) ).

% abs_if
tff(fact_7246_abs__if,axiom,
    ! [A: code_integer] :
      abs_abs_Code_integer(A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) ).

% abs_if
tff(fact_7247_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_7248_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(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_7249_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(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(C),D)))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),C))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_7250_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_7251_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_7252_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(aa_rea7340605660300375871l_real(minus_minus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq4
tff(fact_7253_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(minus_minus_rat(A),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq4
tff(fact_7254_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_7255_abs__diff__le__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,A))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R2)),X)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),plus_p5714425477246183910nteger(A,R2)) ) ) ).

% abs_diff_le_iff
tff(fact_7256_abs__diff__le__iff,axiom,
    ! [X: real,A: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),X)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_7257_abs__diff__le__iff,axiom,
    ! [X: 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(minus_minus_rat(X),A))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),R2)),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(plus_plus_rat(A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_7258_abs__diff__le__iff,axiom,
    ! [X: 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,X),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)),X)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_7259_sum_Oinsert__if,axiom,
    ! [A2: set_real,G: fun_real_real,X: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups8097168146408367636l_real(G,insert_real(X,A2)) = $ite(aa_set_real_o(member_real(X),A2),groups8097168146408367636l_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X)),groups8097168146408367636l_real(G,A2))) ) ) ).

% sum.insert_if
tff(fact_7260_sum_Oinsert__if,axiom,
    ! [A2: set_int,G: fun_int_real,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups8778361861064173332t_real(G,insert_int(X,A2)) = $ite(aa_set_int_o(member_int(X),A2),groups8778361861064173332t_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X)),groups8778361861064173332t_real(G,A2))) ) ) ).

% sum.insert_if
tff(fact_7261_sum_Oinsert__if,axiom,
    ! [A2: set_complex,G: fun_complex_real,X: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(G,insert_complex(X,A2)) = $ite(aa_set_complex_o(member_complex(X),A2),groups5808333547571424918x_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X)),groups5808333547571424918x_real(G,A2))) ) ) ).

% sum.insert_if
tff(fact_7262_sum_Oinsert__if,axiom,
    ! [A2: set_real,G: fun_real_rat,X: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1300246762558778688al_rat(G,insert_real(X,A2)) = $ite(aa_set_real_o(member_real(X),A2),groups1300246762558778688al_rat(G,A2),aa_rat_rat(plus_plus_rat(aa_real_rat(G,X)),groups1300246762558778688al_rat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_7263_sum_Oinsert__if,axiom,
    ! [A2: set_nat,G: fun_nat_rat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups2906978787729119204at_rat(G,insert_nat(X,A2)) = $ite(aa_set_nat_o(member_nat(X),A2),groups2906978787729119204at_rat(G,A2),aa_rat_rat(plus_plus_rat(aa_nat_rat(G,X)),groups2906978787729119204at_rat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_7264_sum_Oinsert__if,axiom,
    ! [A2: set_int,G: fun_int_rat,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(G,insert_int(X,A2)) = $ite(aa_set_int_o(member_int(X),A2),groups3906332499630173760nt_rat(G,A2),aa_rat_rat(plus_plus_rat(aa_int_rat(G,X)),groups3906332499630173760nt_rat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_7265_sum_Oinsert__if,axiom,
    ! [A2: set_complex,G: fun_complex_rat,X: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(G,insert_complex(X,A2)) = $ite(aa_set_complex_o(member_complex(X),A2),groups5058264527183730370ex_rat(G,A2),aa_rat_rat(plus_plus_rat(aa_complex_rat(G,X)),groups5058264527183730370ex_rat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_7266_sum_Oinsert__if,axiom,
    ! [A2: set_real,G: fun_real_nat,X: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1935376822645274424al_nat(G,insert_real(X,A2)) = $ite(aa_set_real_o(member_real(X),A2),groups1935376822645274424al_nat(G,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X)),groups1935376822645274424al_nat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_7267_sum_Oinsert__if,axiom,
    ! [A2: set_int,G: fun_int_nat,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(G,insert_int(X,A2)) = $ite(aa_set_int_o(member_int(X),A2),groups4541462559716669496nt_nat(G,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(G,X)),groups4541462559716669496nt_nat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_7268_sum_Oinsert__if,axiom,
    ! [A2: set_complex,G: fun_complex_nat,X: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5693394587270226106ex_nat(G,insert_complex(X,A2)) = $ite(aa_set_complex_o(member_complex(X),A2),groups5693394587270226106ex_nat(G,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X)),groups5693394587270226106ex_nat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_7269_abs__diff__less__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,A))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R2)),X)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),plus_p5714425477246183910nteger(A,R2)) ) ) ).

% abs_diff_less_iff
tff(fact_7270_abs__diff__less__iff,axiom,
    ! [X: real,A: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),X)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_7271_abs__diff__less__iff,axiom,
    ! [X: 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(minus_minus_rat(X),A))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),R2)),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(plus_plus_rat(A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_7272_abs__diff__less__iff,axiom,
    ! [X: 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,X),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)),X)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_7273_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T5: set_real,S3: set_real,I: fun_real_real,J: fun_real_real,T3: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,S6)
     => ( aa_set_real_o(finite_finite_real,T5)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),minus_minus_set_real(S3,S6))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(S3,S6))
               => aa_set_real_o(member_real(aa_real_real(J,A3)),minus_minus_set_real(T3,T5)) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(T3,T5))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( aa_set_real_o(member_real(B3),minus_minus_set_real(T3,T5))
                   => aa_set_real_o(member_real(aa_real_real(I,B3)),minus_minus_set_real(S3,S6)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real(A3),S6)
                     => ( aa_real_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( aa_set_real_o(member_real(B3),T5)
                       => ( aa_real_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real(A3),S3)
                         => ( aa_real_complex(H,aa_real_real(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups5754745047067104278omplex(G,S3) = groups5754745047067104278omplex(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7274_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T5: set_int,S3: set_real,I: fun_int_real,J: fun_real_int,T3: set_int,G: fun_real_complex,H: fun_int_complex] :
      ( aa_set_real_o(finite_finite_real,S6)
     => ( aa_set_int_o(finite_finite_int,T5)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),minus_minus_set_real(S3,S6))
             => ( aa_int_real(I,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(S3,S6))
               => aa_set_int_o(member_int(aa_real_int(J,A3)),minus_minus_set_int(T3,T5)) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(T3,T5))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( aa_set_int_o(member_int(B3),minus_minus_set_int(T3,T5))
                   => aa_set_real_o(member_real(aa_int_real(I,B3)),minus_minus_set_real(S3,S6)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real(A3),S6)
                     => ( aa_real_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( aa_set_int_o(member_int(B3),T5)
                       => ( aa_int_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real(A3),S3)
                         => ( aa_int_complex(H,aa_real_int(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups5754745047067104278omplex(G,S3) = groups3049146728041665814omplex(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7275_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_int,T5: set_real,S3: set_int,I: fun_real_int,J: fun_int_real,T3: set_real,G: fun_int_complex,H: fun_real_complex] :
      ( aa_set_int_o(finite_finite_int,S6)
     => ( aa_set_real_o(finite_finite_real,T5)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),minus_minus_set_int(S3,S6))
             => ( aa_real_int(I,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(S3,S6))
               => aa_set_real_o(member_real(aa_int_real(J,A3)),minus_minus_set_real(T3,T5)) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(T3,T5))
                 => ( aa_int_real(J,aa_real_int(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( aa_set_real_o(member_real(B3),minus_minus_set_real(T3,T5))
                   => aa_set_int_o(member_int(aa_real_int(I,B3)),minus_minus_set_int(S3,S6)) )
               => ( ! [A3: int] :
                      ( aa_set_int_o(member_int(A3),S6)
                     => ( aa_int_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( aa_set_real_o(member_real(B3),T5)
                       => ( aa_real_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( aa_set_int_o(member_int(A3),S3)
                         => ( aa_real_complex(H,aa_int_real(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups3049146728041665814omplex(G,S3) = groups5754745047067104278omplex(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7276_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_int,T5: set_int,S3: set_int,I: fun_int_int,J: fun_int_int,T3: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,S6)
     => ( aa_set_int_o(finite_finite_int,T5)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),minus_minus_set_int(S3,S6))
             => ( aa_int_int(I,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(S3,S6))
               => aa_set_int_o(member_int(aa_int_int(J,A3)),minus_minus_set_int(T3,T5)) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(T3,T5))
                 => ( aa_int_int(J,aa_int_int(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( aa_set_int_o(member_int(B3),minus_minus_set_int(T3,T5))
                   => aa_set_int_o(member_int(aa_int_int(I,B3)),minus_minus_set_int(S3,S6)) )
               => ( ! [A3: int] :
                      ( aa_set_int_o(member_int(A3),S6)
                     => ( aa_int_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( aa_set_int_o(member_int(B3),T5)
                       => ( aa_int_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( aa_set_int_o(member_int(A3),S3)
                         => ( aa_int_complex(H,aa_int_int(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups3049146728041665814omplex(G,S3) = groups3049146728041665814omplex(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7277_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T5: set_real,S3: set_real,I: fun_real_real,J: fun_real_real,T3: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S6)
     => ( aa_set_real_o(finite_finite_real,T5)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),minus_minus_set_real(S3,S6))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(S3,S6))
               => aa_set_real_o(member_real(aa_real_real(J,A3)),minus_minus_set_real(T3,T5)) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(T3,T5))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( aa_set_real_o(member_real(B3),minus_minus_set_real(T3,T5))
                   => aa_set_real_o(member_real(aa_real_real(I,B3)),minus_minus_set_real(S3,S6)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real(A3),S6)
                     => ( aa_real_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: real] :
                        ( aa_set_real_o(member_real(B3),T5)
                       => ( aa_real_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real(A3),S3)
                         => ( aa_real_real(H,aa_real_real(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups8097168146408367636l_real(G,S3) = groups8097168146408367636l_real(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7278_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T5: set_int,S3: set_real,I: fun_int_real,J: fun_real_int,T3: set_int,G: fun_real_real,H: fun_int_real] :
      ( aa_set_real_o(finite_finite_real,S6)
     => ( aa_set_int_o(finite_finite_int,T5)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),minus_minus_set_real(S3,S6))
             => ( aa_int_real(I,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(S3,S6))
               => aa_set_int_o(member_int(aa_real_int(J,A3)),minus_minus_set_int(T3,T5)) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(T3,T5))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( aa_set_int_o(member_int(B3),minus_minus_set_int(T3,T5))
                   => aa_set_real_o(member_real(aa_int_real(I,B3)),minus_minus_set_real(S3,S6)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real(A3),S6)
                     => ( aa_real_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: int] :
                        ( aa_set_int_o(member_int(B3),T5)
                       => ( aa_int_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real(A3),S3)
                         => ( aa_int_real(H,aa_real_int(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups8097168146408367636l_real(G,S3) = groups8778361861064173332t_real(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7279_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T5: set_complex,S3: set_real,I: fun_complex_real,J: fun_real_complex,T3: set_complex,G: fun_real_real,H: fun_complex_real] :
      ( aa_set_real_o(finite_finite_real,S6)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T5)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),minus_minus_set_real(S3,S6))
             => ( aa_complex_real(I,aa_real_complex(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(S3,S6))
               => aa_set_complex_o(member_complex(aa_real_complex(J,A3)),minus_811609699411566653omplex(T3,T5)) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T3,T5))
                 => ( aa_real_complex(J,aa_complex_real(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T3,T5))
                   => aa_set_real_o(member_real(aa_complex_real(I,B3)),minus_minus_set_real(S3,S6)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real(A3),S6)
                     => ( aa_real_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: complex] :
                        ( aa_set_complex_o(member_complex(B3),T5)
                       => ( aa_complex_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real(A3),S3)
                         => ( aa_complex_real(H,aa_real_complex(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups8097168146408367636l_real(G,S3) = groups5808333547571424918x_real(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7280_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_int,T5: set_real,S3: set_int,I: fun_real_int,J: fun_int_real,T3: set_real,G: fun_int_real,H: fun_real_real] :
      ( aa_set_int_o(finite_finite_int,S6)
     => ( aa_set_real_o(finite_finite_real,T5)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),minus_minus_set_int(S3,S6))
             => ( aa_real_int(I,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(S3,S6))
               => aa_set_real_o(member_real(aa_int_real(J,A3)),minus_minus_set_real(T3,T5)) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(T3,T5))
                 => ( aa_int_real(J,aa_real_int(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( aa_set_real_o(member_real(B3),minus_minus_set_real(T3,T5))
                   => aa_set_int_o(member_int(aa_real_int(I,B3)),minus_minus_set_int(S3,S6)) )
               => ( ! [A3: int] :
                      ( aa_set_int_o(member_int(A3),S6)
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: real] :
                        ( aa_set_real_o(member_real(B3),T5)
                       => ( aa_real_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( aa_set_int_o(member_int(A3),S3)
                         => ( aa_real_real(H,aa_int_real(J,A3)) = aa_int_real(G,A3) ) )
                     => ( groups8778361861064173332t_real(G,S3) = groups8097168146408367636l_real(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7281_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_int,T5: set_int,S3: set_int,I: fun_int_int,J: fun_int_int,T3: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S6)
     => ( aa_set_int_o(finite_finite_int,T5)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),minus_minus_set_int(S3,S6))
             => ( aa_int_int(I,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(S3,S6))
               => aa_set_int_o(member_int(aa_int_int(J,A3)),minus_minus_set_int(T3,T5)) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(T3,T5))
                 => ( aa_int_int(J,aa_int_int(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( aa_set_int_o(member_int(B3),minus_minus_set_int(T3,T5))
                   => aa_set_int_o(member_int(aa_int_int(I,B3)),minus_minus_set_int(S3,S6)) )
               => ( ! [A3: int] :
                      ( aa_set_int_o(member_int(A3),S6)
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: int] :
                        ( aa_set_int_o(member_int(B3),T5)
                       => ( aa_int_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( aa_set_int_o(member_int(A3),S3)
                         => ( aa_int_real(H,aa_int_int(J,A3)) = aa_int_real(G,A3) ) )
                     => ( groups8778361861064173332t_real(G,S3) = groups8778361861064173332t_real(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7282_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_int,T5: set_complex,S3: set_int,I: fun_complex_int,J: fun_int_complex,T3: set_complex,G: fun_int_real,H: fun_complex_real] :
      ( aa_set_int_o(finite_finite_int,S6)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T5)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),minus_minus_set_int(S3,S6))
             => ( aa_complex_int(I,aa_int_complex(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(S3,S6))
               => aa_set_complex_o(member_complex(aa_int_complex(J,A3)),minus_811609699411566653omplex(T3,T5)) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T3,T5))
                 => ( aa_int_complex(J,aa_complex_int(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T3,T5))
                   => aa_set_int_o(member_int(aa_complex_int(I,B3)),minus_minus_set_int(S3,S6)) )
               => ( ! [A3: int] :
                      ( aa_set_int_o(member_int(A3),S6)
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: complex] :
                        ( aa_set_complex_o(member_complex(B3),T5)
                       => ( aa_complex_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( aa_set_int_o(member_int(A3),S3)
                         => ( aa_complex_real(H,aa_int_complex(J,A3)) = aa_int_real(G,A3) ) )
                     => ( groups8778361861064173332t_real(G,S3) = groups5808333547571424918x_real(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7283_sum__eq__Suc0__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( groups4541462559716669496nt_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A2)
            & ( aa_int_nat(F,X4) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),A2)
               => ( ( X4 != Xa4 )
                 => ( aa_int_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_7284_sum__eq__Suc0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( groups5693394587270226106ex_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A2)
            & ( aa_complex_nat(F,X4) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: complex] :
                ( aa_set_complex_o(member_complex(Xa4),A2)
               => ( ( X4 != Xa4 )
                 => ( aa_complex_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_7285_sum__eq__Suc0__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( ( groups977919841031483927at_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A2)
            & ( aa_Pro4243173770044397973at_nat(F,X4) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa4),A2)
               => ( ( X4 != Xa4 )
                 => ( aa_Pro4243173770044397973at_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_7286_sum__eq__Suc0__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A2)
            & ( aa_nat_nat(F,X4) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: nat] :
                ( aa_set_nat_o(member_nat(Xa4),A2)
               => ( ( X4 != Xa4 )
                 => ( aa_nat_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_7287_sum__SucD,axiom,
    ! [F: fun_nat_nat,A2: set_nat,Na: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,Na) )
     => ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A2)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X3)) ) ) ).

% sum_SucD
tff(fact_7288_sum__eq__1__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( groups4541462559716669496nt_nat(F,A2) = one_one_nat )
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A2)
            & ( aa_int_nat(F,X4) = one_one_nat )
            & ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),A2)
               => ( ( X4 != Xa4 )
                 => ( aa_int_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_7289_sum__eq__1__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( groups5693394587270226106ex_nat(F,A2) = one_one_nat )
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A2)
            & ( aa_complex_nat(F,X4) = one_one_nat )
            & ! [Xa4: complex] :
                ( aa_set_complex_o(member_complex(Xa4),A2)
               => ( ( X4 != Xa4 )
                 => ( aa_complex_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_7290_sum__eq__1__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A2)
     => ( ( groups977919841031483927at_nat(F,A2) = one_one_nat )
      <=> ? [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A2)
            & ( aa_Pro4243173770044397973at_nat(F,X4) = one_one_nat )
            & ! [Xa4: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa4),A2)
               => ( ( X4 != Xa4 )
                 => ( aa_Pro4243173770044397973at_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_7291_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = one_one_nat )
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A2)
            & ( aa_nat_nat(F,X4) = one_one_nat )
            & ! [Xa4: nat] :
                ( aa_set_nat_o(member_nat(Xa4),A2)
               => ( ( X4 != Xa4 )
                 => ( aa_nat_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_7292_abs__real__def,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,A) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real),aa_real_real(uminus_uminus_real,A),A) ).

% abs_real_def
tff(fact_7293_zabs__def,axiom,
    ! [I: int] :
      aa_int_int(abs_abs_int,I) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int),aa_int_int(uminus_uminus_int,I),I) ).

% zabs_def
tff(fact_7294_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_real,B2: real,I: real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3)) )
       => ( ( groups8097168146408367636l_real(F,S) = B2 )
         => ( aa_set_real_o(member_real(I),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7295_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_real,B2: real,I: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3)) )
       => ( ( groups8778361861064173332t_real(F,S) = B2 )
         => ( aa_set_int_o(member_int(I),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7296_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_real,B2: real,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3)) )
       => ( ( groups5808333547571424918x_real(F,S) = B2 )
         => ( aa_set_complex_o(member_complex(I),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7297_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_rat,B2: rat,I: real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3)) )
       => ( ( groups1300246762558778688al_rat(F,S) = B2 )
         => ( aa_set_real_o(member_real(I),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7298_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: fun_nat_rat,B2: rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3)) )
       => ( ( groups2906978787729119204at_rat(F,S) = B2 )
         => ( aa_set_nat_o(member_nat(I),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7299_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_rat,B2: rat,I: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3)) )
       => ( ( groups3906332499630173760nt_rat(F,S) = B2 )
         => ( aa_set_int_o(member_int(I),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7300_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_rat,B2: rat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3)) )
       => ( ( groups5058264527183730370ex_rat(F,S) = B2 )
         => ( aa_set_complex_o(member_complex(I),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7301_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_nat,B2: nat,I: real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3)) )
       => ( ( groups1935376822645274424al_nat(F,S) = B2 )
         => ( aa_set_real_o(member_real(I),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7302_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_nat,B2: nat,I: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I3)) )
       => ( ( groups4541462559716669496nt_nat(F,S) = B2 )
         => ( aa_set_int_o(member_int(I),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7303_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_nat,B2: nat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I3)) )
       => ( ( groups5693394587270226106ex_nat(F,S) = B2 )
         => ( aa_set_complex_o(member_complex(I),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7304_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_real,I: real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3)) )
       => ( ( groups8097168146408367636l_real(F,S) = zero_zero_real )
         => ( aa_set_real_o(member_real(I),S)
           => ( aa_real_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7305_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_real,I: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3)) )
       => ( ( groups8778361861064173332t_real(F,S) = zero_zero_real )
         => ( aa_set_int_o(member_int(I),S)
           => ( aa_int_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7306_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_real,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3)) )
       => ( ( groups5808333547571424918x_real(F,S) = zero_zero_real )
         => ( aa_set_complex_o(member_complex(I),S)
           => ( aa_complex_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7307_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_rat,I: real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3)) )
       => ( ( groups1300246762558778688al_rat(F,S) = zero_zero_rat )
         => ( aa_set_real_o(member_real(I),S)
           => ( aa_real_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7308_sum__nonneg__0,axiom,
    ! [S: set_nat,F: fun_nat_rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3)) )
       => ( ( groups2906978787729119204at_rat(F,S) = zero_zero_rat )
         => ( aa_set_nat_o(member_nat(I),S)
           => ( aa_nat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7309_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_rat,I: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3)) )
       => ( ( groups3906332499630173760nt_rat(F,S) = zero_zero_rat )
         => ( aa_set_int_o(member_int(I),S)
           => ( aa_int_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7310_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_rat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3)) )
       => ( ( groups5058264527183730370ex_rat(F,S) = zero_zero_rat )
         => ( aa_set_complex_o(member_complex(I),S)
           => ( aa_complex_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7311_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_nat,I: real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3)) )
       => ( ( groups1935376822645274424al_nat(F,S) = zero_zero_nat )
         => ( aa_set_real_o(member_real(I),S)
           => ( aa_real_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7312_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_nat,I: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I3)) )
       => ( ( groups4541462559716669496nt_nat(F,S) = zero_zero_nat )
         => ( aa_set_int_o(member_int(I),S)
           => ( aa_int_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7313_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_nat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),S)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I3)) )
       => ( ( groups5693394587270226106ex_nat(F,S) = zero_zero_nat )
         => ( aa_set_complex_o(member_complex(I),S)
           => ( aa_complex_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7314_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),I)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I)) ) ) ).

% dvd_imp_le_int
tff(fact_7315_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(modulo_modulo_int(K),L))),aa_int_int(abs_abs_int,L)) ) ).

% abs_mod_less
tff(fact_7316_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups5754745047067104278omplex(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_nq(G)))) = groups5754745047067104278omplex(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7317_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3049146728041665814omplex(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_nr(G)))) = groups3049146728041665814omplex(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7318_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups8097168146408367636l_real(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_ns(G)))) = groups8097168146408367636l_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7319_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups8778361861064173332t_real(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_nt(G)))) = groups8778361861064173332t_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7320_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_nu(G)))) = groups5808333547571424918x_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7321_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1300246762558778688al_rat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_nv(G)))) = groups1300246762558778688al_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7322_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_nw(G)))) = groups3906332499630173760nt_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7323_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_nx(G)))) = groups5058264527183730370ex_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7324_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1935376822645274424al_nat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_ny(G)))) = groups1935376822645274424al_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7325_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_nz(G)))) = groups4541462559716669496nt_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7326_sum__power__add,axiom,
    ! [X: complex,Mb: nat,I5: set_nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_oa(X),Mb),I5) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),Mb)),groups2073611262835488442omplex(power_power_complex(X),I5)) ).

% sum_power_add
tff(fact_7327_sum__power__add,axiom,
    ! [X: rat,Mb: nat,I5: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_ob(X),Mb),I5) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),Mb)),groups2906978787729119204at_rat(power_power_rat(X),I5)) ).

% sum_power_add
tff(fact_7328_sum__power__add,axiom,
    ! [X: int,Mb: nat,I5: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_oc(X),Mb),I5) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),Mb)),groups3539618377306564664at_int(power_power_int(X),I5)) ).

% sum_power_add
tff(fact_7329_sum__power__add,axiom,
    ! [X: real,Mb: nat,I5: set_nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_od(X),Mb),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Mb)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),I5)) ).

% sum_power_add
tff(fact_7330_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Na: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Na,Mb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_oe(G),Na),Mb)),set_or1269000886237332187st_nat(Na,Mb)) ).

% sum.atLeastAtMost_rev
tff(fact_7331_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,Na: nat,Mb: nat] : groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(Na,Mb)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_of(G),Na),Mb),set_or1269000886237332187st_nat(Na,Mb)) ).

% sum.atLeastAtMost_rev
tff(fact_7332_tanh__real__gt__neg1,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(tanh_real,X)) ).

% tanh_real_gt_neg1
tff(fact_7333_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_7334_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_7335_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_7336_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_7337_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_7338_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_7339_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_7340_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_7341_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I),I5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,I))
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),I5)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I3)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7342_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(I),I5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I))
         => ( ! [I3: complex] :
                ( aa_set_complex_o(member_complex(I3),I5)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I3)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7343_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_7344_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_7345_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_7346_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_7347_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_7348_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_7349_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_7350_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( aa_set_complex_o(member_complex(I3),I5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7351_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_7352_sum__pos,axiom,
    ! [I5: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,I3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7353_abs__add__one__gt__zero,axiom,
    ! [X: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(X))) ).

% abs_add_one_gt_zero
tff(fact_7354_abs__add__one__gt__zero,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(abs_abs_real,X))) ).

% abs_add_one_gt_zero
tff(fact_7355_abs__add__one__gt__zero,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(abs_abs_rat,X))) ).

% abs_add_one_gt_zero
tff(fact_7356_abs__add__one__gt__zero,axiom,
    ! [X: 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,X))) ).

% abs_add_one_gt_zero
tff(fact_7357_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,T3)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T3)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),minus_minus_set_real(T3,S3))
             => ( aa_real_complex(G,X3) = zero_zero_complex ) )
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S3)
               => ( aa_real_complex(G,X3) = aa_real_complex(H,X3) ) )
           => ( groups5754745047067104278omplex(G,T3) = groups5754745047067104278omplex(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7358_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T3)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T3)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),minus_minus_set_real(T3,S3))
             => ( aa_real_real(G,X3) = zero_zero_real ) )
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S3)
               => ( aa_real_real(G,X3) = aa_real_real(H,X3) ) )
           => ( groups8097168146408367636l_real(G,T3) = groups8097168146408367636l_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7359_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T3)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T3,S3))
             => ( aa_complex_real(G,X3) = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S3)
               => ( aa_complex_real(G,X3) = aa_complex_real(H,X3) ) )
           => ( groups5808333547571424918x_real(G,T3) = groups5808333547571424918x_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7360_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,T3)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T3)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),minus_minus_set_real(T3,S3))
             => ( aa_real_rat(G,X3) = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S3)
               => ( aa_real_rat(G,X3) = aa_real_rat(H,X3) ) )
           => ( groups1300246762558778688al_rat(G,T3) = groups1300246762558778688al_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7361_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T3)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T3,S3))
             => ( aa_complex_rat(G,X3) = zero_zero_rat ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S3)
               => ( aa_complex_rat(G,X3) = aa_complex_rat(H,X3) ) )
           => ( groups5058264527183730370ex_rat(G,T3) = groups5058264527183730370ex_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7362_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,T3)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T3)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),minus_minus_set_real(T3,S3))
             => ( aa_real_nat(G,X3) = zero_zero_nat ) )
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S3)
               => ( aa_real_nat(G,X3) = aa_real_nat(H,X3) ) )
           => ( groups1935376822645274424al_nat(G,T3) = groups1935376822645274424al_nat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7363_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T3)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T3,S3))
             => ( aa_complex_nat(G,X3) = zero_zero_nat ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S3)
               => ( aa_complex_nat(G,X3) = aa_complex_nat(H,X3) ) )
           => ( groups5693394587270226106ex_nat(G,T3) = groups5693394587270226106ex_nat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7364_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: fun_real_int,H: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,T3)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T3)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),minus_minus_set_real(T3,S3))
             => ( aa_real_int(G,X3) = zero_zero_int ) )
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S3)
               => ( aa_real_int(G,X3) = aa_real_int(H,X3) ) )
           => ( groups1932886352136224148al_int(G,T3) = groups1932886352136224148al_int(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7365_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T3)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T3,S3))
             => ( aa_complex_int(G,X3) = zero_zero_int ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S3)
               => ( aa_complex_int(G,X3) = aa_complex_int(H,X3) ) )
           => ( groups5690904116761175830ex_int(G,T3) = groups5690904116761175830ex_int(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7366_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),T3)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T3,S3))
             => ( aa_nat_complex(G,X3) = zero_zero_complex ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S3)
               => ( aa_nat_complex(G,X3) = aa_nat_complex(H,X3) ) )
           => ( groups2073611262835488442omplex(G,T3) = groups2073611262835488442omplex(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7367_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,T3)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T3)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),minus_minus_set_real(T3,S3))
             => ( aa_real_complex(H,X3) = zero_zero_complex ) )
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S3)
               => ( aa_real_complex(G,X3) = aa_real_complex(H,X3) ) )
           => ( groups5754745047067104278omplex(G,S3) = groups5754745047067104278omplex(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7368_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T3)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T3)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),minus_minus_set_real(T3,S3))
             => ( aa_real_real(H,X3) = zero_zero_real ) )
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S3)
               => ( aa_real_real(G,X3) = aa_real_real(H,X3) ) )
           => ( groups8097168146408367636l_real(G,S3) = groups8097168146408367636l_real(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7369_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T3)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T3,S3))
             => ( aa_complex_real(H,X3) = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S3)
               => ( aa_complex_real(G,X3) = aa_complex_real(H,X3) ) )
           => ( groups5808333547571424918x_real(G,S3) = groups5808333547571424918x_real(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7370_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,T3)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T3)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),minus_minus_set_real(T3,S3))
             => ( aa_real_rat(H,X3) = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S3)
               => ( aa_real_rat(G,X3) = aa_real_rat(H,X3) ) )
           => ( groups1300246762558778688al_rat(G,S3) = groups1300246762558778688al_rat(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7371_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T3)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T3,S3))
             => ( aa_complex_rat(H,X3) = zero_zero_rat ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S3)
               => ( aa_complex_rat(G,X3) = aa_complex_rat(H,X3) ) )
           => ( groups5058264527183730370ex_rat(G,S3) = groups5058264527183730370ex_rat(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7372_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,T3)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T3)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),minus_minus_set_real(T3,S3))
             => ( aa_real_nat(H,X3) = zero_zero_nat ) )
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S3)
               => ( aa_real_nat(G,X3) = aa_real_nat(H,X3) ) )
           => ( groups1935376822645274424al_nat(G,S3) = groups1935376822645274424al_nat(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7373_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T3)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T3,S3))
             => ( aa_complex_nat(H,X3) = zero_zero_nat ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S3)
               => ( aa_complex_nat(G,X3) = aa_complex_nat(H,X3) ) )
           => ( groups5693394587270226106ex_nat(G,S3) = groups5693394587270226106ex_nat(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7374_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H: fun_real_int,G: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,T3)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T3)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),minus_minus_set_real(T3,S3))
             => ( aa_real_int(H,X3) = zero_zero_int ) )
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S3)
               => ( aa_real_int(G,X3) = aa_real_int(H,X3) ) )
           => ( groups1932886352136224148al_int(G,S3) = groups1932886352136224148al_int(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7375_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T3)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T3,S3))
             => ( aa_complex_int(H,X3) = zero_zero_int ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S3)
               => ( aa_complex_int(G,X3) = aa_complex_int(H,X3) ) )
           => ( groups5690904116761175830ex_int(G,S3) = groups5690904116761175830ex_int(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7376_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_nat,S3: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),T3)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T3,S3))
             => ( aa_nat_complex(H,X3) = zero_zero_complex ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S3)
               => ( aa_nat_complex(G,X3) = aa_nat_complex(H,X3) ) )
           => ( groups2073611262835488442omplex(G,S3) = groups2073611262835488442omplex(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7377_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T3)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T3,S3))
             => ( aa_complex_real(G,X3) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T3) = groups5808333547571424918x_real(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7378_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T3)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T3,S3))
             => ( aa_complex_rat(G,X3) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,T3) = groups5058264527183730370ex_rat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7379_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T3)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T3,S3))
             => ( aa_complex_nat(G,X3) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T3) = groups5693394587270226106ex_nat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7380_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T3)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T3,S3))
             => ( aa_complex_int(G,X3) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T3) = groups5690904116761175830ex_int(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7381_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),T3)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T3,S3))
             => ( aa_nat_complex(G,X3) = zero_zero_complex ) )
         => ( groups2073611262835488442omplex(G,T3) = groups2073611262835488442omplex(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7382_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),T3)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T3,S3))
             => ( aa_nat_rat(G,X3) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,T3) = groups2906978787729119204at_rat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7383_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),T3)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T3,S3))
             => ( aa_nat_int(G,X3) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G,T3) = groups3539618377306564664at_int(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7384_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S3: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T3)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T3)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T3,S3))
             => ( aa_int_complex(G,X3) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,T3) = groups3049146728041665814omplex(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7385_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S3: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T3)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T3)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T3,S3))
             => ( aa_int_real(G,X3) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,T3) = groups8778361861064173332t_real(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7386_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T3)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T3)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T3,S3))
             => ( aa_int_rat(G,X3) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,T3) = groups3906332499630173760nt_rat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7387_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T3)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T3,S3))
             => ( aa_complex_real(G,X3) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S3) = groups5808333547571424918x_real(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7388_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T3)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T3,S3))
             => ( aa_complex_rat(G,X3) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,S3) = groups5058264527183730370ex_rat(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7389_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T3)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T3,S3))
             => ( aa_complex_nat(G,X3) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S3) = groups5693394587270226106ex_nat(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7390_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T3)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T3,S3))
             => ( aa_complex_int(G,X3) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S3) = groups5690904116761175830ex_int(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7391_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),T3)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T3,S3))
             => ( aa_nat_complex(G,X3) = zero_zero_complex ) )
         => ( groups2073611262835488442omplex(G,S3) = groups2073611262835488442omplex(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7392_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),T3)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T3,S3))
             => ( aa_nat_rat(G,X3) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,S3) = groups2906978787729119204at_rat(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7393_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),T3)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T3,S3))
             => ( aa_nat_int(G,X3) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G,S3) = groups3539618377306564664at_int(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7394_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S3: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T3)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T3)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T3,S3))
             => ( aa_int_complex(G,X3) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,S3) = groups3049146728041665814omplex(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7395_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S3: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T3)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T3)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T3,S3))
             => ( aa_int_real(G,X3) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,S3) = groups8778361861064173332t_real(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7396_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T3)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T3)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T3,S3))
             => ( aa_int_rat(G,X3) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,S3) = groups3906332499630173760nt_rat(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7397_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,C4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C4)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C4)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C4,B2))
                 => ( aa_real_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,C4) = groups5754745047067104278omplex(H,C4) )
               => ( groups5754745047067104278omplex(G,A2) = groups5754745047067104278omplex(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7398_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,C4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C4)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C4)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C4,B2))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) )
               => ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7399_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A2))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B2))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) )
               => ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7400_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,C4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C4)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C4)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C4,B2))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) )
               => ( groups1300246762558778688al_rat(G,A2) = groups1300246762558778688al_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7401_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A2))
               => ( aa_complex_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B2))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,C4) = groups5058264527183730370ex_rat(H,C4) )
               => ( groups5058264527183730370ex_rat(G,A2) = groups5058264527183730370ex_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7402_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,C4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C4)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C4)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C4,B2))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,C4) = groups1935376822645274424al_nat(H,C4) )
               => ( groups1935376822645274424al_nat(G,A2) = groups1935376822645274424al_nat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7403_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A2))
               => ( aa_complex_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B2))
                 => ( aa_complex_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups5693394587270226106ex_nat(G,C4) = groups5693394587270226106ex_nat(H,C4) )
               => ( groups5693394587270226106ex_nat(G,A2) = groups5693394587270226106ex_nat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7404_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_int,H: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,C4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C4)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C4)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C4,B2))
                 => ( aa_real_int(H,B3) = zero_zero_int ) )
             => ( ( groups1932886352136224148al_int(G,C4) = groups1932886352136224148al_int(H,C4) )
               => ( groups1932886352136224148al_int(G,A2) = groups1932886352136224148al_int(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7405_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A2))
               => ( aa_complex_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B2))
                 => ( aa_complex_int(H,B3) = zero_zero_int ) )
             => ( ( groups5690904116761175830ex_int(G,C4) = groups5690904116761175830ex_int(H,C4) )
               => ( groups5690904116761175830ex_int(G,A2) = groups5690904116761175830ex_int(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7406_sum_Osame__carrierI,axiom,
    ! [C4: set_nat,A2: set_nat,B2: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,C4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C4)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C4)
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),minus_minus_set_nat(C4,A2))
               => ( aa_nat_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),minus_minus_set_nat(C4,B2))
                 => ( aa_nat_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups2073611262835488442omplex(G,C4) = groups2073611262835488442omplex(H,C4) )
               => ( groups2073611262835488442omplex(G,A2) = groups2073611262835488442omplex(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7407_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,C4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C4)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C4)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C4,B2))
                 => ( aa_real_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,A2) = groups5754745047067104278omplex(H,B2) )
              <=> ( groups5754745047067104278omplex(G,C4) = groups5754745047067104278omplex(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7408_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,C4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C4)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C4)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C4,B2))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B2) )
              <=> ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7409_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A2))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B2))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B2) )
              <=> ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7410_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,C4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C4)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C4)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C4,B2))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,A2) = groups1300246762558778688al_rat(H,B2) )
              <=> ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7411_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A2))
               => ( aa_complex_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B2))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,A2) = groups5058264527183730370ex_rat(H,B2) )
              <=> ( groups5058264527183730370ex_rat(G,C4) = groups5058264527183730370ex_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7412_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,C4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C4)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C4)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C4,B2))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,A2) = groups1935376822645274424al_nat(H,B2) )
              <=> ( groups1935376822645274424al_nat(G,C4) = groups1935376822645274424al_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7413_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A2))
               => ( aa_complex_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B2))
                 => ( aa_complex_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups5693394587270226106ex_nat(G,A2) = groups5693394587270226106ex_nat(H,B2) )
              <=> ( groups5693394587270226106ex_nat(G,C4) = groups5693394587270226106ex_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7414_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_int,H: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,C4)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C4)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C4)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C4,B2))
                 => ( aa_real_int(H,B3) = zero_zero_int ) )
             => ( ( groups1932886352136224148al_int(G,A2) = groups1932886352136224148al_int(H,B2) )
              <=> ( groups1932886352136224148al_int(G,C4) = groups1932886352136224148al_int(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7415_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),C4)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C4,A2))
               => ( aa_complex_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B2))
                 => ( aa_complex_int(H,B3) = zero_zero_int ) )
             => ( ( groups5690904116761175830ex_int(G,A2) = groups5690904116761175830ex_int(H,B2) )
              <=> ( groups5690904116761175830ex_int(G,C4) = groups5690904116761175830ex_int(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7416_sum_Osame__carrier,axiom,
    ! [C4: set_nat,A2: set_nat,B2: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,C4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C4)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C4)
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),minus_minus_set_nat(C4,A2))
               => ( aa_nat_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),minus_minus_set_nat(C4,B2))
                 => ( aa_nat_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups2073611262835488442omplex(G,A2) = groups2073611262835488442omplex(H,B2) )
              <=> ( groups2073611262835488442omplex(G,C4) = groups2073611262835488442omplex(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7417_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
       => ( groups5808333547571424918x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,B2))),groups5808333547571424918x_real(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_7418_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
       => ( groups5058264527183730370ex_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,B2))),groups5058264527183730370ex_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_7419_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
       => ( groups5693394587270226106ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,B2))),groups5693394587270226106ex_nat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_7420_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
       => ( groups5690904116761175830ex_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,B2))),groups5690904116761175830ex_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_7421_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( groups2906978787729119204at_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,minus_minus_set_nat(A2,B2))),groups2906978787729119204at_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_7422_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( groups3539618377306564664at_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,minus_minus_set_nat(A2,B2))),groups3539618377306564664at_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_7423_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_real] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A2)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ( groups8778361861064173332t_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G,minus_minus_set_int(A2,B2))),groups8778361861064173332t_real(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_7424_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A2)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ( groups3906332499630173760nt_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,B2))),groups3906332499630173760nt_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_7425_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A2)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ( groups4541462559716669496nt_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,B2))),groups4541462559716669496nt_nat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_7426_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A2)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ( groups4538972089207619220nt_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,minus_minus_set_int(A2,B2))),groups4538972089207619220nt_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_7427_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A2)
       => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B2)) ) ) ) ).

% sum_diff
tff(fact_7428_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A2)
       => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A2,B2)) = aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(F,B2)) ) ) ) ).

% sum_diff
tff(fact_7429_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A2)
       => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A2,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(F,B2)) ) ) ) ).

% sum_diff
tff(fact_7430_sum__diff,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
       => ( groups2906978787729119204at_rat(F,minus_minus_set_nat(A2,B2)) = aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(F,B2)) ) ) ) ).

% sum_diff
tff(fact_7431_sum__diff,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
       => ( groups3539618377306564664at_int(F,minus_minus_set_nat(A2,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,A2)),groups3539618377306564664at_int(F,B2)) ) ) ) ).

% sum_diff
tff(fact_7432_sum__diff,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),A2)
       => ( groups8778361861064173332t_real(F,minus_minus_set_int(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(F,B2)) ) ) ) ).

% sum_diff
tff(fact_7433_sum__diff,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),A2)
       => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A2,B2)) = aa_rat_rat(minus_minus_rat(groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(F,B2)) ) ) ) ).

% sum_diff
tff(fact_7434_sum__diff,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),A2)
       => ( groups4538972089207619220nt_int(F,minus_minus_set_int(A2,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(F,B2)) ) ) ) ).

% sum_diff
tff(fact_7435_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A2)
       => ( groups7754918857620584856omplex(F,minus_811609699411566653omplex(A2,B2)) = aa_complex_complex(minus_minus_complex(groups7754918857620584856omplex(F,A2)),groups7754918857620584856omplex(F,B2)) ) ) ) ).

% sum_diff
tff(fact_7436_sum__diff,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
       => ( groups6591440286371151544t_real(F,minus_minus_set_nat(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(F,A2)),groups6591440286371151544t_real(F,B2)) ) ) ) ).

% sum_diff
tff(fact_7437_of__int__leD,axiom,
    ! [Na: int,X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Na))),X)
     => ( ( Na = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),X) ) ) ).

% of_int_leD
tff(fact_7438_of__int__leD,axiom,
    ! [Na: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Na))),X)
     => ( ( Na = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ) ).

% of_int_leD
tff(fact_7439_of__int__leD,axiom,
    ! [Na: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Na))),X)
     => ( ( Na = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X) ) ) ).

% of_int_leD
tff(fact_7440_of__int__leD,axiom,
    ! [Na: int,X: 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,Na))),X)
     => ( ( Na = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),X) ) ) ).

% of_int_leD
tff(fact_7441_sum__diff__nat,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A2)
       => ( groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B2)) ) ) ) ).

% sum_diff_nat
tff(fact_7442_sum__diff__nat,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B2)
     => ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(B2),A2)
       => ( groups977919841031483927at_nat(F,minus_1356011639430497352at_nat(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A2)),groups977919841031483927at_nat(F,B2)) ) ) ) ).

% sum_diff_nat
tff(fact_7443_sum__diff__nat,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),A2)
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(F,B2)) ) ) ) ).

% sum_diff_nat
tff(fact_7444_sum__diff__nat,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B2)) ) ) ) ).

% sum_diff_nat
tff(fact_7445_of__int__lessD,axiom,
    ! [Na: int,X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Na))),X)
     => ( ( Na = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),X) ) ) ).

% of_int_lessD
tff(fact_7446_of__int__lessD,axiom,
    ! [Na: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Na))),X)
     => ( ( Na = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X) ) ) ).

% of_int_lessD
tff(fact_7447_of__int__lessD,axiom,
    ! [Na: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Na))),X)
     => ( ( Na = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),X) ) ) ).

% of_int_lessD
tff(fact_7448_of__int__lessD,axiom,
    ! [Na: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Na))),X)
     => ( ( Na = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),X) ) ) ).

% of_int_lessD
tff(fact_7449_sum__diff1__nat,axiom,
    ! [F: fun_Pr1253225528153422666at_nat,A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      groups3860910324918113789at_nat(F,minus_3314409938677909166at_nat(A2,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) = $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups3860910324918113789at_nat(F,A2)),aa_Pro6313114768868464763at_nat(F,A)),groups3860910324918113789at_nat(F,A2)) ).

% sum_diff1_nat
tff(fact_7450_sum__diff1__nat,axiom,
    ! [F: fun_set_nat_nat,A2: set_set_nat,A: set_nat] :
      groups8294997508430121362at_nat(F,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = $ite(aa_set_set_nat_o(member_set_nat(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A2)),aa_set_nat_nat(F,A)),groups8294997508430121362at_nat(F,A2)) ).

% sum_diff1_nat
tff(fact_7451_sum__diff1__nat,axiom,
    ! [F: fun_Pr2602117396471556196at_nat,A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      groups977919841031483927at_nat(F,minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A2)),aa_Pro4243173770044397973at_nat(F,A)),groups977919841031483927at_nat(F,A2)) ).

% sum_diff1_nat
tff(fact_7452_sum__diff1__nat,axiom,
    ! [F: fun_real_nat,A2: set_real,A: real] :
      groups1935376822645274424al_nat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A2)),aa_real_nat(F,A)),groups1935376822645274424al_nat(F,A2)) ).

% sum_diff1_nat
tff(fact_7453_sum__diff1__nat,axiom,
    ! [F: fun_int_nat,A2: set_int,A: int] :
      groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),aa_int_nat(F,A)),groups4541462559716669496nt_nat(F,A2)) ).

% sum_diff1_nat
tff(fact_7454_sum__diff1__nat,axiom,
    ! [F: fun_nat_nat,A2: set_nat,A: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_nat_nat(F,A)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) ).

% sum_diff1_nat
tff(fact_7455_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( groups2073611262835488442omplex(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups2073611262835488442omplex(F,set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7456_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_rat,K: nat] :
      ( ( aa_nat_rat(F,zero_zero_nat) = zero_zero_rat )
     => ( groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7457_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_int,K: nat] :
      ( ( aa_nat_int(F,zero_zero_nat) = zero_zero_int )
     => ( groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7458_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_nat,K: nat] :
      ( ( aa_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7459_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( groups6591440286371151544t_real(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups6591440286371151544t_real(F,set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7460_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,Na))),aa_nat_rat(G,aa_nat_nat(suc,Na))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7461_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,Na: nat] : groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,Na))),aa_nat_int(G,aa_nat_nat(suc,Na))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7462_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(zero_zero_nat,Na))),aa_nat_nat(G,aa_nat_nat(suc,Na))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7463_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,Na: nat] : groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(zero_zero_nat,Na))),aa_nat_real(G,aa_nat_nat(suc,Na))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7464_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Na: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,Na)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Mb)),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Na))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7465_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Na: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Mb)),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Na))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7466_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Na: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Na))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7467_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Na: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(Mb,Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,Mb)),groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Na))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7468_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Na: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Na))
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Na))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,aa_nat_nat(suc,Na))),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,Na))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7469_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Na: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Na))
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,aa_nat_nat(suc,Na))),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,Na))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7470_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Na: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Na))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,Na))),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,Na))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7471_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Na: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Na))
     => ( groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,aa_nat_nat(suc,Na))),groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(Mb,Na))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7472_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(aa_rea7340605660300375871l_real(minus_minus_real,Z),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Z))))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),aa_int_real(ring_1_of_int_real,Mb)))) ).

% round_diff_minimal
tff(fact_7473_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(minus_minus_rat(Z),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Z))))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z),aa_int_rat(ring_1_of_int_rat,Mb)))) ).

% round_diff_minimal
tff(fact_7474_zdvd__mult__cancel1,axiom,
    ! [Mb: int,Na: int] :
      ( ( Mb != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Na)),Mb)
      <=> ( aa_int_int(abs_abs_int,Na) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_7475_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Na: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,Na))),aa_nat_rat(G,aa_nat_nat(suc,Na))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Mb)),groups2906978787729119204at_rat(aTP_Lamm_og(G),set_or1269000886237332187st_nat(Mb,Na))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7476_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Na: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,Na))),aa_nat_int(G,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Mb)),groups3539618377306564664at_int(aTP_Lamm_oh(G),set_or1269000886237332187st_nat(Mb,Na))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7477_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Na: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,Na))),aa_nat_nat(G,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_nm(G)),set_or1269000886237332187st_nat(Mb,Na))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7478_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Na: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(Mb,Na))),aa_nat_real(G,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,Mb)),groups6591440286371151544t_real(aTP_Lamm_nn(G),set_or1269000886237332187st_nat(Mb,Na))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7479_sum__Suc__diff,axiom,
    ! [Mb: nat,Na: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Na))
     => ( groups2906978787729119204at_rat(aTP_Lamm_oi(F),set_or1269000886237332187st_nat(Mb,Na)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,Na))),aa_nat_rat(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_7480_sum__Suc__diff,axiom,
    ! [Mb: nat,Na: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Na))
     => ( groups3539618377306564664at_int(aTP_Lamm_oj(F),set_or1269000886237332187st_nat(Mb,Na)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,Na))),aa_nat_int(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_7481_sum__Suc__diff,axiom,
    ! [Mb: nat,Na: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Na))
     => ( groups6591440286371151544t_real(aTP_Lamm_ok(F),set_or1269000886237332187st_nat(Mb,Na)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,aa_nat_nat(suc,Na))),aa_nat_real(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_7482_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
       => ( ! [B3: real] :
              ( aa_set_real_o(member_real(B3),minus_minus_set_real(B2,A2))
             => 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,A2)),groups8097168146408367636l_real(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_7483_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B2,A2))
             => 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,A2)),groups5808333547571424918x_real(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_7484_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
       => ( ! [B3: real] :
              ( aa_set_real_o(member_real(B3),minus_minus_set_real(B2,A2))
             => 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,A2)),groups1300246762558778688al_rat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_7485_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B2,A2))
             => 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,A2)),groups5058264527183730370ex_rat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_7486_sum__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
       => ( ! [B3: nat] :
              ( aa_set_nat_o(member_nat(B3),minus_minus_set_nat(B2,A2))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,B3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_7487_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
       => ( ! [B3: real] :
              ( aa_set_real_o(member_real(B3),minus_minus_set_real(B2,A2))
             => 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,A2)),groups1935376822645274424al_nat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_7488_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B2,A2))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,B3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_7489_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
       => ( ! [B3: real] :
              ( aa_set_real_o(member_real(B3),minus_minus_set_real(B2,A2))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,B3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,A2)),groups1932886352136224148al_int(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_7490_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B2,A2))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,B3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_7491_sum__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
       => ( ! [B3: nat] :
              ( aa_set_nat_o(member_nat(B3),minus_minus_set_nat(B2,A2))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,B3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,A2)),groups3539618377306564664at_int(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_7492_abs__le__square__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),abs_abs_Code_integer(Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_7493_abs__le__square__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_7494_abs__le__square__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),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(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_7495_abs__le__square__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,X)),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(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_7496_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(X),A2)
       => ( groups5808333547571424918x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_7497_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(X),A2)
       => ( groups5058264527183730370ex_rat(G,A2) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,X)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_7498_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(X),A2)
       => ( groups5693394587270226106ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_7499_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(X),A2)
       => ( groups5690904116761175830ex_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,X)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_7500_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(X),A2)
       => ( groups8097168146408367636l_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X)),groups8097168146408367636l_real(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_7501_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(X),A2)
       => ( groups1300246762558778688al_rat(G,A2) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,X)),groups1300246762558778688al_rat(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_7502_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(X),A2)
       => ( groups1935376822645274424al_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X)),groups1935376822645274424al_nat(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_7503_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(X),A2)
       => ( groups1932886352136224148al_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(G,X)),groups1932886352136224148al_int(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_7504_sum_Oremove,axiom,
    ! [A2: set_int,X: int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(X),A2)
       => ( groups8778361861064173332t_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X)),groups8778361861064173332t_real(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% sum.remove
tff(fact_7505_sum_Oremove,axiom,
    ! [A2: set_int,X: int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(X),A2)
       => ( groups3906332499630173760nt_rat(G,A2) = aa_rat_rat(plus_plus_rat(aa_int_rat(G,X)),groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% sum.remove
tff(fact_7506_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_real,X: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(G,insert_complex(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_7507_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_rat,X: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(G,insert_complex(X,A2)) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,X)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_7508_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_nat,X: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5693394587270226106ex_nat(G,insert_complex(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_7509_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_int,X: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5690904116761175830ex_int(G,insert_complex(X,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,X)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_7510_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_real,X: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups8097168146408367636l_real(G,insert_real(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X)),groups8097168146408367636l_real(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_7511_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_rat,X: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1300246762558778688al_rat(G,insert_real(X,A2)) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,X)),groups1300246762558778688al_rat(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_7512_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_nat,X: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1935376822645274424al_nat(G,insert_real(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X)),groups1935376822645274424al_nat(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_7513_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_int,X: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1932886352136224148al_int(G,insert_real(X,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(G,X)),groups1932886352136224148al_int(G,minus_minus_set_real(A2,insert_real(X,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_7514_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: fun_int_real,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups8778361861064173332t_real(G,insert_int(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X)),groups8778361861064173332t_real(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ).

% sum.insert_remove
tff(fact_7515_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: fun_int_rat,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(G,insert_int(X,A2)) = aa_rat_rat(plus_plus_rat(aa_int_rat(G,X)),groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ).

% sum.insert_remove
tff(fact_7516_sum__diff1,axiom,
    ! [A2: set_complex,F: fun_complex_real,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(F,A2)),aa_complex_real(F,A)),groups5808333547571424918x_real(F,A2)) ) ) ).

% sum_diff1
tff(fact_7517_sum__diff1,axiom,
    ! [A2: set_real,F: fun_real_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups8097168146408367636l_real(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8097168146408367636l_real(F,A2)),aa_real_real(F,A)),groups8097168146408367636l_real(F,A2)) ) ) ).

% sum_diff1
tff(fact_7518_sum__diff1,axiom,
    ! [A2: set_int,F: fun_int_real,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups8778361861064173332t_real(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(F,A2)),aa_int_real(F,A)),groups8778361861064173332t_real(F,A2)) ) ) ).

% sum_diff1
tff(fact_7519_sum__diff1,axiom,
    ! [A2: set_complex,F: fun_complex_rat,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A2),aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(F,A2)),aa_complex_rat(F,A)),groups5058264527183730370ex_rat(F,A2)) ) ) ).

% sum_diff1
tff(fact_7520_sum__diff1,axiom,
    ! [A2: set_real,F: fun_real_rat,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1300246762558778688al_rat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A2),aa_rat_rat(minus_minus_rat(groups1300246762558778688al_rat(F,A2)),aa_real_rat(F,A)),groups1300246762558778688al_rat(F,A2)) ) ) ).

% sum_diff1
tff(fact_7521_sum__diff1,axiom,
    ! [A2: set_int,F: fun_int_rat,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A2),aa_rat_rat(minus_minus_rat(groups3906332499630173760nt_rat(F,A2)),aa_int_rat(F,A)),groups3906332499630173760nt_rat(F,A2)) ) ) ).

% sum_diff1
tff(fact_7522_sum__diff1,axiom,
    ! [A2: set_complex,F: fun_complex_int,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A2)),aa_complex_int(F,A)),groups5690904116761175830ex_int(F,A2)) ) ) ).

% sum_diff1
tff(fact_7523_sum__diff1,axiom,
    ! [A2: set_real,F: fun_real_int,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1932886352136224148al_int(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups1932886352136224148al_int(F,A2)),aa_real_int(F,A)),groups1932886352136224148al_int(F,A2)) ) ) ).

% sum_diff1
tff(fact_7524_sum__diff1,axiom,
    ! [A2: set_int,F: fun_int_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4538972089207619220nt_int(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A2)),aa_int_int(F,A)),groups4538972089207619220nt_int(F,A2)) ) ) ).

% sum_diff1
tff(fact_7525_sum__diff1,axiom,
    ! [A2: set_nat,F: fun_nat_rat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups2906978787729119204at_rat(F,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat(A),A2),aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(F,A2)),aa_nat_rat(F,A)),groups2906978787729119204at_rat(F,A2)) ) ) ).

% sum_diff1
tff(fact_7526_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Na: 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,Na),one_one_nat))
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P2))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,Na))),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7527_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Na: 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,Na),one_one_nat))
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,Na))),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7528_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Na: 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,Na),one_one_nat))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,Na))),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7529_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Na: 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,Na),one_one_nat))
     => ( groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(Mb,Na))),groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7530_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real,C: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_ol(A),B),C),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(B,A)),groups5808333547571424918x_real(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))),groups5808333547571424918x_real(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_7531_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_rat,C: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_om(A),B),C),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_rat_rat(plus_plus_rat(aa_complex_rat(B,A)),groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))),groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_7532_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_nat,C: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_on(A),B),C),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(B,A)),groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))),groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_7533_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_int,C: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_oo(A),B),C),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(B,A)),groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))),groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_7534_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: fun_real_real,C: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_op(A),B),C),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(B,A)),groups8097168146408367636l_real(C,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real)))),groups8097168146408367636l_real(C,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real)))) ) ) ).

% sum.delta_remove
tff(fact_7535_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: fun_real_rat,C: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_oq(A),B),C),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_rat_rat(plus_plus_rat(aa_real_rat(B,A)),groups1300246762558778688al_rat(C,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real)))),groups1300246762558778688al_rat(C,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real)))) ) ) ).

% sum.delta_remove
tff(fact_7536_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: fun_real_nat,C: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_or(A),B),C),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(B,A)),groups1935376822645274424al_nat(C,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real)))),groups1935376822645274424al_nat(C,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real)))) ) ) ).

% sum.delta_remove
tff(fact_7537_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: fun_real_int,C: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_os(A),B),C),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(B,A)),groups1932886352136224148al_int(C,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real)))),groups1932886352136224148al_int(C,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real)))) ) ) ).

% sum.delta_remove
tff(fact_7538_sum_Odelta__remove,axiom,
    ! [S3: set_int,A: int,B: fun_int_real,C: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_ot(A),B),C),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(B,A)),groups8778361861064173332t_real(C,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int)))),groups8778361861064173332t_real(C,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int)))) ) ) ).

% sum.delta_remove
tff(fact_7539_sum_Odelta__remove,axiom,
    ! [S3: set_int,A: int,B: fun_int_rat,C: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_ou(A),B),C),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_rat_rat(plus_plus_rat(aa_int_rat(B,A)),groups3906332499630173760nt_rat(C,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int)))),groups3906332499630173760nt_rat(C,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int)))) ) ) ).

% sum.delta_remove
tff(fact_7540_sum__count__set,axiom,
    ! [Xs: list_complex,X5: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Xs)),X5)
     => ( aa_set_complex_o(finite3207457112153483333omplex,X5)
       => ( groups5693394587270226106ex_nat(count_list_complex(Xs),X5) = size_s3451745648224563538omplex(Xs) ) ) ) ).

% sum_count_set
tff(fact_7541_sum__count__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat,X5: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Xs)),X5)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,X5)
       => ( groups977919841031483927at_nat(count_4203492906077236349at_nat(Xs),X5) = size_s5460976970255530739at_nat(Xs) ) ) ) ).

% sum_count_set
tff(fact_7542_sum__count__set,axiom,
    ! [Xs: list_VEBT_VEBT,X5: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),X5)
     => ( finite5795047828879050333T_VEBT(X5)
       => ( groups771621172384141258BT_nat(count_list_VEBT_VEBT(Xs),X5) = aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Xs) ) ) ) ).

% sum_count_set
tff(fact_7543_sum__count__set,axiom,
    ! [Xs: list_o,X5: set_o] :
      ( ord_less_eq_set_o(set_o2(Xs),X5)
     => ( finite_finite_o(X5)
       => ( groups8507830703676809646_o_nat(count_list_o(Xs),X5) = aa_list_o_nat(size_size_list_o,Xs) ) ) ) ).

% sum_count_set
tff(fact_7544_sum__count__set,axiom,
    ! [Xs: list_int,X5: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),X5)
     => ( aa_set_int_o(finite_finite_int,X5)
       => ( groups4541462559716669496nt_nat(count_list_int(Xs),X5) = aa_list_int_nat(size_size_list_int,Xs) ) ) ) ).

% sum_count_set
tff(fact_7545_sum__count__set,axiom,
    ! [Xs: list_nat,X5: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Xs)),X5)
     => ( aa_set_nat_o(finite_finite_nat,X5)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(count_list_nat(Xs)),X5) = aa_list_nat_nat(size_size_list_nat,Xs) ) ) ) ).

% sum_count_set
tff(fact_7546_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
       => ( aa_set_real_o(member_real(B),minus_minus_set_real(B2,A2))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,B))
           => ( ! [X3: real] :
                  ( aa_set_real_o(member_real(X3),B2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7547_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B2,A2))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,B))
           => ( ! [X3: complex] :
                  ( aa_set_complex_o(member_complex(X3),B2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X3)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7548_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
       => ( aa_set_real_o(member_real(B),minus_minus_set_real(B2,A2))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,B))
           => ( ! [X3: real] :
                  ( aa_set_real_o(member_real(X3),B2)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X3)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(F,A2)),groups1300246762558778688al_rat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7549_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B2,A2))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,B))
           => ( ! [X3: complex] :
                  ( aa_set_complex_o(member_complex(X3),B2)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X3)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7550_sum__strict__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,B: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
       => ( aa_set_nat_o(member_nat(B),minus_minus_set_nat(B2,A2))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,B))
           => ( ! [X3: nat] :
                  ( aa_set_nat_o(member_nat(X3),B2)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7551_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
       => ( aa_set_real_o(member_real(B),minus_minus_set_real(B2,A2))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,B))
           => ( ! [X3: real] :
                  ( aa_set_real_o(member_real(X3),B2)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X3)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7552_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B2,A2))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,B))
           => ( ! [X3: complex] :
                  ( aa_set_complex_o(member_complex(X3),B2)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X3)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7553_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
       => ( aa_set_real_o(member_real(B),minus_minus_set_real(B2,A2))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(F,B))
           => ( ! [X3: real] :
                  ( aa_set_real_o(member_real(X3),B2)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,X3)) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,groups1932886352136224148al_int(F,A2)),groups1932886352136224148al_int(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7554_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B2,A2))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_complex_int(F,B))
           => ( ! [X3: complex] :
                  ( aa_set_complex_o(member_complex(X3),B2)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,X3)) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7555_sum__strict__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,B: nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
       => ( aa_set_nat_o(member_nat(B),minus_minus_set_nat(B2,A2))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,B))
           => ( ! [X3: nat] :
                  ( aa_set_nat_o(member_nat(X3),B2)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X3)) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,A2)),groups3539618377306564664at_int(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7556_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(member_complex(I),A2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(A2,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,X3)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I)),groups5808333547571424918x_real(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_7557_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: fun_real_real] :
      ( aa_set_real_o(member_real(I),A2)
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),minus_minus_set_real(A2,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,X3)) )
       => ( aa_set_real_o(finite_finite_real,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I)),groups8097168146408367636l_real(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_7558_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: fun_int_real] :
      ( aa_set_int_o(member_int(I),A2)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),minus_minus_set_int(A2,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,X3)) )
       => ( aa_set_int_o(finite_finite_int,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I)),groups8778361861064173332t_real(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_7559_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(member_complex(I),A2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(A2,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,X3)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I)),groups5058264527183730370ex_rat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_7560_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: fun_real_rat] :
      ( aa_set_real_o(member_real(I),A2)
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),minus_minus_set_real(A2,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,X3)) )
       => ( aa_set_real_o(finite_finite_real,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I)),groups1300246762558778688al_rat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_7561_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(member_int(I),A2)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),minus_minus_set_int(A2,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,X3)) )
       => ( aa_set_int_o(finite_finite_int,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I)),groups3906332499630173760nt_rat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_7562_member__le__sum,axiom,
    ! [I: nat,A2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(member_nat(I),A2)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(A2,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,X3)) )
       => ( aa_set_nat_o(finite_finite_nat,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I)),groups2906978787729119204at_rat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_7563_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(member_complex(I),A2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(A2,insert_complex(I,bot_bot_set_complex)))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X3)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I)),groups5693394587270226106ex_nat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_7564_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: fun_real_nat] :
      ( aa_set_real_o(member_real(I),A2)
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),minus_minus_set_real(A2,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,X3)) )
       => ( aa_set_real_o(finite_finite_real,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I)),groups1935376822645274424al_nat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_7565_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(member_int(I),A2)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),minus_minus_set_int(A2,insert_int(I,bot_bot_set_int)))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X3)) )
       => ( aa_set_int_o(finite_finite_int,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I)),groups4541462559716669496nt_nat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_7566_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,X: 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(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7567_power2__le__iff__abs__le,axiom,
    ! [Y: real,X: 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(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_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,X)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7568_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X: 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(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7569_power2__le__iff__abs__le,axiom,
    ! [Y: int,X: 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(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,X)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7570_abs__sqrt__wlog,axiom,
    ! [P: fun_Co2081214437782444048eger_o,X: code_integer] :
      ( ! [X3: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,X3),aa_nat_Code_integer(power_8256067586552552935nteger(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,abs_abs_Code_integer(X)),aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7571_abs__sqrt__wlog,axiom,
    ! [P: fun_real_fun_real_o,X: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
         => aa_real_o(aa_real_fun_real_o(P,X3),aa_nat_real(aa_real_fun_nat_real(power_power_real,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_real_o(aa_real_fun_real_o(P,aa_real_real(abs_abs_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7572_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,X: rat] :
      ( ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X3)
         => aa_rat_o(aa_rat_fun_rat_o(P,X3),aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_rat_o(aa_rat_fun_rat_o(P,aa_rat_rat(abs_abs_rat,X)),aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7573_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_o,X: int] :
      ( ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
         => aa_int_o(aa_int_fun_int_o(P,X3),aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_int_o(aa_int_fun_int_o(P,aa_int_int(abs_abs_int,X)),aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7574_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),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(X)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_7575_abs__square__le__1,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),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,X)),one_one_real) ) ).

% abs_square_le_1
tff(fact_7576_abs__square__le__1,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_7577_abs__square__le__1,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,X)),one_one_int) ) ).

% abs_square_le_1
tff(fact_7578_abs__square__less__1,axiom,
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),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(X)),one_one_Code_integer) ) ).

% abs_square_less_1
tff(fact_7579_abs__square__less__1,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),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,X)),one_one_real) ) ).

% abs_square_less_1
tff(fact_7580_abs__square__less__1,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(X),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,X)),one_one_rat) ) ).

% abs_square_less_1
tff(fact_7581_abs__square__less__1,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(X),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,X)),one_one_int) ) ).

% abs_square_less_1
tff(fact_7582_power__mono__even,axiom,
    ! [Na: 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))),Na)
     => ( 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),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Na)) ) ) ).

% power_mono_even
tff(fact_7583_power__mono__even,axiom,
    ! [Na: 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))),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na)) ) ) ).

% power_mono_even
tff(fact_7584_power__mono__even,axiom,
    ! [Na: 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))),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na)) ) ) ).

% power_mono_even
tff(fact_7585_power__mono__even,axiom,
    ! [Na: 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))),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na)) ) ) ).

% power_mono_even
tff(fact_7586_nat__intermed__int__val,axiom,
    ! [Mb: nat,Na: nat,F: fun_nat_int,K: 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),Na) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I3))),aa_nat_int(F,I3)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Mb)),K)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
           => ? [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),Na)
                & ( aa_nat_int(F,I3) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_7587_incr__lemma,axiom,
    ! [D: int,Z: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D))) ) ).

% incr_lemma
tff(fact_7588_decr__lemma,axiom,
    ! [D: int,X: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D))),Z) ) ).

% decr_lemma
tff(fact_7589_sum__natinterval__diff,axiom,
    ! [F: fun_nat_complex,Mb: nat,Na: nat] :
      groups2073611262835488442omplex(aTP_Lamm_ov(F),set_or1269000886237332187st_nat(Mb,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na),aa_complex_complex(minus_minus_complex(aa_nat_complex(F,Mb)),aa_nat_complex(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))),zero_zero_complex) ).

% sum_natinterval_diff
tff(fact_7590_sum__natinterval__diff,axiom,
    ! [F: fun_nat_rat,Mb: nat,Na: nat] :
      groups2906978787729119204at_rat(aTP_Lamm_ow(F),set_or1269000886237332187st_nat(Mb,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na),aa_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,Na),one_one_nat))),zero_zero_rat) ).

% sum_natinterval_diff
tff(fact_7591_sum__natinterval__diff,axiom,
    ! [F: fun_nat_int,Mb: nat,Na: nat] :
      groups3539618377306564664at_int(aTP_Lamm_ox(F),set_or1269000886237332187st_nat(Mb,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na),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,Na),one_one_nat))),zero_zero_int) ).

% sum_natinterval_diff
tff(fact_7592_sum__natinterval__diff,axiom,
    ! [F: fun_nat_real,Mb: nat,Na: nat] :
      groups6591440286371151544t_real(aTP_Lamm_oy(F),set_or1269000886237332187st_nat(Mb,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,Mb)),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))),zero_zero_real) ).

% sum_natinterval_diff
tff(fact_7593_sum__telescope_H_H,axiom,
    ! [Mb: nat,Na: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( groups2906978787729119204at_rat(aTP_Lamm_oz(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Na)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,Na)),aa_nat_rat(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_7594_sum__telescope_H_H,axiom,
    ! [Mb: nat,Na: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( groups3539618377306564664at_int(aTP_Lamm_pa(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Na)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Na)),aa_nat_int(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_7595_sum__telescope_H_H,axiom,
    ! [Mb: nat,Na: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( groups6591440286371151544t_real(aTP_Lamm_pb(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Na)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,Na)),aa_nat_real(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_7596_mask__eq__sum__exp,axiom,
    ! [Na: 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))),Na)),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_by,Na))) ).

% mask_eq_sum_exp
tff(fact_7597_mask__eq__sum__exp,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),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_by,Na))) ).

% mask_eq_sum_exp
tff(fact_7598_nat__ivt__aux,axiom,
    ! [Na: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,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)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
         => ? [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Na)
              & ( aa_nat_int(F,I3) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_7599_divmod__nat__def,axiom,
    ! [Mb: nat,Na: nat] : divmod_nat(Mb,Na) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Na)) ).

% divmod_nat_def
tff(fact_7600_sum__gp__multiplied,axiom,
    ! [Mb: nat,Na: nat,X: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(power_power_complex(X),set_or1269000886237332187st_nat(Mb,Na))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),Mb)),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,Na))) ) ) ).

% sum_gp_multiplied
tff(fact_7601_sum__gp__multiplied,axiom,
    ! [Mb: nat,Na: nat,X: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X)),groups2906978787729119204at_rat(power_power_rat(X),set_or1269000886237332187st_nat(Mb,Na))) = aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),Mb)),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,Na))) ) ) ).

% sum_gp_multiplied
tff(fact_7602_sum__gp__multiplied,axiom,
    ! [Mb: nat,Na: nat,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),groups3539618377306564664at_int(power_power_int(X),set_or1269000886237332187st_nat(Mb,Na))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),Mb)),aa_nat_int(power_power_int(X),aa_nat_nat(suc,Na))) ) ) ).

% sum_gp_multiplied
tff(fact_7603_sum__gp__multiplied,axiom,
    ! [Mb: nat,Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),set_or1269000886237332187st_nat(Mb,Na))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,Na))) ) ) ).

% sum_gp_multiplied
tff(fact_7604_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,Mb: nat,Na: nat] : groups2906978787729119204at_rat(G,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))),Na)))) = groups2906978787729119204at_rat(aTP_Lamm_pc(G),set_or1269000886237332187st_nat(Mb,Na)) ).

% sum.in_pairs
tff(fact_7605_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,Mb: nat,Na: nat] : groups3539618377306564664at_int(G,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))),Na)))) = groups3539618377306564664at_int(aTP_Lamm_pd(G),set_or1269000886237332187st_nat(Mb,Na)) ).

% sum.in_pairs
tff(fact_7606_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,Mb: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),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))),Na)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pe(G)),set_or1269000886237332187st_nat(Mb,Na)) ).

% sum.in_pairs
tff(fact_7607_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,Mb: nat,Na: nat] : groups6591440286371151544t_real(G,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))),Na)))) = groups6591440286371151544t_real(aTP_Lamm_pf(G),set_or1269000886237332187st_nat(Mb,Na)) ).

% sum.in_pairs
tff(fact_7608_nat0__intermed__int__val,axiom,
    ! [Na: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,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)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
         => ? [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Na)
              & ( aa_nat_int(F,I3) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_7609_mask__eq__sum__exp__nat,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),aa_nat_nat(suc,zero_zero_nat)) = 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_by,Na))) ).

% mask_eq_sum_exp_nat
tff(fact_7610_gauss__sum__nat,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_dr),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_7611_of__int__round__abs__le,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X))),X))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_7612_of__int__round__abs__le,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))),X))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_7613_listrel1p__def,axiom,
    ! [R2: fun_int_fun_int_o,Xs: list_int,Ys: list_int] :
      ( aa_list_int_o(listrel1p_int(R2,Xs),Ys)
    <=> member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),listrel1_int(collec213857154873943460nt_int(produc4947309494688390418_int_o(R2)))) ) ).

% listrel1p_def
tff(fact_7614_lemma__interval,axiom,
    ! [A: real,X: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),B)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [Y4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y4))),D4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),B) ) ) ) ) ) ).

% lemma_interval
tff(fact_7615_sum__gp,axiom,
    ! [X: complex,Mb: nat,Na: nat] :
      groups2073611262835488442omplex(power_power_complex(X),set_or1269000886237332187st_nat(Mb,Na)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb),
        zero_zero_complex,
        $ite(X = 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,Na),one_one_nat)),Mb)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),Mb)),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,Na)))),aa_complex_complex(minus_minus_complex(one_one_complex),X))) ) ).

% sum_gp
tff(fact_7616_sum__gp,axiom,
    ! [X: rat,Mb: nat,Na: nat] :
      groups2906978787729119204at_rat(power_power_rat(X),set_or1269000886237332187st_nat(Mb,Na)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb),
        zero_zero_rat,
        $ite(X = 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,Na),one_one_nat)),Mb)),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),Mb)),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,Na)))),aa_rat_rat(minus_minus_rat(one_one_rat),X))) ) ).

% sum_gp
tff(fact_7617_sum__gp,axiom,
    ! [X: real,Mb: nat,Na: nat] :
      groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),set_or1269000886237332187st_nat(Mb,Na)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb),
        zero_zero_real,
        $ite(X = 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,Na),one_one_nat)),Mb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,Na)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X))) ) ).

% sum_gp
tff(fact_7618_gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Na),one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7619_gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7620_gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7621_lemma__interval__lt,axiom,
    ! [A: real,X: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),B)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [Y4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y4))),D4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),B) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_7622_arctan__double,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,X)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_7623_of__nat__eq__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( Mb = Na ) ) ).

% of_nat_eq_iff
tff(fact_7624_of__nat__eq__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = aa_nat_real(semiri5074537144036343181t_real,Na) )
    <=> ( Mb = Na ) ) ).

% of_nat_eq_iff
tff(fact_7625_of__nat__eq__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Mb) = aa_nat_nat(semiri1316708129612266289at_nat,Na) )
    <=> ( Mb = Na ) ) ).

% of_nat_eq_iff
tff(fact_7626_split__part,axiom,
    ! [P: $o,Q: fun_int_fun_int_o,X2: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_pg((P)),Q)),X2)
    <=> ( (P)
        & aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Q),X2) ) ) ).

% split_part
tff(fact_7627_int__eq__iff__numeral,axiom,
    ! [Mb: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_num_int(numeral_numeral_int,V) )
    <=> ( Mb = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_7628_abs__of__nat,axiom,
    ! [Na: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)) = aa_nat_rat(semiri681578069525770553at_rat,Na) ).

% abs_of_nat
tff(fact_7629_abs__of__nat,axiom,
    ! [Na: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) = aa_nat_Code_integer(semiri4939895301339042750nteger,Na) ).

% abs_of_nat
tff(fact_7630_abs__of__nat,axiom,
    ! [Na: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_int(semiri1314217659103216013at_int,Na) ).

% abs_of_nat
tff(fact_7631_abs__of__nat,axiom,
    ! [Na: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,Na)) = aa_nat_real(semiri5074537144036343181t_real,Na) ).

% abs_of_nat
tff(fact_7632_negative__eq__positive,axiom,
    ! [Na: nat,Mb: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_int(semiri1314217659103216013at_int,Mb) )
    <=> ( ( Na = zero_zero_nat )
        & ( Mb = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_7633_of__int__of__nat__eq,axiom,
    ! [Na: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_rat(semiri681578069525770553at_rat,Na) ).

% of_int_of_nat_eq
tff(fact_7634_of__int__of__nat__eq,axiom,
    ! [Na: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_int(semiri1314217659103216013at_int,Na) ).

% of_int_of_nat_eq
tff(fact_7635_of__int__of__nat__eq,axiom,
    ! [Na: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_real(semiri5074537144036343181t_real,Na) ).

% of_int_of_nat_eq
tff(fact_7636_negative__zle,axiom,
    ! [Na: nat,Mb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% negative_zle
tff(fact_7637_int__dvd__int__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Na) ) ).

% int_dvd_int_iff
tff(fact_7638_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Mb) = zero_zero_complex )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7639_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_7640_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_7641_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_7642_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_7643_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_7644_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_7645_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_7646_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_7647_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_7648_of__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% of_nat_0
tff(fact_7649_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_7650_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_7651_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_7652_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_7653_of__nat__less__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ).

% of_nat_less_iff
tff(fact_7654_of__nat__less__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ).

% of_nat_less_iff
tff(fact_7655_of__nat__less__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ).

% of_nat_less_iff
tff(fact_7656_of__nat__less__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ).

% of_nat_less_iff
tff(fact_7657_of__nat__le__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% of_nat_le_iff
tff(fact_7658_of__nat__le__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% of_nat_le_iff
tff(fact_7659_of__nat__le__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% of_nat_le_iff
tff(fact_7660_of__nat__le__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% of_nat_le_iff
tff(fact_7661_of__nat__add,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ).

% of_nat_add
tff(fact_7662_of__nat__add,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_add
tff(fact_7663_of__nat__add,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% of_nat_add
tff(fact_7664_of__nat__add,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_add
tff(fact_7665_of__nat__mult,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri8010041392384452111omplex,Na)) ).

% of_nat_mult
tff(fact_7666_of__nat__mult,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ).

% of_nat_mult
tff(fact_7667_of__nat__mult,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_mult
tff(fact_7668_of__nat__mult,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% of_nat_mult
tff(fact_7669_of__nat__mult,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_mult
tff(fact_7670_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_7671_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_7672_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_7673_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_7674_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_7675_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7676_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7677_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7678_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7679_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7680_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Na) = one_one_complex )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7681_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Na) = one_one_rat )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7682_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Na) = one_one_int )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7683_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Na) = one_one_real )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7684_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Na) = one_one_nat )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7685_negative__zless,axiom,
    ! [Na: 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,Na)))),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% negative_zless
tff(fact_7686_arctan__less__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,X)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real) ) ).

% arctan_less_zero_iff
tff(fact_7687_zero__less__arctan__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arctan,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X) ) ).

% zero_less_arctan_iff
tff(fact_7688_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_real(semiri5074537144036343181t_real,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = zero_n3304061248610475627l_real((P)) ).

% of_nat_of_bool
tff(fact_7689_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_7690_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_7691_of__nat__sum,axiom,
    ! [F: fun_complex_nat,A2: set_complex] : aa_nat_complex(semiri8010041392384452111omplex,groups5693394587270226106ex_nat(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_ph(F),A2) ).

% of_nat_sum
tff(fact_7692_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups3539618377306564664at_int(aTP_Lamm_pi(F),A2) ).

% of_nat_sum
tff(fact_7693_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pj(F)),A2) ).

% of_nat_sum
tff(fact_7694_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups6591440286371151544t_real(aTP_Lamm_pk(F),A2) ).

% of_nat_sum
tff(fact_7695_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_7696_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_7697_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_7698_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_7699_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_7700_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Mb)) = aa_rat_rat(plus_plus_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Mb)) ).

% of_nat_Suc
tff(fact_7701_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_7702_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Mb)) ).

% of_nat_Suc
tff(fact_7703_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_7704_real__of__nat__less__numeral__iff,axiom,
    ! [Na: nat,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_num_real(numeral_numeral_real,W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(numeral_numeral_nat,W2)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_7705_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W2)),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,W2)),Na) ) ).

% numeral_less_real_of_nat_iff
tff(fact_7706_numeral__le__real__of__nat__iff,axiom,
    ! [Na: num,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Na)),aa_nat_real(semiri5074537144036343181t_real,Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Na)),Mb) ) ).

% numeral_le_real_of_nat_iff
tff(fact_7707_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_7708_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_7709_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_7710_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_7711_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7712_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7713_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7714_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7715_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)),aa_nat_rat(semiri681578069525770553at_rat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7716_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2)),aa_nat_int(semiri1314217659103216013at_int,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7717_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W2)),aa_nat_real(semiri5074537144036343181t_real,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7718_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: 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)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7719_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7720_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7721_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7722_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7723_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W2)),aa_nat_real(semiri5074537144036343181t_real,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7724_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: 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)),W2)),aa_nat_rat(semiri681578069525770553at_rat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7725_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: 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)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7726_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: 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)),W2)),aa_nat_int(semiri1314217659103216013at_int,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7727_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7728_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,X)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7729_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,X)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7730_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7731_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,X: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Na)),aa_nat_rat(semiri681578069525770553at_rat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7732_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,X: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Na)),aa_nat_int(semiri1314217659103216013at_int,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7733_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,X: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Na)),aa_nat_real(semiri5074537144036343181t_real,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7734_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),aa_nat_nat(semiri1316708129612266289at_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7735_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7736_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7737_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7738_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7739_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7740_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7741_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7742_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7743_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,X: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Na)),aa_nat_real(semiri5074537144036343181t_real,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7744_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,X: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Na)),aa_nat_rat(semiri681578069525770553at_rat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7745_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),aa_nat_nat(semiri1316708129612266289at_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7746_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,X: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Na)),aa_nat_int(semiri1314217659103216013at_int,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7747_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_pl),Prod) ).

% prod.disc_eq_case
tff(fact_7748_int__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups3539618377306564664at_int(aTP_Lamm_pi(F),A2) ).

% int_sum
tff(fact_7749_real__arch__simple,axiom,
    ! [X: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% real_arch_simple
tff(fact_7750_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% real_arch_simple
tff(fact_7751_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% reals_Archimedean2
tff(fact_7752_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% reals_Archimedean2
tff(fact_7753_mult__of__nat__commute,axiom,
    ! [X: nat,Y: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,X)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(semiri8010041392384452111omplex,X)) ).

% mult_of_nat_commute
tff(fact_7754_mult__of__nat__commute,axiom,
    ! [X: nat,Y: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(semiri681578069525770553at_rat,X)) ).

% mult_of_nat_commute
tff(fact_7755_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,X)) ).

% mult_of_nat_commute
tff(fact_7756_mult__of__nat__commute,axiom,
    ! [X: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(semiri5074537144036343181t_real,X)) ).

% mult_of_nat_commute
tff(fact_7757_mult__of__nat__commute,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,X)) ).

% mult_of_nat_commute
tff(fact_7758_int__cases2,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,N)) ) ).

% int_cases2
tff(fact_7759_arctan__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ).

% arctan_less_iff
tff(fact_7760_arctan__monotone,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y)) ) ).

% arctan_monotone
tff(fact_7761_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M: nat,N: nat] : Z != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% int_diff_cases
tff(fact_7762_of__nat__less__of__int__iff,axiom,
    ! [Na: nat,X: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_int_rat(ring_1_of_int_rat,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),X) ) ).

% of_nat_less_of_int_iff
tff(fact_7763_of__nat__less__of__int__iff,axiom,
    ! [Na: nat,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(ring_1_of_int_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),X) ) ).

% of_nat_less_of_int_iff
tff(fact_7764_of__nat__less__of__int__iff,axiom,
    ! [Na: nat,X: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_int_real(ring_1_of_int_real,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),X) ) ).

% of_nat_less_of_int_iff
tff(fact_7765_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% of_nat_0_le_iff
tff(fact_7766_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Na)) ).

% of_nat_0_le_iff
tff(fact_7767_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_0_le_iff
tff(fact_7768_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_0_le_iff
tff(fact_7769_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_7770_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_7771_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_7772_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_7773_of__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Na)) != zero_zero_complex ).

% of_nat_neq_0
tff(fact_7774_of__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Na)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_7775_of__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_7776_of__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_7777_of__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Na)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_7778_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ).

% of_nat_less_imp_less
tff(fact_7779_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ).

% of_nat_less_imp_less
tff(fact_7780_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ).

% of_nat_less_imp_less
tff(fact_7781_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ).

% of_nat_less_imp_less
tff(fact_7782_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ) ).

% less_imp_of_nat_less
tff(fact_7783_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% less_imp_of_nat_less
tff(fact_7784_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ).

% less_imp_of_nat_less
tff(fact_7785_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ) ).

% less_imp_of_nat_less
tff(fact_7786_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_7787_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_7788_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_7789_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_7790_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_7791_nat__int__comparison_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(2)
tff(fact_7792_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_7793_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_7794_zle__int,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% zle_int
tff(fact_7795_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_7796_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ? [N: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N) ) ).

% zero_le_imp_eq_int
tff(fact_7797_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ~ ! [N: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N) ) ).

% nonneg_int_cases
tff(fact_7798_zadd__int__left,axiom,
    ! [Mb: nat,Na: 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,Na)),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),Na))),Z) ).

% zadd_int_left
tff(fact_7799_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_7800_int__plus,axiom,
    ! [Na: nat,Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% int_plus
tff(fact_7801_zle__iff__zadd,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z)
    <=> ? [N3: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,N3)) ) ).

% zle_iff_zadd
tff(fact_7802_not__int__zless__negative,axiom,
    ! [Na: nat,Mb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Mb))) ).

% not_int_zless_negative
tff(fact_7803_nat__less__as__int,axiom,
    ! [X2: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_less_as_int
tff(fact_7804_nat__leq__as__int,axiom,
    ! [X2: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_leq_as_int
tff(fact_7805_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),X)) ) ).

% ex_less_of_nat_mult
tff(fact_7806_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X)) ) ).

% ex_less_of_nat_mult
tff(fact_7807_of__nat__diff,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ) ) ).

% of_nat_diff
tff(fact_7808_of__nat__diff,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ) ).

% of_nat_diff
tff(fact_7809_of__nat__diff,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% of_nat_diff
tff(fact_7810_of__nat__diff,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ) ) ).

% of_nat_diff
tff(fact_7811_reals__Archimedean3,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ! [Y4: real] :
        ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X)) ) ).

% reals_Archimedean3
tff(fact_7812_int__cases4,axiom,
    ! [Mb: int] :
      ( ! [N: nat] : Mb != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( Mb != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ) ).

% int_cases4
tff(fact_7813_int__zle__neg,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)))
    <=> ( ( Na = zero_zero_nat )
        & ( Mb = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_7814_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_7815_int__Suc,axiom,
    ! [Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int) ).

% int_Suc
tff(fact_7816_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z)
    <=> ? [N3: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3))) ) ).

% zless_iff_Suc_zadd
tff(fact_7817_negative__zle__0,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),zero_zero_int) ).

% negative_zle_0
tff(fact_7818_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ~ ! [N: nat] : K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% nonpos_int_cases
tff(fact_7819_sum__roots__unity,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( groups7754918857620584856omplex(aTP_Lamm_do,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_he,Na))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_7820_sum__nth__roots,axiom,
    ! [Na: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( groups7754918857620584856omplex(aTP_Lamm_do,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_hc(Na),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_7821_mod__mult2__eq_H,axiom,
    ! [A: code_integer,Mb: nat,Na: nat] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))) = 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,Na))),modulo364778990260209775nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb))) ).

% mod_mult2_eq'
tff(fact_7822_mod__mult2__eq_H,axiom,
    ! [A: int,Mb: nat,Na: nat] : aa_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,Na))) = 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(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_int(semiri1314217659103216013at_int,Na)))),aa_int_int(modulo_modulo_int(A),aa_nat_int(semiri1314217659103216013at_int,Mb))) ).

% mod_mult2_eq'
tff(fact_7823_mod__mult2__eq_H,axiom,
    ! [A: nat,Mb: nat,Na: 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,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(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,Na)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Mb))) ).

% mod_mult2_eq'
tff(fact_7824_pos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ~ ! [N: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% pos_int_cases
tff(fact_7825_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_7826_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) )
       => ~ ! [N: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ) ).

% int_cases3
tff(fact_7827_nat__less__real__le,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ).

% nat_less_real_le
tff(fact_7828_nat__le__real__less,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_7829_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),I)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_7830_not__zle__0__negative,axiom,
    ! [Na: 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,Na)))) ).

% not_zle_0_negative
tff(fact_7831_negative__zless__0,axiom,
    ! [Na: 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,Na)))),zero_zero_int) ).

% negative_zless_0
tff(fact_7832_negD,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),zero_zero_int)
     => ? [N: nat] : X = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% negD
tff(fact_7833_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ).

% int_ops(6)
tff(fact_7834_nat__approx__posE,axiom,
    ! [E2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
     => ~ ! [N: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)))),E2) ) ).

% nat_approx_posE
tff(fact_7835_nat__approx__posE,axiom,
    ! [E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
     => ~ ! [N: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)))),E2) ) ).

% nat_approx_posE
tff(fact_7836_of__nat__less__two__power,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Na)) ).

% of_nat_less_two_power
tff(fact_7837_of__nat__less__two__power,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% of_nat_less_two_power
tff(fact_7838_of__nat__less__two__power,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)) ).

% of_nat_less_two_power
tff(fact_7839_inverse__of__nat__le,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
     => ( ( Na != zero_zero_nat )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Na))) ) ) ).

% inverse_of_nat_le
tff(fact_7840_inverse__of__nat__le,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
     => ( ( Na != zero_zero_nat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Mb))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Na))) ) ) ).

% inverse_of_nat_le
tff(fact_7841_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),X)),C) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_7842_neg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ~ ! [N: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% neg_int_cases
tff(fact_7843_zdiff__int__split,axiom,
    ! [P: fun_int_o,X: nat,Y: nat] :
      ( aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Y)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_7844_ln__realpow,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(ln_ln_real,X)) ) ) ).

% ln_realpow
tff(fact_7845_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)))) ) ) ) ).

% arctan_add
tff(fact_7846_double__gauss__sum,axiom,
    ! [Na: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,set_or1269000886237332187st_nat(zero_zero_nat,Na))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),one_one_complex)) ).

% double_gauss_sum
tff(fact_7847_double__gauss__sum,axiom,
    ! [Na: nat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,set_or1269000886237332187st_nat(zero_zero_nat,Na))) = times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Na),plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Na),one_on7984719198319812577d_enat)) ).

% double_gauss_sum
tff(fact_7848_double__gauss__sum,axiom,
    ! [Na: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,set_or1269000886237332187st_nat(zero_zero_nat,Na))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),one_one_rat)) ).

% double_gauss_sum
tff(fact_7849_double__gauss__sum,axiom,
    ! [Na: 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,set_or1269000886237332187st_nat(zero_zero_nat,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int)) ).

% double_gauss_sum
tff(fact_7850_double__gauss__sum,axiom,
    ! [Na: 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),set_or1269000886237332187st_nat(zero_zero_nat,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),one_one_nat)) ).

% double_gauss_sum
tff(fact_7851_double__gauss__sum,axiom,
    ! [Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),groups6591440286371151544t_real(semiri5074537144036343181t_real,set_or1269000886237332187st_nat(zero_zero_nat,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real)) ).

% double_gauss_sum
tff(fact_7852_double__arith__series,axiom,
    ! [A: complex,D: complex,Na: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_pm(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Na))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),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,Na)),D))) ).

% double_arith_series
tff(fact_7853_double__arith__series,axiom,
    ! [A: extended_enat,D: extended_enat,Na: nat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),groups7108830773950497114d_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_pn(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Na))) = times_7803423173614009249d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Na),one_on7984719198319812577d_enat),plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),A),times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Na),D))) ).

% double_arith_series
tff(fact_7854_double__arith__series,axiom,
    ! [A: rat,D: rat,Na: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_po(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Na))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),one_one_rat)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),D))) ).

% double_arith_series
tff(fact_7855_double__arith__series,axiom,
    ! [A: int,D: int,Na: 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_pp(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Na))) = 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,Na)),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,Na)),D))) ).

% double_arith_series
tff(fact_7856_double__arith__series,axiom,
    ! [A: nat,D: nat,Na: 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_pq(A),D)),set_or1269000886237332187st_nat(zero_zero_nat,Na))) = 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,Na)),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,Na)),D))) ).

% double_arith_series
tff(fact_7857_double__arith__series,axiom,
    ! [A: real,D: real,Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_pr(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),D))) ).

% double_arith_series
tff(fact_7858_gauss__sum,axiom,
    ! [Na: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(zero_zero_nat,Na)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Na),one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7859_gauss__sum,axiom,
    ! [Na: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7860_gauss__sum,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7861_arith__series,axiom,
    ! [A: code_integer,D: code_integer,Na: nat] : groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_ps(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Na),one_one_Code_integer)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),D))),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7862_arith__series,axiom,
    ! [A: int,D: int,Na: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_pp(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),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,Na)),D)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7863_arith__series,axiom,
    ! [A: nat,D: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pq(A),D)),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),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,Na)),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7864_double__gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),one_one_complex)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7865_double__gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na))) = times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Na),plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Na),one_on7984719198319812577d_enat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7866_double__gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7867_double__gauss__sum__from__Suc__0,axiom,
    ! [Na: 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,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7868_double__gauss__sum__from__Suc__0,axiom,
    ! [Na: 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),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),one_one_nat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7869_double__gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),groups6591440286371151544t_real(semiri5074537144036343181t_real,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7870_sum__gp__offset,axiom,
    ! [X: complex,Mb: nat,Na: nat] :
      groups2073611262835488442omplex(power_power_complex(X),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na))) = $ite(X = one_one_complex,aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),one_one_complex),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),Mb)),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,Na))))),aa_complex_complex(minus_minus_complex(one_one_complex),X))) ).

% sum_gp_offset
tff(fact_7871_sum__gp__offset,axiom,
    ! [X: rat,Mb: nat,Na: nat] :
      groups2906978787729119204at_rat(power_power_rat(X),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na))) = $ite(X = one_one_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),one_one_rat),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),Mb)),aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,Na))))),aa_rat_rat(minus_minus_rat(one_one_rat),X))) ).

% sum_gp_offset
tff(fact_7872_sum__gp__offset,axiom,
    ! [X: real,Mb: nat,Na: nat] :
      groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na))) = $ite(X = one_one_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Mb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,Na))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X))) ).

% sum_gp_offset
tff(fact_7873_of__nat__code__if,axiom,
    ! [Na: nat] :
      aa_nat_complex(semiri8010041392384452111omplex,Na) = $ite(Na = zero_zero_nat,zero_zero_complex,produc1917071388513777916omplex(aTP_Lamm_pt,divmod_nat(Na,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7874_of__nat__code__if,axiom,
    ! [Na: nat] :
      aa_nat_Extended_enat(semiri4216267220026989637d_enat,Na) = $ite(Na = zero_zero_nat,zero_z5237406670263579293d_enat,produc2676513652042109336d_enat(aTP_Lamm_pu,divmod_nat(Na,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7875_of__nat__code__if,axiom,
    ! [Na: nat] :
      aa_nat_rat(semiri681578069525770553at_rat,Na) = $ite(Na = zero_zero_nat,zero_zero_rat,produc6207742614233964070at_rat(aTP_Lamm_pv,divmod_nat(Na,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7876_of__nat__code__if,axiom,
    ! [Na: nat] :
      aa_nat_int(semiri1314217659103216013at_int,Na) = $ite(Na = zero_zero_nat,zero_zero_int,produc6840382203811409530at_int(aTP_Lamm_pw,divmod_nat(Na,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7877_of__nat__code__if,axiom,
    ! [Na: nat] :
      aa_nat_real(semiri5074537144036343181t_real,Na) = $ite(Na = zero_zero_nat,zero_zero_real,produc1703576794950452218t_real(aTP_Lamm_px,divmod_nat(Na,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7878_of__nat__code__if,axiom,
    ! [Na: nat] :
      aa_nat_nat(semiri1316708129612266289at_nat,Na) = $ite(Na = zero_zero_nat,zero_zero_nat,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_py),divmod_nat(Na,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7879_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_pz,X)) ) ).

% monoseq_arctan_series
tff(fact_7880_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K5: real,Na: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H))),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Na))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_7881_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K5: real,Na: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),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)),Na)),aa_nat_complex(power_power_complex(Z),Na))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_7882_ln__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => ( aa_real_real(ln_ln_real,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_qa,X)) ) ) ) ).

% ln_series
tff(fact_7883_lemma__termdiff2,axiom,
    ! [H: complex,Z: complex,Na: 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)),Na)),aa_nat_complex(power_power_complex(Z),Na))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) = aa_complex_complex(times_times_complex(H),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_qc(H),Z),Na),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_7884_lemma__termdiff2,axiom,
    ! [H: rat,Z: rat,Na: nat] :
      ( ( H != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Z),H)),Na)),aa_nat_rat(power_power_rat(Z),Na))),H)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_nat_rat(power_power_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) = aa_rat_rat(times_times_rat(H),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_qe(H),Z),Na),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_7885_lemma__termdiff2,axiom,
    ! [H: real,Z: real,Na: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Na))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,H),groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qg(H),Z),Na),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_7886_arctan__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_real(arctan,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_qh,X)) ) ) ).

% arctan_series
tff(fact_7887_lessThan__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(I),set_or890127255671739683et_nat(K))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,I),K) ) ).

% lessThan_iff
tff(fact_7888_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( aa_set_rat_o(member_rat(I),set_ord_lessThan_rat(K))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),K) ) ).

% lessThan_iff
tff(fact_7889_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( aa_set_num_o(member_num(I),set_ord_lessThan_num(K))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,I),K) ) ).

% lessThan_iff
tff(fact_7890_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( aa_set_nat_o(member_nat(I),set_ord_lessThan_nat(K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K) ) ).

% lessThan_iff
tff(fact_7891_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( aa_set_int_o(member_int(I),set_ord_lessThan_int(K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) ) ).

% lessThan_iff
tff(fact_7892_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( aa_set_real_o(member_real(I),set_or5984915006950818249n_real(K))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,I),K) ) ).

% lessThan_iff
tff(fact_7893_finite__lessThan,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,set_ord_lessThan_nat(K)) ).

% finite_lessThan
tff(fact_7894_lessThan__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_ord_lessThan_rat(X)),set_ord_lessThan_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% lessThan_subset_iff
tff(fact_7895_lessThan__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_ord_lessThan_num(X)),set_ord_lessThan_num(Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y) ) ).

% lessThan_subset_iff
tff(fact_7896_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_ord_lessThan_nat(X)),set_ord_lessThan_nat(Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ).

% lessThan_subset_iff
tff(fact_7897_lessThan__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_ord_lessThan_int(X)),set_ord_lessThan_int(Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ).

% lessThan_subset_iff
tff(fact_7898_lessThan__subset__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_or5984915006950818249n_real(X)),set_or5984915006950818249n_real(Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ).

% lessThan_subset_iff
tff(fact_7899_lessThan__0,axiom,
    set_ord_lessThan_nat(zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_7900_single__Diff__lessThan,axiom,
    ! [K: nat] : minus_minus_set_nat(insert_nat(K,bot_bot_set_nat),set_ord_lessThan_nat(K)) = insert_nat(K,bot_bot_set_nat) ).

% single_Diff_lessThan
tff(fact_7901_single__Diff__lessThan,axiom,
    ! [K: int] : minus_minus_set_int(insert_int(K,bot_bot_set_int),set_ord_lessThan_int(K)) = insert_int(K,bot_bot_set_int) ).

% single_Diff_lessThan
tff(fact_7902_single__Diff__lessThan,axiom,
    ! [K: real] : minus_minus_set_real(insert_real(K,bot_bot_set_real),set_or5984915006950818249n_real(K)) = insert_real(K,bot_bot_set_real) ).

% single_Diff_lessThan
tff(fact_7903_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups2906978787729119204at_rat(G,set_ord_lessThan_nat(aa_nat_nat(suc,Na))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_ord_lessThan_nat(Na))),aa_nat_rat(G,Na)) ).

% sum.lessThan_Suc
tff(fact_7904_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Na: nat] : groups3539618377306564664at_int(G,set_ord_lessThan_nat(aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_ord_lessThan_nat(Na))),aa_nat_int(G,Na)) ).

% sum.lessThan_Suc
tff(fact_7905_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_ord_lessThan_nat(aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_ord_lessThan_nat(Na))),aa_nat_nat(G,Na)) ).

% sum.lessThan_Suc
tff(fact_7906_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Na: nat] : groups6591440286371151544t_real(G,set_ord_lessThan_nat(aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,set_ord_lessThan_nat(Na))),aa_nat_real(G,Na)) ).

% sum.lessThan_Suc
tff(fact_7907_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_kr(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_7908_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_kt(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_7909_int__int__eq,axiom,
    ! [Mb: nat,Na: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( Mb = Na ) ) ).

% int_int_eq
tff(fact_7910_lessThan__non__empty,axiom,
    ! [X: int] : set_ord_lessThan_int(X) != bot_bot_set_int ).

% lessThan_non_empty
tff(fact_7911_lessThan__non__empty,axiom,
    ! [X: real] : set_or5984915006950818249n_real(X) != bot_bot_set_real ).

% lessThan_non_empty
tff(fact_7912_infinite__Iio,axiom,
    ! [A: int] : ~ aa_set_int_o(finite_finite_int,set_ord_lessThan_int(A)) ).

% infinite_Iio
tff(fact_7913_infinite__Iio,axiom,
    ! [A: real] : ~ aa_set_real_o(finite_finite_real,set_or5984915006950818249n_real(A)) ).

% infinite_Iio
tff(fact_7914_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_qi,U)) ).

% lessThan_def
tff(fact_7915_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_qj,U)) ).

% lessThan_def
tff(fact_7916_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_qk,U)) ).

% lessThan_def
tff(fact_7917_lessThan__def,axiom,
    ! [U: nat] : set_ord_lessThan_nat(U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_by,U)) ).

% lessThan_def
tff(fact_7918_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_ql,U)) ).

% lessThan_def
tff(fact_7919_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_real_fun_real_o(aTP_Lamm_qm,U)) ).

% lessThan_def
tff(fact_7920_Iio__eq__empty__iff,axiom,
    ! [Na: nat] :
      ( ( set_ord_lessThan_nat(Na) = bot_bot_set_nat )
    <=> ( Na = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
tff(fact_7921_lessThan__strict__subset__iff,axiom,
    ! [Mb: rat,Na: rat] :
      ( aa_set_rat_o(ord_less_set_rat(set_ord_lessThan_rat(Mb)),set_ord_lessThan_rat(Na))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Mb),Na) ) ).

% lessThan_strict_subset_iff
tff(fact_7922_lessThan__strict__subset__iff,axiom,
    ! [Mb: num,Na: num] :
      ( aa_set_num_o(ord_less_set_num(set_ord_lessThan_num(Mb)),set_ord_lessThan_num(Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Na) ) ).

% lessThan_strict_subset_iff
tff(fact_7923_lessThan__strict__subset__iff,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,set_ord_lessThan_nat(Mb)),set_ord_lessThan_nat(Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ).

% lessThan_strict_subset_iff
tff(fact_7924_lessThan__strict__subset__iff,axiom,
    ! [Mb: int,Na: int] :
      ( aa_set_int_o(ord_less_set_int(set_ord_lessThan_int(Mb)),set_ord_lessThan_int(Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Mb),Na) ) ).

% lessThan_strict_subset_iff
tff(fact_7925_lessThan__strict__subset__iff,axiom,
    ! [Mb: real,Na: real] :
      ( aa_set_real_o(ord_less_set_real(set_or5984915006950818249n_real(Mb)),set_or5984915006950818249n_real(Na))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Mb),Na) ) ).

% lessThan_strict_subset_iff
tff(fact_7926_lessThan__Suc,axiom,
    ! [K: nat] : set_ord_lessThan_nat(aa_nat_nat(suc,K)) = insert_nat(K,set_ord_lessThan_nat(K)) ).

% lessThan_Suc
tff(fact_7927_lessThan__empty__iff,axiom,
    ! [Na: nat] :
      ( ( set_ord_lessThan_nat(Na) = bot_bot_set_nat )
    <=> ( Na = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_7928_finite__nat__iff__bounded,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
    <=> ? [K3: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),set_ord_lessThan_nat(K3)) ) ).

% finite_nat_iff_bounded
tff(fact_7929_finite__nat__bounded,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ? [K2: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),set_ord_lessThan_nat(K2)) ) ).

% finite_nat_bounded
tff(fact_7930_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qn(G),Na)),set_ord_lessThan_nat(Na)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),set_ord_lessThan_nat(Na)) ).

% sum.nat_diff_reindex
tff(fact_7931_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_real,Na: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_qo(G),Na),set_ord_lessThan_nat(Na)) = groups6591440286371151544t_real(G,set_ord_lessThan_nat(Na)) ).

% sum.nat_diff_reindex
tff(fact_7932_sum__diff__distrib,axiom,
    ! [Q: fun_int_nat,P: fun_int_nat,Na: int] :
      ( ! [X3: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Q,X3)),aa_int_nat(P,X3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(P,set_ord_lessThan_int(Na))),groups4541462559716669496nt_nat(Q,set_ord_lessThan_int(Na))) = groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_na(Q),P),set_ord_lessThan_int(Na)) ) ) ).

% sum_diff_distrib
tff(fact_7933_sum__diff__distrib,axiom,
    ! [Q: fun_real_nat,P: fun_real_nat,Na: real] :
      ( ! [X3: real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(Q,X3)),aa_real_nat(P,X3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(P,set_or5984915006950818249n_real(Na))),groups1935376822645274424al_nat(Q,set_or5984915006950818249n_real(Na))) = groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_my(Q),P),set_or5984915006950818249n_real(Na)) ) ) ).

% sum_diff_distrib
tff(fact_7934_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,P: fun_nat_nat,Na: nat] :
      ( ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Q,X3)),aa_nat_nat(P,X3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(P),set_ord_lessThan_nat(Na))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),set_ord_lessThan_nat(Na))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nb(Q),P)),set_ord_lessThan_nat(Na)) ) ) ).

% sum_diff_distrib
tff(fact_7935_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups2906978787729119204at_rat(G,set_ord_lessThan_nat(aa_nat_nat(suc,Na))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_og(G),set_ord_lessThan_nat(Na))) ).

% sum.lessThan_Suc_shift
tff(fact_7936_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,Na: nat] : groups3539618377306564664at_int(G,set_ord_lessThan_nat(aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_oh(G),set_ord_lessThan_nat(Na))) ).

% sum.lessThan_Suc_shift
tff(fact_7937_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_ord_lessThan_nat(aa_nat_nat(suc,Na))) = 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_nm(G)),set_ord_lessThan_nat(Na))) ).

% sum.lessThan_Suc_shift
tff(fact_7938_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,Na: nat] : groups6591440286371151544t_real(G,set_ord_lessThan_nat(aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),groups6591440286371151544t_real(aTP_Lamm_nn(G),set_ord_lessThan_nat(Na))) ).

% sum.lessThan_Suc_shift
tff(fact_7939_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_oi(F),set_ord_lessThan_nat(Mb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,Mb)),aa_nat_rat(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_7940_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,Mb: nat] : groups3539618377306564664at_int(aTP_Lamm_oj(F),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_7941_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,Mb: nat] : groups6591440286371151544t_real(aTP_Lamm_ok(F),set_ord_lessThan_nat(Mb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,Mb)),aa_nat_real(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_7942_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_qp(F),set_ord_lessThan_nat(Mb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_7943_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,Mb: nat] : groups3539618377306564664at_int(aTP_Lamm_qq(F),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_7944_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,Mb: nat] : groups6591440286371151544t_real(aTP_Lamm_qr(F),set_ord_lessThan_nat(Mb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_7945_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_nm(G)),set_ord_lessThan_nat(Na)) ).

% sum.atLeast1_atMost_eq
tff(fact_7946_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_real,Na: nat] : groups6591440286371151544t_real(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na)) = groups6591440286371151544t_real(aTP_Lamm_nn(G),set_ord_lessThan_nat(Na)) ).

% sum.atLeast1_atMost_eq
tff(fact_7947_lemma__termdiff1,axiom,
    ! [Z: complex,H: complex,Mb: nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_qs(Z),H),Mb),set_ord_lessThan_nat(Mb)) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_qt(Z),H),Mb),set_ord_lessThan_nat(Mb)) ).

% lemma_termdiff1
tff(fact_7948_lemma__termdiff1,axiom,
    ! [Z: rat,H: rat,Mb: nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_qu(Z),H),Mb),set_ord_lessThan_nat(Mb)) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_qv(Z),H),Mb),set_ord_lessThan_nat(Mb)) ).

% lemma_termdiff1
tff(fact_7949_lemma__termdiff1,axiom,
    ! [Z: int,H: int,Mb: nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_qw(Z),H),Mb),set_ord_lessThan_nat(Mb)) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_qx(Z),H),Mb),set_ord_lessThan_nat(Mb)) ).

% lemma_termdiff1
tff(fact_7950_lemma__termdiff1,axiom,
    ! [Z: real,H: real,Mb: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qy(Z),H),Mb),set_ord_lessThan_nat(Mb)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qz(Z),H),Mb),set_ord_lessThan_nat(Mb)) ).

% lemma_termdiff1
tff(fact_7951_diff__power__eq__sum,axiom,
    ! [X: complex,Na: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,Na))),aa_nat_complex(power_power_complex(Y),aa_nat_nat(suc,Na))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ra(X),Na),Y),set_ord_lessThan_nat(aa_nat_nat(suc,Na)))) ).

% diff_power_eq_sum
tff(fact_7952_diff__power__eq__sum,axiom,
    ! [X: rat,Na: nat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,Na))),aa_nat_rat(power_power_rat(Y),aa_nat_nat(suc,Na))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_rb(X),Na),Y),set_ord_lessThan_nat(aa_nat_nat(suc,Na)))) ).

% diff_power_eq_sum
tff(fact_7953_diff__power__eq__sum,axiom,
    ! [X: int,Na: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),aa_nat_nat(suc,Na))),aa_nat_int(power_power_int(Y),aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_rc(X),Na),Y),set_ord_lessThan_nat(aa_nat_nat(suc,Na)))) ).

% diff_power_eq_sum
tff(fact_7954_diff__power__eq__sum,axiom,
    ! [X: real,Na: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_rd(X),Na),Y),set_ord_lessThan_nat(aa_nat_nat(suc,Na)))) ).

% diff_power_eq_sum
tff(fact_7955_power__diff__sumr2,axiom,
    ! [X: complex,Na: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),Na)),aa_nat_complex(power_power_complex(Y),Na)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_re(X),Na),Y),set_ord_lessThan_nat(Na))) ).

% power_diff_sumr2
tff(fact_7956_power__diff__sumr2,axiom,
    ! [X: rat,Na: nat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),Na)),aa_nat_rat(power_power_rat(Y),Na)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_rf(X),Na),Y),set_ord_lessThan_nat(Na))) ).

% power_diff_sumr2
tff(fact_7957_power__diff__sumr2,axiom,
    ! [X: int,Na: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),Na)),aa_nat_int(power_power_int(Y),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_rg(X),Na),Y),set_ord_lessThan_nat(Na))) ).

% power_diff_sumr2
tff(fact_7958_power__diff__sumr2,axiom,
    ! [X: real,Na: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_rh(X),Na),Y),set_ord_lessThan_nat(Na))) ).

% power_diff_sumr2
tff(fact_7959_real__sum__nat__ivl__bounded2,axiom,
    ! [Na: nat,F: fun_nat_rat,K5: rat,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Na)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,P4)),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,set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7960_real__sum__nat__ivl__bounded2,axiom,
    ! [Na: nat,F: fun_nat_int,K5: int,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,P4)),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,set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7961_real__sum__nat__ivl__bounded2,axiom,
    ! [Na: nat,F: fun_nat_nat,K5: nat,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,P4)),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),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7962_real__sum__nat__ivl__bounded2,axiom,
    ! [Na: nat,F: fun_nat_real,K5: real,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Na)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,P4)),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,set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7963_one__diff__power__eq_H,axiom,
    ! [X: complex,Na: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),Na)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_ri(X),Na),set_ord_lessThan_nat(Na))) ).

% one_diff_power_eq'
tff(fact_7964_one__diff__power__eq_H,axiom,
    ! [X: rat,Na: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),Na)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_rj(X),Na),set_ord_lessThan_nat(Na))) ).

% one_diff_power_eq'
tff(fact_7965_one__diff__power__eq_H,axiom,
    ! [X: int,Na: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_rk(X),Na),set_ord_lessThan_nat(Na))) ).

% one_diff_power_eq'
tff(fact_7966_one__diff__power__eq_H,axiom,
    ! [X: real,Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)),groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_rl(X),Na),set_ord_lessThan_nat(Na))) ).

% one_diff_power_eq'
tff(fact_7967_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,Na: nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_rm(F),G),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_rn(F),set_ord_lessThan_nat(Na))),groups6591440286371151544t_real(aTP_Lamm_ro(G),set_ord_lessThan_nat(Na))) ).

% sum_split_even_odd
tff(fact_7968_norm__le__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,X)),zero_zero_real)
    <=> ( X = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_7969_norm__le__zero__iff,axiom,
    ! [X: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),zero_zero_real)
    <=> ( X = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_7970_zero__less__norm__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,X))
    <=> ( X != zero_zero_real ) ) ).

% zero_less_norm_iff
tff(fact_7971_zero__less__norm__iff,axiom,
    ! [X: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,X))
    <=> ( X != zero_zero_complex ) ) ).

% zero_less_norm_iff
tff(fact_7972_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(aa_real_fun_nat_real(power_power_real,C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),C)) ) ) ).

% suminf_geometric
tff(fact_7973_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_7974_norm__zero,axiom,
    aa_real_real(real_V7735802525324610683m_real,zero_zero_real) = zero_zero_real ).

% norm_zero
tff(fact_7975_norm__zero,axiom,
    aa_complex_real(real_V1022390504157884413omplex,zero_zero_complex) = zero_zero_real ).

% norm_zero
tff(fact_7976_norm__eq__zero,axiom,
    ! [X: real] :
      ( ( aa_real_real(real_V7735802525324610683m_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% norm_eq_zero
tff(fact_7977_norm__eq__zero,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,X) = zero_zero_real )
    <=> ( X = zero_zero_complex ) ) ).

% norm_eq_zero
tff(fact_7978_suminf__zero,axiom,
    suminf_complex(aTP_Lamm_rp) = zero_zero_complex ).

% suminf_zero
tff(fact_7979_suminf__zero,axiom,
    suminf_real(aTP_Lamm_jt) = zero_zero_real ).

% suminf_zero
tff(fact_7980_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_dw) = zero_zero_nat ).

% suminf_zero
tff(fact_7981_suminf__zero,axiom,
    suminf_int(aTP_Lamm_rq) = zero_zero_int ).

% suminf_zero
tff(fact_7982_norm__not__less__zero,axiom,
    ! [X: complex] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),zero_zero_real) ).

% norm_not_less_zero
tff(fact_7983_norm__uminus__minus,axiom,
    ! [X: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X)),Y)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) ).

% norm_uminus_minus
tff(fact_7984_norm__uminus__minus,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),Y)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y)) ).

% norm_uminus_minus
tff(fact_7985_nonzero__norm__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ) ) ).

% nonzero_norm_divide
tff(fact_7986_nonzero__norm__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ) ) ).

% nonzero_norm_divide
tff(fact_7987_power__eq__imp__eq__norm,axiom,
    ! [W2: real,Na: nat,Z: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W2),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_real_real(real_V7735802525324610683m_real,W2) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_7988_power__eq__imp__eq__norm,axiom,
    ! [W2: complex,Na: nat,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W2),Na) = aa_nat_complex(power_power_complex(Z),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_complex_real(real_V1022390504157884413omplex,W2) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_7989_norm__mult__less,axiom,
    ! [X: real,R2: real,Y: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S)) ) ) ).

% norm_mult_less
tff(fact_7990_norm__mult__less,axiom,
    ! [X: complex,R2: real,Y: complex,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S)) ) ) ).

% norm_mult_less
tff(fact_7991_norm__triangle__lt,axiom,
    ! [X: real,Y: real,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),E2) ) ).

% norm_triangle_lt
tff(fact_7992_norm__triangle__lt,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E2) ) ).

% norm_triangle_lt
tff(fact_7993_norm__add__less,axiom,
    ! [X: real,R2: real,Y: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S)) ) ) ).

% norm_add_less
tff(fact_7994_norm__add__less,axiom,
    ! [X: complex,R2: real,Y: complex,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S)) ) ) ).

% norm_add_less
tff(fact_7995_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B: real,S: 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)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S)) ) ) ).

% norm_triangle_mono
tff(fact_7996_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B: complex,S: 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)),S)
       => 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(aa_rea7340605660300375871l_real(plus_plus_real,R2),S)) ) ) ).

% norm_triangle_mono
tff(fact_7997_norm__triangle__ineq,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))) ).

% norm_triangle_ineq
tff(fact_7998_norm__triangle__ineq,axiom,
    ! [X: 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(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))) ).

% norm_triangle_ineq
tff(fact_7999_norm__triangle__le,axiom,
    ! [X: real,Y: real,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),E2) ) ).

% norm_triangle_le
tff(fact_8000_norm__triangle__le,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E2) ) ).

% norm_triangle_le
tff(fact_8001_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),C)) ) ).

% norm_add_leD
tff(fact_8002_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(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),C)) ) ).

% norm_add_leD
tff(fact_8003_norm__diff__triangle__less,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_8004_norm__diff__triangle__less,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_8005_norm__diff__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ).

% norm_diff_ineq
tff(fact_8006_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))) ).

% norm_diff_ineq
tff(fact_8007_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N4)
           => ( aa_nat_complex(F,N) = zero_zero_complex ) )
       => ( suminf_complex(F) = groups2073611262835488442omplex(F,N4) ) ) ) ).

% suminf_finite
tff(fact_8008_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N4)
           => ( aa_nat_int(F,N) = zero_zero_int ) )
       => ( suminf_int(F) = groups3539618377306564664at_int(F,N4) ) ) ) ).

% suminf_finite
tff(fact_8009_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N4)
           => ( aa_nat_nat(F,N) = zero_zero_nat ) )
       => ( suminf_nat(F) = aa_set_nat_nat(groups3542108847815614940at_nat(F),N4) ) ) ) ).

% suminf_finite
tff(fact_8010_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N4)
           => ( aa_nat_real(F,N) = zero_zero_real ) )
       => ( suminf_real(F) = groups6591440286371151544t_real(F,N4) ) ) ) ).

% suminf_finite
tff(fact_8011_power__eq__1__iff,axiom,
    ! [W2: real,Na: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W2),Na) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W2) = one_one_real )
        | ( Na = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_8012_power__eq__1__iff,axiom,
    ! [W2: complex,Na: nat] :
      ( ( aa_nat_complex(power_power_complex(W2),Na) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W2) = one_one_real )
        | ( Na = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_8013_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(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_8014_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(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),C))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_8015_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_nm(F)),set_ord_lessThan_nat(Mm)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(one_one_nat,Mm)) ).

% sum_bounds_lt_plus1
tff(fact_8016_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_real,Mm: nat] : groups6591440286371151544t_real(aTP_Lamm_nn(F),set_ord_lessThan_nat(Mm)) = groups6591440286371151544t_real(F,set_or1269000886237332187st_nat(one_one_nat,Mm)) ).

% sum_bounds_lt_plus1
tff(fact_8017_sumr__cos__zero__one,axiom,
    ! [Na: nat] : groups6591440286371151544t_real(aTP_Lamm_rr,set_ord_lessThan_nat(aa_nat_nat(suc,Na))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_8018_pi__series,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_rs) ).

% pi_series
tff(fact_8019_summable__arctan__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_qh,X)) ) ).

% summable_arctan_series
tff(fact_8020_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_8021_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(aa_real_fun_nat_real(aTP_Lamm_rt,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% geometric_deriv_sums
tff(fact_8022_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_ru(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_8023_summable__zero,axiom,
    summable_complex(aTP_Lamm_rp) ).

% summable_zero
tff(fact_8024_summable__zero,axiom,
    summable_real(aTP_Lamm_jt) ).

% summable_zero
tff(fact_8025_summable__zero,axiom,
    summable_nat(aTP_Lamm_dw) ).

% summable_zero
tff(fact_8026_summable__zero,axiom,
    summable_int(aTP_Lamm_rq) ).

% summable_zero
tff(fact_8027_summable__single,axiom,
    ! [I: nat,F: fun_nat_complex] : summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_jx(I),F)) ).

% summable_single
tff(fact_8028_summable__single,axiom,
    ! [I: nat,F: fun_nat_real] : summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_rv(I),F)) ).

% summable_single
tff(fact_8029_summable__single,axiom,
    ! [I: nat,F: fun_nat_nat] : summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rw(I),F)) ).

% summable_single
tff(fact_8030_summable__single,axiom,
    ! [I: nat,F: fun_nat_int] : summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_rx(I),F)) ).

% summable_single
tff(fact_8031_sums__zero,axiom,
    aa_complex_o(sums_complex(aTP_Lamm_rp),zero_zero_complex) ).

% sums_zero
tff(fact_8032_sums__zero,axiom,
    aa_real_o(sums_real(aTP_Lamm_jt),zero_zero_real) ).

% sums_zero
tff(fact_8033_sums__zero,axiom,
    aa_nat_o(sums_nat(aTP_Lamm_dw),zero_zero_nat) ).

% sums_zero
tff(fact_8034_sums__zero,axiom,
    aa_int_o(sums_int(aTP_Lamm_rq),zero_zero_int) ).

% sums_zero
tff(fact_8035_summable__iff__shift,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(aa_nat_fun_nat_real(aTP_Lamm_np(F),K))
    <=> summable_real(F) ) ).

% summable_iff_shift
tff(fact_8036_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_8037_summable__cmult__iff,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ry(C),F))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_cmult_iff
tff(fact_8038_summable__cmult__iff,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_rz(C),F))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_8039_summable__divide__iff,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_sa(F),C))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_divide_iff
tff(fact_8040_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_sb(F),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_8041_summable__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sc(P),F)) ) ).

% summable_If_finite
tff(fact_8042_summable__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_sd(P),F)) ) ).

% summable_If_finite
tff(fact_8043_summable__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_se(P),F)) ) ).

% summable_If_finite
tff(fact_8044_summable__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_sf(P),F)) ) ).

% summable_If_finite
tff(fact_8045_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sg(A2),F)) ) ).

% summable_If_finite_set
tff(fact_8046_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_sh(A2),F)) ) ).

% summable_If_finite_set
tff(fact_8047_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_si(A2),F)) ) ).

% summable_If_finite_set
tff(fact_8048_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_sj(A2),F)) ) ).

% summable_If_finite_set
tff(fact_8049_log__eq__one,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),A) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_8050_log__less__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( 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,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_8051_log__less__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),A) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_8052_one__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),X) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_8053_log__less__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_8054_zero__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_8055_log__le__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( 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(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_8056_log__le__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),A) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_8057_one__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_8058_log__le__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_8059_zero__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_8060_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_complex,X: complex] :
      ( aa_complex_o(sums_complex(aTP_Lamm_kr(A)),X)
    <=> ( aa_nat_complex(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_8061_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_real,X: real] :
      ( aa_real_o(sums_real(aTP_Lamm_kt(A)),X)
    <=> ( aa_nat_real(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_8062_summable__geometric__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_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_8063_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_8064_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(aa_rea7340605660300375871l_real(log,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_8065_sums__0,axiom,
    ! [F: fun_nat_complex] :
      ( ! [N: nat] : aa_nat_complex(F,N) = zero_zero_complex
     => aa_complex_o(sums_complex(F),zero_zero_complex) ) ).

% sums_0
tff(fact_8066_sums__0,axiom,
    ! [F: fun_nat_real] :
      ( ! [N: nat] : aa_nat_real(F,N) = zero_zero_real
     => aa_real_o(sums_real(F),zero_zero_real) ) ).

% sums_0
tff(fact_8067_sums__0,axiom,
    ! [F: fun_nat_nat] :
      ( ! [N: nat] : aa_nat_nat(F,N) = zero_zero_nat
     => aa_nat_o(sums_nat(F),zero_zero_nat) ) ).

% sums_0
tff(fact_8068_sums__0,axiom,
    ! [F: fun_nat_int] :
      ( ! [N: nat] : aa_nat_int(F,N) = zero_zero_int
     => aa_int_o(sums_int(F),zero_zero_int) ) ).

% sums_0
tff(fact_8069_sums__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,S: real,T: 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),S)
       => ( aa_real_o(sums_real(G),T)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,S),T) ) ) ) ).

% sums_le
tff(fact_8070_sums__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,S: nat,T: 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),S)
       => ( aa_nat_o(sums_nat(G),T)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,S),T) ) ) ) ).

% sums_le
tff(fact_8071_sums__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,S: int,T: 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),S)
       => ( aa_int_o(sums_int(G),T)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,S),T) ) ) ) ).

% sums_le
tff(fact_8072_sums__single,axiom,
    ! [I: nat,F: fun_nat_complex] : aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_jx(I),F)),aa_nat_complex(F,I)) ).

% sums_single
tff(fact_8073_sums__single,axiom,
    ! [I: nat,F: fun_nat_real] : aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_rv(I),F)),aa_nat_real(F,I)) ).

% sums_single
tff(fact_8074_sums__single,axiom,
    ! [I: nat,F: fun_nat_nat] : aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rw(I),F)),aa_nat_nat(F,I)) ).

% sums_single
tff(fact_8075_sums__single,axiom,
    ! [I: nat,F: fun_nat_int] : aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_rx(I),F)),aa_nat_int(F,I)) ).

% sums_single
tff(fact_8076_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_kz(F),G)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% sums_add
tff(fact_8077_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_kx(F),G)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% sums_add
tff(fact_8078_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_sk(F),G)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% sums_add
tff(fact_8079_summable__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),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_8080_summable__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),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_8081_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_8082_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_8083_summable__const__iff,axiom,
    ! [C: complex] :
      ( summable_complex(aTP_Lamm_sl(C))
    <=> ( C = zero_zero_complex ) ) ).

% summable_const_iff
tff(fact_8084_summable__const__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_sm,C))
    <=> ( C = zero_zero_real ) ) ).

% summable_const_iff
tff(fact_8085_summable__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_kz(F),G)) ) ) ).

% summable_add
tff(fact_8086_summable__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kx(F),G)) ) ) ).

% summable_add
tff(fact_8087_summable__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_sk(F),G)) ) ) ).

% summable_add
tff(fact_8088_summable__Suc__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_nn(F))
    <=> summable_real(F) ) ).

% summable_Suc_iff
tff(fact_8089_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => summable_real(aa_nat_fun_nat_real(aTP_Lamm_np(F),K)) ) ).

% summable_ignore_initial_segment
tff(fact_8090_powser__insidea,axiom,
    ! [F: fun_nat_real,X: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_sn(F),X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,X))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_so(F),Z)) ) ) ).

% powser_insidea
tff(fact_8091_powser__insidea,axiom,
    ! [F: fun_nat_complex,X: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_sp(F),X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,X))
       => summable_real(aa_com20334259897932645t_real(aTP_Lamm_sq(F),Z)) ) ) ).

% powser_insidea
tff(fact_8092_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_8093_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_8094_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_8095_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N4)
           => ( aa_nat_complex(F,N) = zero_zero_complex ) )
       => summable_complex(F) ) ) ).

% summable_finite
tff(fact_8096_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N4)
           => ( aa_nat_real(F,N) = zero_zero_real ) )
       => summable_real(F) ) ) ).

% summable_finite
tff(fact_8097_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N4)
           => ( aa_nat_nat(F,N) = zero_zero_nat ) )
       => summable_nat(F) ) ) ).

% summable_finite
tff(fact_8098_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N4)
           => ( aa_nat_int(F,N) = zero_zero_int ) )
       => summable_int(F) ) ) ).

% summable_finite
tff(fact_8099_sums__mult__iff,axiom,
    ! [C: complex,F: fun_nat_complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ry(C),F)),aa_complex_complex(times_times_complex(C),D))
      <=> aa_complex_o(sums_complex(F),D) ) ) ).

% sums_mult_iff
tff(fact_8100_sums__mult__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_rz(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D))
      <=> aa_real_o(sums_real(F),D) ) ) ).

% sums_mult_iff
tff(fact_8101_sums__mult2__iff,axiom,
    ! [C: complex,F: fun_nat_complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sr(C),F)),aa_complex_complex(times_times_complex(D),C))
      <=> aa_complex_o(sums_complex(F),D) ) ) ).

% sums_mult2_iff
tff(fact_8102_sums__mult2__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ss(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,D),C))
      <=> aa_real_o(sums_real(F),D) ) ) ).

% sums_mult2_iff
tff(fact_8103_summable__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ry(C),F))
     => ( ( C != zero_zero_complex )
       => summable_complex(F) ) ) ).

% summable_mult_D
tff(fact_8104_summable__mult__D,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_rz(C),F))
     => ( ( C != zero_zero_real )
       => summable_real(F) ) ) ).

% summable_mult_D
tff(fact_8105_summable__zero__power,axiom,
    summable_real(aa_real_fun_nat_real(power_power_real,zero_zero_real)) ).

% summable_zero_power
tff(fact_8106_summable__zero__power,axiom,
    summable_int(power_power_int(zero_zero_int)) ).

% summable_zero_power
tff(fact_8107_summable__zero__power,axiom,
    summable_complex(power_power_complex(zero_zero_complex)) ).

% summable_zero_power
tff(fact_8108_pi__not__less__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,pi),zero_zero_real) ).

% pi_not_less_zero
tff(fact_8109_pi__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),pi) ).

% pi_gt_zero
tff(fact_8110_suminf__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_kz(F),G)) ) ) ) ).

% suminf_add
tff(fact_8111_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_kx(F),G)) ) ) ) ).

% suminf_add
tff(fact_8112_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_sk(F),G)) ) ) ) ).

% suminf_add
tff(fact_8113_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_8114_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_8115_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_8116_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_8117_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_8118_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_8119_suminf__pos,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_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)) ) ) ).

% suminf_pos
tff(fact_8120_suminf__pos,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_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)) ) ) ).

% suminf_pos
tff(fact_8121_suminf__pos,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_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)) ) ) ).

% suminf_pos
tff(fact_8122_sums__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex,A: complex] :
      ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ry(C),F)),A)
     => ( ( C != zero_zero_complex )
       => aa_complex_o(sums_complex(F),aa_complex_complex(divide1717551699836669952omplex(A),C)) ) ) ).

% sums_mult_D
tff(fact_8123_sums__mult__D,axiom,
    ! [C: real,F: fun_nat_real,A: real] :
      ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_rz(C),F)),A)
     => ( ( C != zero_zero_real )
       => aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)) ) ) ).

% sums_mult_D
tff(fact_8124_sums__Suc__imp,axiom,
    ! [F: fun_nat_complex,S: complex] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( aa_complex_o(sums_complex(aTP_Lamm_st(F)),S)
       => aa_complex_o(sums_complex(F),S) ) ) ).

% sums_Suc_imp
tff(fact_8125_sums__Suc__imp,axiom,
    ! [F: fun_nat_real,S: real] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( aa_real_o(sums_real(aTP_Lamm_nn(F)),S)
       => aa_real_o(sums_real(F),S) ) ) ).

% sums_Suc_imp
tff(fact_8126_sums__Suc,axiom,
    ! [F: fun_nat_real,L: real] :
      ( aa_real_o(sums_real(aTP_Lamm_nn(F)),L)
     => aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,L),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_8127_sums__Suc,axiom,
    ! [F: fun_nat_nat,L: nat] :
      ( aa_nat_o(sums_nat(aTP_Lamm_nm(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_8128_sums__Suc,axiom,
    ! [F: fun_nat_int,L: int] :
      ( aa_int_o(sums_int(aTP_Lamm_oh(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_8129_sums__Suc__iff,axiom,
    ! [F: fun_nat_real,S: real] :
      ( aa_real_o(sums_real(aTP_Lamm_nn(F)),S)
    <=> aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc_iff
tff(fact_8130_sums__zero__iff__shift,axiom,
    ! [Na: nat,F: fun_nat_complex,S: complex] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Na)
         => ( aa_nat_complex(F,I3) = zero_zero_complex ) )
     => ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_su(Na),F)),S)
      <=> aa_complex_o(sums_complex(F),S) ) ) ).

% sums_zero_iff_shift
tff(fact_8131_sums__zero__iff__shift,axiom,
    ! [Na: nat,F: fun_nat_real,S: real] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Na)
         => ( aa_nat_real(F,I3) = zero_zero_real ) )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_sv(Na),F)),S)
      <=> aa_real_o(sums_real(F),S) ) ) ).

% sums_zero_iff_shift
tff(fact_8132_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_kr(F)) ).

% summable_0_powser
tff(fact_8133_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_kt(F)) ).

% summable_0_powser
tff(fact_8134_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_kr(F)) ).

% summable_zero_power'
tff(fact_8135_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_kt(F)) ).

% summable_zero_power'
tff(fact_8136_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_sw(F)) ).

% summable_zero_power'
tff(fact_8137_summable__powser__split__head,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_sx(F),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_sp(F),Z)) ) ).

% summable_powser_split_head
tff(fact_8138_summable__powser__split__head,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_sy(F),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_sn(F),Z)) ) ).

% summable_powser_split_head
tff(fact_8139_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_sp(F),Z))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_sx(F),Z)) ) ).

% powser_split_head(3)
tff(fact_8140_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_sn(F),Z))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_sy(F),Z)) ) ).

% powser_split_head(3)
tff(fact_8141_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,Mb: nat,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_sz(F),Mb),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_sp(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_8142_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_ta(F),Mb),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_sn(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_8143_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N4)
           => ( aa_nat_complex(F,N) = zero_zero_complex ) )
       => aa_complex_o(sums_complex(F),groups2073611262835488442omplex(F,N4)) ) ) ).

% sums_finite
tff(fact_8144_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N4)
           => ( aa_nat_int(F,N) = zero_zero_int ) )
       => aa_int_o(sums_int(F),groups3539618377306564664at_int(F,N4)) ) ) ).

% sums_finite
tff(fact_8145_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N4)
           => ( aa_nat_nat(F,N) = zero_zero_nat ) )
       => aa_nat_o(sums_nat(F),aa_set_nat_nat(groups3542108847815614940at_nat(F),N4)) ) ) ).

% sums_finite
tff(fact_8146_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,N4)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N4)
           => ( aa_nat_real(F,N) = zero_zero_real ) )
       => aa_real_o(sums_real(F),groups6591440286371151544t_real(F,N4)) ) ) ).

% sums_finite
tff(fact_8147_sums__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sc(P),F)),groups2073611262835488442omplex(F,collect_nat(P))) ) ).

% sums_If_finite
tff(fact_8148_sums__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_sf(P),F)),groups3539618377306564664at_int(F,collect_nat(P))) ) ).

% sums_If_finite
tff(fact_8149_sums__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_se(P),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),collect_nat(P))) ) ).

% sums_If_finite
tff(fact_8150_sums__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_sd(P),F)),groups6591440286371151544t_real(F,collect_nat(P))) ) ).

% sums_If_finite
tff(fact_8151_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sg(A2),F)),groups2073611262835488442omplex(F,A2)) ) ).

% sums_If_finite_set
tff(fact_8152_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_sj(A2),F)),groups3539618377306564664at_int(F,A2)) ) ).

% sums_If_finite_set
tff(fact_8153_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_si(A2),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) ) ).

% sums_If_finite_set
tff(fact_8154_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_sh(A2),F)),groups6591440286371151544t_real(F,A2)) ) ).

% sums_If_finite_set
tff(fact_8155_summable__norm__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),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_tb(F)) ) ) ).

% summable_norm_comparison_test
tff(fact_8156_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),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_kq(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_8157_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))
        <=> ? [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I2)) ) ) ) ).

% suminf_pos_iff
tff(fact_8158_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))
        <=> ? [I2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I2)) ) ) ) ).

% suminf_pos_iff
tff(fact_8159_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))
        <=> ? [I2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I2)) ) ) ) ).

% suminf_pos_iff
tff(fact_8160_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_8161_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_8162_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_8163_suminf__le__const,axiom,
    ! [F: fun_nat_int,X: int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,set_ord_lessThan_nat(N))),X)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,suminf_int(F)),X) ) ) ).

% suminf_le_const
tff(fact_8164_suminf__le__const,axiom,
    ! [F: fun_nat_nat,X: 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),set_ord_lessThan_nat(N))),X)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,suminf_nat(F)),X) ) ) ).

% suminf_le_const
tff(fact_8165_suminf__le__const,axiom,
    ! [F: fun_nat_real,X: real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(F,set_ord_lessThan_nat(N))),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(F)),X) ) ) ).

% suminf_le_const
tff(fact_8166_powser__sums__if,axiom,
    ! [Mb: nat,Z: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_tc(Mb),Z)),aa_nat_complex(power_power_complex(Z),Mb)) ).

% powser_sums_if
tff(fact_8167_powser__sums__if,axiom,
    ! [Mb: nat,Z: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_td(Mb),Z)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Mb)) ).

% powser_sums_if
tff(fact_8168_powser__sums__if,axiom,
    ! [Mb: nat,Z: int] : aa_int_o(sums_int(aa_int_fun_nat_int(aTP_Lamm_te(Mb),Z)),aa_nat_int(power_power_int(Z),Mb)) ).

% powser_sums_if
tff(fact_8169_powser__sums__zero,axiom,
    ! [A: fun_nat_complex] : aa_complex_o(sums_complex(aTP_Lamm_kr(A)),aa_nat_complex(A,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_8170_powser__sums__zero,axiom,
    ! [A: fun_nat_real] : aa_real_o(sums_real(aTP_Lamm_kt(A)),aa_nat_real(A,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_8171_log__base__change,axiom,
    ! [A: real,B: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),B)) ) ) ) ).

% log_base_change
tff(fact_8172_log__of__power__eq,axiom,
    ! [Mb: nat,B: real,Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_real(semiri5074537144036343181t_real,Na) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ) ) ).

% log_of_power_eq
tff(fact_8173_less__log__of__power,axiom,
    ! [B: real,Na: nat,Mb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na)),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,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Mb)) ) ) ).

% less_log_of_power
tff(fact_8174_powser__inside,axiom,
    ! [F: fun_nat_real,X: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_sn(F),X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,X))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_sn(F),Z)) ) ) ).

% powser_inside
tff(fact_8175_powser__inside,axiom,
    ! [F: fun_nat_complex,X: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_sp(F),X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,X))
       => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_sp(F),Z)) ) ) ).

% powser_inside
tff(fact_8176_sums__iff__shift,axiom,
    ! [F: fun_nat_real,Na: nat,S: real] :
      ( aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_np(F),Na)),S)
    <=> aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S),groups6591440286371151544t_real(F,set_ord_lessThan_nat(Na)))) ) ).

% sums_iff_shift
tff(fact_8177_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_int,X: 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,set_ord_lessThan_nat(N))),X)
       => summable_int(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8178_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_nat,X: 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),set_ord_lessThan_nat(N))),X)
       => summable_nat(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8179_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_real,X: 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,set_ord_lessThan_nat(N))),X)
       => summable_real(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8180_sums__split__initial__segment,axiom,
    ! [F: fun_nat_real,S: real,Na: nat] :
      ( aa_real_o(sums_real(F),S)
     => aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_np(F),Na)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S),groups6591440286371151544t_real(F,set_ord_lessThan_nat(Na)))) ) ).

% sums_split_initial_segment
tff(fact_8181_sums__iff__shift_H,axiom,
    ! [F: fun_nat_real,Na: nat,S: real] :
      ( aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_np(F),Na)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S),groups6591440286371151544t_real(F,set_ord_lessThan_nat(Na))))
    <=> aa_real_o(sums_real(F),S) ) ).

% sums_iff_shift'
tff(fact_8182_complete__algebra__summable__geometric,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),one_one_real)
     => summable_real(aa_real_fun_nat_real(power_power_real,X)) ) ).

% complete_algebra_summable_geometric
tff(fact_8183_complete__algebra__summable__geometric,axiom,
    ! [X: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),one_one_real)
     => summable_complex(power_power_complex(X)) ) ).

% complete_algebra_summable_geometric
tff(fact_8184_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(aa_real_fun_nat_real(power_power_real,C)) ) ).

% summable_geometric
tff(fact_8185_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_8186_sums__If__finite__set_H,axiom,
    ! [G: fun_nat_real,S3: real,A2: set_nat,S6: real,F: fun_nat_real] :
      ( aa_real_o(sums_real(G),S3)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( ( S6 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S3),groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_tf(G),F),A2)) )
         => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_tg(G),A2),F)),S6) ) ) ) ).

% sums_If_finite_set'
tff(fact_8187_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_nn(F)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),aa_nat_real(F,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_8188_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_8189_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_8190_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_8191_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_8192_log__mult,axiom,
    ! [A: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_8193_log__divide,axiom,
    ! [A: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_8194_le__log__of__power,axiom,
    ! [B: real,Na: nat,Mb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na)),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,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Mb)) ) ) ).

% le_log_of_power
tff(fact_8195_log__base__pow,axiom,
    ! [A: real,Na: nat,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log_base_pow
tff(fact_8196_log__nat__power,axiom,
    ! [X: real,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ).

% log_nat_power
tff(fact_8197_suminf__split__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(F) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_np(F),K))),groups6591440286371151544t_real(F,set_ord_lessThan_nat(K))) ) ) ).

% suminf_split_initial_segment
tff(fact_8198_suminf__minus__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(aa_nat_fun_nat_real(aTP_Lamm_np(F),K)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),groups6591440286371151544t_real(F,set_ord_lessThan_nat(K))) ) ) ).

% suminf_minus_initial_segment
tff(fact_8199_pi__half__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% pi_half_less_two
tff(fact_8200_sum__less__suminf,axiom,
    ! [F: fun_nat_int,Na: nat] :
      ( summable_int(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),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,set_ord_lessThan_nat(Na))),suminf_int(F)) ) ) ).

% sum_less_suminf
tff(fact_8201_sum__less__suminf,axiom,
    ! [F: fun_nat_nat,Na: nat] :
      ( summable_nat(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),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),set_ord_lessThan_nat(Na))),suminf_nat(F)) ) ) ).

% sum_less_suminf
tff(fact_8202_sum__less__suminf,axiom,
    ! [F: fun_nat_real,Na: nat] :
      ( summable_real(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),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,set_ord_lessThan_nat(Na))),suminf_real(F)) ) ) ).

% sum_less_suminf
tff(fact_8203_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_sp(F),Z))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_sp(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_sx(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_8204_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_sn(F),Z))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_sn(F),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,zero_zero_nat)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_sy(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_8205_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_sp(F),Z))
     => ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_sx(F),Z))),Z) = aa_complex_complex(minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_sp(F),Z))),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_8206_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_sn(F),Z))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_sy(F),Z))),Z) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_sn(F),Z))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_8207_log__of__power__less,axiom,
    ! [Mb: nat,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na))
     => ( 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(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_of_power_less
tff(fact_8208_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( ( B != one_one_real )
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
             => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_real_real(ln_ln_real,A))),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_8209_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_complex,E2: real] :
      ( summable_complex(F)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
       => ~ ! [N7: nat] :
              ~ ! [M4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),M4)
                 => ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,set_or1269000886237332187st_nat(M4,N5)))),E2) ) ) ) ).

% summable_partial_sum_bound
tff(fact_8210_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_real,E2: real] :
      ( summable_real(F)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
       => ~ ! [N7: nat] :
              ~ ! [M4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),M4)
                 => ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,groups6591440286371151544t_real(F,set_or1269000886237332187st_nat(M4,N5)))),E2) ) ) ) ).

% summable_partial_sum_bound
tff(fact_8211_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)
       => ? [N7: nat] :
          ! [N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N5)
           => 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_np(F),N5)))),R2) ) ) ) ).

% suminf_exist_split
tff(fact_8212_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)
       => ? [N7: nat] :
          ! [N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N5)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_th(F),N5)))),R2) ) ) ) ).

% suminf_exist_split
tff(fact_8213_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_sn(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_8214_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(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(A,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,R0),N))),M6)
         => summable_real(aa_fun7054779078796324018t_real(aTP_Lamm_ti(R2),A)) ) ) ) ).

% Abel_lemma
tff(fact_8215_pi__half__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_gt_zero
tff(fact_8216_m2pi__less__pi,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))),pi) ).

% m2pi_less_pi
tff(fact_8217_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(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N)))) )
       => summable_real(F) ) ) ).

% summable_ratio_test
tff(fact_8218_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(aa_rea7340605660300375871l_real(times_times_real,C),aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N)))) )
       => summable_complex(F) ) ) ).

% summable_ratio_test
tff(fact_8219_sum__less__suminf2,axiom,
    ! [F: fun_nat_int,Na: nat,I: nat] :
      ( summable_int(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),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,Na),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,set_ord_lessThan_nat(Na))),suminf_int(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8220_sum__less__suminf2,axiom,
    ! [F: fun_nat_nat,Na: nat,I: nat] :
      ( summable_nat(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),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,Na),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),set_ord_lessThan_nat(Na))),suminf_nat(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8221_sum__less__suminf2,axiom,
    ! [F: fun_nat_real,Na: nat,I: nat] :
      ( summable_real(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),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,Na),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,set_ord_lessThan_nat(Na))),suminf_real(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8222_arctan__ubound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% arctan_ubound
tff(fact_8223_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(aa_real_fun_nat_real(power_power_real,C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),C))) ) ).

% geometric_sums
tff(fact_8224_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_8225_power__half__series,axiom,
    aa_real_o(sums_real(aTP_Lamm_tj),one_one_real) ).

% power_half_series
tff(fact_8226_log__of__power__le,axiom,
    ! [Mb: nat,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na))
     => ( 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(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_of_power_le
tff(fact_8227_minus__pi__half__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),zero_zero_real) ).

% minus_pi_half_less_zero
tff(fact_8228_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(aa_rea7340605660300375871l_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_8229_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% arctan_bounded
tff(fact_8230_less__log2__of__power,axiom,
    ! [Na: 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))),Na)),Mb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ).

% less_log2_of_power
tff(fact_8231_le__log2__of__power,axiom,
    ! [Na: 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))),Na)),Mb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ).

% le_log2_of_power
tff(fact_8232_log2__of__power__less,axiom,
    ! [Mb: nat,Na: 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))),Na))
     => ( 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(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log2_of_power_less
tff(fact_8233_log2__of__power__le,axiom,
    ! [Mb: nat,Na: 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))),Na))
     => ( 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(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log2_of_power_le
tff(fact_8234_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D4)))),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D4)),one_one_nat)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,groups6591440286371151544t_real(F,set_ord_lessThan_nat(K))),suminf_real(F)) ) ) ).

% sum_pos_lt_pair
tff(fact_8235_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_8236_ceiling__log2__div2,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_8237_sin__cos__npi,axiom,
    ! [Na: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na) ).

% sin_cos_npi
tff(fact_8238_diffs__equiv,axiom,
    ! [C: fun_nat_complex,X: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_tk(C),X))
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_tl(C),X)),suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_tk(C),X))) ) ).

% diffs_equiv
tff(fact_8239_diffs__equiv,axiom,
    ! [C: fun_nat_real,X: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_tm(C),X))
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_tn(C),X)),suminf_real(aa_real_fun_nat_real(aTP_Lamm_tm(C),X))) ) ).

% diffs_equiv
tff(fact_8240_cos__pi__eq__zero,axiom,
    ! [Mb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_8241_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),X)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_8242_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Na) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_8243_sin__zero,axiom,
    aa_complex_complex(sin_complex,zero_zero_complex) = zero_zero_complex ).

% sin_zero
tff(fact_8244_sin__zero,axiom,
    aa_real_real(sin_real,zero_zero_real) = zero_zero_real ).

% sin_zero
tff(fact_8245_powr__eq__0__iff,axiom,
    ! [W2: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,W2),Z) = zero_zero_real )
    <=> ( W2 = zero_zero_real ) ) ).

% powr_eq_0_iff
tff(fact_8246_powr__0,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,zero_zero_real),Z) = zero_zero_real ).

% powr_0
tff(fact_8247_cos__zero,axiom,
    aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_8248_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_8249_powr__zero__eq__one,axiom,
    ! [X: real] :
      aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),zero_zero_real) = $ite(X = zero_zero_real,zero_zero_real,one_one_real) ).

% powr_zero_eq_one
tff(fact_8250_floor__zero,axiom,
    aa_real_int(archim6058952711729229775r_real,zero_zero_real) = zero_zero_int ).

% floor_zero
tff(fact_8251_floor__zero,axiom,
    aa_rat_int(archim3151403230148437115or_rat,zero_zero_rat) = zero_zero_int ).

% floor_zero
tff(fact_8252_powr__gt__zero,axiom,
    ! [X: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A))
    <=> ( X != zero_zero_real ) ) ).

% powr_gt_zero
tff(fact_8253_powr__less__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% powr_less_cancel_iff
tff(fact_8254_powr__eq__one__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X) = one_one_real )
      <=> ( X = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_8255_powr__le__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% powr_le_cancel_iff
tff(fact_8256_zero__le__floor,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) ).

% zero_le_floor
tff(fact_8257_zero__le__floor,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X) ) ).

% zero_le_floor
tff(fact_8258_sin__cos__squared__add3,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,X))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(sin_complex,X))) = one_one_complex ).

% sin_cos_squared_add3
tff(fact_8259_sin__cos__squared__add3,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,X))) = one_one_real ).

% sin_cos_squared_add3
tff(fact_8260_floor__less__zero,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real) ) ).

% floor_less_zero
tff(fact_8261_floor__less__zero,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat) ) ).

% floor_less_zero
tff(fact_8262_numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,V)),X) ) ).

% numeral_le_floor
tff(fact_8263_numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,V)),X) ) ).

% numeral_le_floor
tff(fact_8264_zero__less__floor,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ).

% zero_less_floor
tff(fact_8265_zero__less__floor,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X) ) ).

% zero_less_floor
tff(fact_8266_floor__le__zero,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ).

% floor_le_zero
tff(fact_8267_floor__le__zero,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),one_one_rat) ) ).

% floor_le_zero
tff(fact_8268_floor__less__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_num_real(numeral_numeral_real,V)) ) ).

% floor_less_numeral
tff(fact_8269_floor__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_num_rat(numeral_numeral_rat,V)) ) ).

% floor_less_numeral
tff(fact_8270_one__le__floor,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ).

% one_le_floor
tff(fact_8271_one__le__floor,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X) ) ).

% one_le_floor
tff(fact_8272_floor__less__one,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ).

% floor_less_one
tff(fact_8273_floor__less__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),one_one_rat) ) ).

% floor_less_one
tff(fact_8274_powr__log__cancel,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)) = X ) ) ) ) ).

% powr_log_cancel
tff(fact_8275_log__powr__cancel,axiom,
    ! [A: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_8276_numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)),X) ) ).

% numeral_less_floor
tff(fact_8277_numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),X) ) ).

% numeral_less_floor
tff(fact_8278_floor__le__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)) ) ).

% floor_le_numeral
tff(fact_8279_floor__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)) ) ).

% floor_le_numeral
tff(fact_8280_one__less__floor,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X) ) ).

% one_less_floor
tff(fact_8281_one__less__floor,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X) ) ).

% one_less_floor
tff(fact_8282_floor__le__one,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ).

% floor_le_one
tff(fact_8283_floor__le__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ) ).

% floor_le_one
tff(fact_8284_neg__numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),X) ) ).

% neg_numeral_le_floor
tff(fact_8285_neg__numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),X) ) ).

% neg_numeral_le_floor
tff(fact_8286_floor__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) ) ).

% floor_less_neg_numeral
tff(fact_8287_floor__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) ) ).

% floor_less_neg_numeral
tff(fact_8288_sin__cos__squared__add2,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add2
tff(fact_8289_sin__cos__squared__add2,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add2
tff(fact_8290_sin__cos__squared__add,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add
tff(fact_8291_sin__cos__squared__add,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add
tff(fact_8292_neg__numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)),X) ) ).

% neg_numeral_less_floor
tff(fact_8293_neg__numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),X) ) ).

% neg_numeral_less_floor
tff(fact_8294_floor__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)) ) ).

% floor_le_neg_numeral
tff(fact_8295_floor__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)) ) ).

% floor_le_neg_numeral
tff(fact_8296_cos__one__sin__zero,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cos_complex,X) = one_one_complex )
     => ( aa_complex_complex(sin_complex,X) = zero_zero_complex ) ) ).

% cos_one_sin_zero
tff(fact_8297_cos__one__sin__zero,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
     => ( aa_real_real(sin_real,X) = zero_zero_real ) ) ).

% cos_one_sin_zero
tff(fact_8298_sin__add,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(cos_complex,Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(sin_complex,Y))) ).

% sin_add
tff(fact_8299_sin__add,axiom,
    ! [X: real,Y: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(sin_real,Y))) ).

% sin_add
tff(fact_8300_cos__diff,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(X),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(sin_complex,Y))) ).

% cos_diff
tff(fact_8301_cos__diff,axiom,
    ! [X: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))) ).

% cos_diff
tff(fact_8302_cos__add,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(sin_complex,Y))) ).

% cos_add
tff(fact_8303_cos__add,axiom,
    ! [X: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))) ).

% cos_add
tff(fact_8304_sin__zero__norm__cos__one,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(cos_real,X)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_8305_sin__zero__norm__cos__one,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(sin_complex,X) = zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(cos_complex,X)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_8306_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y3)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),pi)
      & ( aa_real_real(sin_real,Y3) = aa_real_real(sin_real,X) )
      & ( aa_real_real(cos_real,Y3) = aa_real_real(cos_real,X) ) ) ).

% sincos_principal_value
tff(fact_8307_floor__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y)) ) ).

% floor_mono
tff(fact_8308_floor__mono,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y)) ) ).

% floor_mono
tff(fact_8309_of__int__floor__le,axiom,
    ! [X: 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,X))),X) ).

% of_int_floor_le
tff(fact_8310_of__int__floor__le,axiom,
    ! [X: 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,X))),X) ).

% of_int_floor_le
tff(fact_8311_floor__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ).

% floor_less_cancel
tff(fact_8312_floor__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ).

% floor_less_cancel
tff(fact_8313_powr__non__neg,axiom,
    ! [A: real,X: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X)),zero_zero_real) ).

% powr_non_neg
tff(fact_8314_powr__less__mono2__neg,axiom,
    ! [A: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_8315_powr__less__cancel,axiom,
    ! [X: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% powr_less_cancel
tff(fact_8316_powr__less__mono,axiom,
    ! [A: real,B: real,X: 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,one_one_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)) ) ) ).

% powr_less_mono
tff(fact_8317_le__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),X) ) ).

% le_floor_iff
tff(fact_8318_le__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X) ) ).

% le_floor_iff
tff(fact_8319_floor__less__iff,axiom,
    ! [X: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z)) ) ).

% floor_less_iff
tff(fact_8320_floor__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% floor_less_iff
tff(fact_8321_powr__mono2_H,axiom,
    ! [A: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)) ) ) ) ).

% powr_mono2'
tff(fact_8322_powr__less__mono2,axiom,
    ! [A: real,X: real,Y: 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),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_less_mono2
tff(fact_8323_le__floor__add,axiom,
    ! [X: 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,X)),aa_real_int(archim6058952711729229775r_real,Y))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))) ).

% le_floor_add
tff(fact_8324_le__floor__add,axiom,
    ! [X: 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,X)),aa_rat_int(archim3151403230148437115or_rat,Y))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),Y))) ).

% le_floor_add
tff(fact_8325_gr__one__powr,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => ( 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,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y)) ) ) ).

% gr_one_powr
tff(fact_8326_powr__inj,axiom,
    ! [A: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y) )
        <=> ( X = Y ) ) ) ) ).

% powr_inj
tff(fact_8327_int__add__floor,axiom,
    ! [Z: int,X: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_real_int(archim6058952711729229775r_real,X)) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),X)) ).

% int_add_floor
tff(fact_8328_int__add__floor,axiom,
    ! [Z: int,X: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_rat_int(archim3151403230148437115or_rat,X)) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),X)) ).

% int_add_floor
tff(fact_8329_floor__add__int,axiom,
    ! [X: real,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),Z) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_int_real(ring_1_of_int_real,Z))) ).

% floor_add_int
tff(fact_8330_floor__add__int,axiom,
    ! [X: rat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),aa_int_rat(ring_1_of_int_rat,Z))) ).

% floor_add_int
tff(fact_8331_sin__gt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,X)) ) ) ).

% sin_gt_zero
tff(fact_8332_sin__times__sin,axiom,
    ! [W2: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W2)),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(W2),Z))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W2),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_8333_sin__times__sin,axiom,
    ! [W2: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,W2)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W2),Z))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_8334_sin__times__cos,axiom,
    ! [W2: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W2)),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(W2),Z))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W2),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_8335_sin__times__cos,axiom,
    ! [W2: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,W2)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W2),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_8336_cos__times__sin,axiom,
    ! [W2: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W2)),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(W2),Z))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W2),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_8337_cos__times__sin,axiom,
    ! [W2: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,W2)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W2),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_8338_sin__plus__sin,axiom,
    ! [W2: complex,Z: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,W2)),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(W2),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W2),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_8339_sin__plus__sin,axiom,
    ! [W2: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,W2)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W2),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_8340_sin__diff__sin,axiom,
    ! [W2: complex,Z: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,W2)),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(W2),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W2),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_8341_sin__diff__sin,axiom,
    ! [W2: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,W2)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W2),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_8342_cos__diff__cos,axiom,
    ! [W2: complex,Z: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,W2)),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(W2),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Z),W2)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_8343_cos__diff__cos,axiom,
    ! [W2: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cos_real,W2)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),W2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_8344_powr__add,axiom,
    ! [X: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)) ).

% powr_add
tff(fact_8345_one__add__floor,axiom,
    ! [X: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) ).

% one_add_floor
tff(fact_8346_one__add__floor,axiom,
    ! [X: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),one_one_rat)) ).

% one_add_floor
tff(fact_8347_floor__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = K )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,K))),X)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_8348_powr__realpow,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_nat_real(semiri5074537144036343181t_real,Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na) ) ) ).

% powr_realpow
tff(fact_8349_less__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( 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,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X) ) ) ) ).

% less_log_iff
tff(fact_8350_log__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( 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,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)) ) ) ) ).

% log_less_iff
tff(fact_8351_less__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( 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,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y) ) ) ) ).

% less_powr_iff
tff(fact_8352_powr__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( 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,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ) ).

% powr_less_iff
tff(fact_8353_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
       => ( 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),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_8354_cos__monotone__0__pi,axiom,
    ! [Y: real,X: 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_real,Y),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_8355_sin__eq__0__pi,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),pi)
       => ( ( aa_real_real(sin_real,X) = zero_zero_real )
         => ( X = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
tff(fact_8356_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real)) ).

% real_of_int_floor_add_one_gt
tff(fact_8357_floor__eq,axiom,
    ! [Na: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Na)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,X) = Na ) ) ) ).

% floor_eq
tff(fact_8358_sin__zero__pi__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),pi)
     => ( ( aa_real_real(sin_real,X) = zero_zero_real )
      <=> ( X = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
tff(fact_8359_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))) ).

% real_of_int_floor_gt_diff_one
tff(fact_8360_diffs__def,axiom,
    ! [C: fun_nat_complex,X2: nat] : aa_nat_complex(diffs_complex(C),X2) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,X2))),aa_nat_complex(C,aa_nat_nat(suc,X2))) ).

% diffs_def
tff(fact_8361_diffs__def,axiom,
    ! [C: fun_nat_rat,X2: nat] : aa_nat_rat(diffs_rat(C),X2) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,X2))),aa_nat_rat(C,aa_nat_nat(suc,X2))) ).

% diffs_def
tff(fact_8362_diffs__def,axiom,
    ! [C: fun_nat_int,X2: nat] : aa_nat_int(diffs_int(C),X2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,X2))),aa_nat_int(C,aa_nat_nat(suc,X2))) ).

% diffs_def
tff(fact_8363_diffs__def,axiom,
    ! [C: fun_nat_real,X2: nat] : aa_nat_real(diffs_real(C),X2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,X2))),aa_nat_real(C,aa_nat_nat(suc,X2))) ).

% diffs_def
tff(fact_8364_sin__expansion__lemma,axiom,
    ! [X: real,Mb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_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_8365_floor__unique,axiom,
    ! [Z: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,X) = Z ) ) ) ).

% floor_unique
tff(fact_8366_floor__unique,axiom,
    ! [Z: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
       => ( aa_rat_int(archim3151403230148437115or_rat,X) = Z ) ) ) ).

% floor_unique
tff(fact_8367_floor__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( aa_real_int(archim6058952711729229775r_real,X) = A )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),X)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,A)),one_one_real)) ) ) ).

% floor_eq_iff
tff(fact_8368_floor__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( aa_rat_int(archim3151403230148437115or_rat,X) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)) ) ) ).

% floor_eq_iff
tff(fact_8369_floor__split,axiom,
    ! [P: fun_int_o,T: real] :
      ( aa_int_o(P,aa_real_int(archim6058952711729229775r_real,T))
    <=> ! [I2: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,I2)),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,I2)),one_one_real)) )
         => aa_int_o(P,I2) ) ) ).

% floor_split
tff(fact_8370_floor__split,axiom,
    ! [P: fun_int_o,T: rat] :
      ( aa_int_o(P,aa_rat_int(archim3151403230148437115or_rat,T))
    <=> ! [I2: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,I2)),T)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,T),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,I2)),one_one_rat)) )
         => aa_int_o(P,I2) ) ) ).

% floor_split
tff(fact_8371_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(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% le_mult_floor
tff(fact_8372_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(times_times_rat(A),B))) ) ) ).

% le_mult_floor
tff(fact_8373_less__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X) ) ).

% less_floor_iff
tff(fact_8374_less__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X) ) ).

% less_floor_iff
tff(fact_8375_floor__le__iff,axiom,
    ! [X: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)) ) ).

% floor_le_iff
tff(fact_8376_floor__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)) ) ).

% floor_le_iff
tff(fact_8377_floor__correct,axiom,
    ! [X: 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,X))),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int))) ) ).

% floor_correct
tff(fact_8378_floor__correct,axiom,
    ! [X: 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,X))),X)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int))) ) ).

% floor_correct
tff(fact_8379_cos__expansion__lemma,axiom,
    ! [X: real,Mb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,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(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_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_8380_powr__neg__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X) ) ) ).

% powr_neg_one
tff(fact_8381_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),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,X)) ) ) ).

% sin_gt_zero_02
tff(fact_8382_le__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( 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,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X) ) ) ) ).

% le_log_iff
tff(fact_8383_log__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( 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,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)) ) ) ) ).

% log_le_iff
tff(fact_8384_le__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( 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,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y) ) ) ) ).

% le_powr_iff
tff(fact_8385_powr__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( 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,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ) ).

% powr_le_iff
tff(fact_8386_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_8387_floor__eq2,axiom,
    ! [Na: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Na)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,X) = Na ) ) ) ).

% floor_eq2
tff(fact_8388_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,X)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_8389_floor__divide__lower,axiom,
    ! [Q4: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q4)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q4)))),Q4)),P2) ) ).

% floor_divide_lower
tff(fact_8390_floor__divide__lower,axiom,
    ! [Q4: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q4)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(P2),Q4)))),Q4)),P2) ) ).

% floor_divide_lower
tff(fact_8391_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T6: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T6)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( ( X = aa_real_real(cos_real,T6) )
               => ( Y != aa_real_real(sin_real,T6) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_8392_ln__powr__bound,axiom,
    ! [X: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),A)) ) ) ).

% ln_powr_bound
tff(fact_8393_ln__powr__bound2,axiom,
    ! [X: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(ln_ln_real,X)),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),A)),X)) ) ) ).

% ln_powr_bound2
tff(fact_8394_log__add__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_8395_add__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_8396_minus__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_8397_sin__pi__divide__n__ge__0,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,Na)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_8398_cos__times__cos,axiom,
    ! [W2: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W2)),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(W2),Z))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W2),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_8399_cos__times__cos,axiom,
    ! [W2: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,W2)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W2),Z))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_8400_cos__plus__cos,axiom,
    ! [W2: complex,Z: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,W2)),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(W2),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W2),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_8401_cos__plus__cos,axiom,
    ! [W2: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,W2)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W2),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_8402_floor__divide__upper,axiom,
    ! [Q4: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q4)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,P2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q4)))),one_one_real)),Q4)) ) ).

% floor_divide_upper
tff(fact_8403_floor__divide__upper,axiom,
    ! [Q4: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q4)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(P2),Q4)))),one_one_rat)),Q4)) ) ).

% floor_divide_upper
tff(fact_8404_round__def,axiom,
    ! [X: real] : aa_real_int(archim8280529875227126926d_real,X) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% round_def
tff(fact_8405_round__def,axiom,
    ! [X: rat] : aa_rat_int(archim7778729529865785530nd_rat,X) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% round_def
tff(fact_8406_sin__gt__zero2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,X)) ) ) ).

% sin_gt_zero2
tff(fact_8407_sin__lt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,pi),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real) ) ) ).

% sin_lt_zero
tff(fact_8408_cos__double__less__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X))),one_one_real) ) ) ).

% cos_double_less_one
tff(fact_8409_cos__gt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,X)) ) ) ).

% cos_gt_zero
tff(fact_8410_log__minus__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_8411_termdiff__converges,axiom,
    ! [X: real,K5: real,C: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),K5)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X3)),K5)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_sn(C),X3)) )
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_to(X),C)) ) ) ).

% termdiff_converges
tff(fact_8412_termdiff__converges,axiom,
    ! [X: complex,K5: real,C: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),K5)
     => ( ! [X3: complex] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X3)),K5)
           => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_sp(C),X3)) )
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_tp(X),C)) ) ) ).

% termdiff_converges
tff(fact_8413_powr__neg__numeral,axiom,
    ! [X: real,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,Na))) ) ) ).

% powr_neg_numeral
tff(fact_8414_sin__le__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,pi),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,X)),zero_zero_real) ) ) ).

% sin_le_zero
tff(fact_8415_sin__less__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real) ) ) ).

% sin_less_zero
tff(fact_8416_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_8417_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X)) ) ) ) ).

% sin_monotone_2pi
tff(fact_8418_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,X)) ) ) ).

% cos_gt_zero_pi
tff(fact_8419_sin__pi__divide__n__gt__0,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,Na)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_8420_floor__log2__div2,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_8421_floor__log__nat__eq__if,axiom,
    ! [B: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Na) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_8422_Maclaurin__minus__cos__expansion,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
       => ? [T6: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),T6)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T6),zero_zero_real)
            & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tq,X),set_ord_lessThan_nat(Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_8423_Maclaurin__cos__expansion2,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ? [T6: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T6)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T6),X)
            & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tq,X),set_ord_lessThan_nat(Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_8424_sin__paired,axiom,
    ! [X: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_tr,X)),aa_real_real(sin_real,X)) ).

% sin_paired
tff(fact_8425_tan__double,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) != zero_zero_complex )
       => ( aa_complex_complex(tan_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(tan_complex,X))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(tan_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_8426_tan__double,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) != zero_zero_real )
       => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(tan_real,X))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_8427_arcosh__def,axiom,
    ! [X: real] : aa_real_real(arcosh_real,X) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% arcosh_def
tff(fact_8428_tan__zero,axiom,
    aa_complex_complex(tan_complex,zero_zero_complex) = zero_zero_complex ).

% tan_zero
tff(fact_8429_tan__zero,axiom,
    aa_real_real(tan_real,zero_zero_real) = zero_zero_real ).

% tan_zero
tff(fact_8430_of__real__eq__0__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(real_V1803761363581548252l_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% of_real_eq_0_iff
tff(fact_8431_of__real__eq__0__iff,axiom,
    ! [X: real] :
      ( ( aa_real_complex(real_V4546457046886955230omplex,X) = zero_zero_complex )
    <=> ( X = zero_zero_real ) ) ).

% of_real_eq_0_iff
tff(fact_8432_of__real__0,axiom,
    aa_real_real(real_V1803761363581548252l_real,zero_zero_real) = zero_zero_real ).

% of_real_0
tff(fact_8433_of__real__0,axiom,
    aa_real_complex(real_V4546457046886955230omplex,zero_zero_real) = zero_zero_complex ).

% of_real_0
tff(fact_8434_fact__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).

% fact_0
tff(fact_8435_fact__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).

% fact_0
tff(fact_8436_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_8437_fact__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).

% fact_0
tff(fact_8438_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_8439_of__real__add,axiom,
    ! [X: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,X)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_add
tff(fact_8440_of__real__add,axiom,
    ! [X: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_add
tff(fact_8441_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_8442_fact__Suc__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).

% fact_Suc_0
tff(fact_8443_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_8444_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_8445_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_8446_fact__Suc,axiom,
    ! [Na: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,Na)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Na))),aa_nat_complex(semiri5044797733671781792omplex,Na)) ).

% fact_Suc
tff(fact_8447_fact__Suc,axiom,
    ! [Na: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,Na)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Na))),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ).

% fact_Suc
tff(fact_8448_fact__Suc,axiom,
    ! [Na: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na))),aa_nat_int(semiri1406184849735516958ct_int,Na)) ).

% fact_Suc
tff(fact_8449_fact__Suc,axiom,
    ! [Na: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na))),aa_nat_real(semiri2265585572941072030t_real,Na)) ).

% fact_Suc
tff(fact_8450_fact__Suc,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Na))),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_Suc
tff(fact_8451_sin__of__real__pi,axiom,
    aa_real_real(sin_real,aa_real_real(real_V1803761363581548252l_real,pi)) = zero_zero_real ).

% sin_of_real_pi
tff(fact_8452_sin__of__real__pi,axiom,
    aa_complex_complex(sin_complex,aa_real_complex(real_V4546457046886955230omplex,pi)) = zero_zero_complex ).

% sin_of_real_pi
tff(fact_8453_norm__of__real__add1,axiom,
    ! [X: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,X)),one_one_real)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) ).

% norm_of_real_add1
tff(fact_8454_norm__of__real__add1,axiom,
    ! [X: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),one_one_complex)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) ).

% norm_of_real_add1
tff(fact_8455_norm__of__real__addn,axiom,
    ! [X: real,B: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,X)),aa_num_real(numeral_numeral_real,B))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_num_real(numeral_numeral_real,B))) ).

% norm_of_real_addn
tff(fact_8456_norm__of__real__addn,axiom,
    ! [X: real,B: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),numera6690914467698888265omplex(B))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_num_real(numeral_numeral_real,B))) ).

% norm_of_real_addn
tff(fact_8457_cos__of__real__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_of_real_pi_half
tff(fact_8458_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_8459_fact__nonzero,axiom,
    ! [Na: nat] : aa_nat_complex(semiri5044797733671781792omplex,Na) != zero_zero_complex ).

% fact_nonzero
tff(fact_8460_fact__nonzero,axiom,
    ! [Na: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Na) != zero_zero_rat ).

% fact_nonzero
tff(fact_8461_fact__nonzero,axiom,
    ! [Na: nat] : aa_nat_int(semiri1406184849735516958ct_int,Na) != zero_zero_int ).

% fact_nonzero
tff(fact_8462_fact__nonzero,axiom,
    ! [Na: nat] : aa_nat_real(semiri2265585572941072030t_real,Na) != zero_zero_real ).

% fact_nonzero
tff(fact_8463_fact__nonzero,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Na) != zero_zero_nat ).

% fact_nonzero
tff(fact_8464_fact__ge__zero,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ).

% fact_ge_zero
tff(fact_8465_fact__ge__zero,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Na)) ).

% fact_ge_zero
tff(fact_8466_fact__ge__zero,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Na)) ).

% fact_ge_zero
tff(fact_8467_fact__ge__zero,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_ge_zero
tff(fact_8468_fact__not__neg,axiom,
    ! [Na: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Na)),zero_zero_rat) ).

% fact_not_neg
tff(fact_8469_fact__not__neg,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Na)),zero_zero_int) ).

% fact_not_neg
tff(fact_8470_fact__not__neg,axiom,
    ! [Na: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Na)),zero_zero_real) ).

% fact_not_neg
tff(fact_8471_fact__not__neg,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),zero_zero_nat) ).

% fact_not_neg
tff(fact_8472_fact__gt__zero,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ).

% fact_gt_zero
tff(fact_8473_fact__gt__zero,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Na)) ).

% fact_gt_zero
tff(fact_8474_fact__gt__zero,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Na)) ).

% fact_gt_zero
tff(fact_8475_fact__gt__zero,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_gt_zero
tff(fact_8476_fact__ge__1,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ).

% fact_ge_1
tff(fact_8477_fact__ge__1,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,Na)) ).

% fact_ge_1
tff(fact_8478_fact__ge__1,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri2265585572941072030t_real,Na)) ).

% fact_ge_1
tff(fact_8479_fact__ge__1,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_ge_1
tff(fact_8480_fact__mono,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ) ).

% fact_mono
tff(fact_8481_fact__mono,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,Na)) ) ).

% fact_mono
tff(fact_8482_fact__mono,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,Na)) ) ).

% fact_mono
tff(fact_8483_fact__mono,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ).

% fact_mono
tff(fact_8484_fact__dvd,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,Na)),aa_nat_int(semiri1406184849735516958ct_int,Mb)) ) ).

% fact_dvd
tff(fact_8485_fact__dvd,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri3624122377584611663nteger,Na)),aa_nat_Code_integer(semiri3624122377584611663nteger,Mb)) ) ).

% fact_dvd
tff(fact_8486_fact__dvd,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(semiri2265585572941072030t_real,Na)),aa_nat_real(semiri2265585572941072030t_real,Mb)) ) ).

% fact_dvd
tff(fact_8487_fact__dvd,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(semiri1408675320244567234ct_nat,Mb)) ) ).

% fact_dvd
tff(fact_8488_fact__less__mono,axiom,
    ! [Mb: nat,Na: 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),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ) ) ).

% fact_less_mono
tff(fact_8489_fact__less__mono,axiom,
    ! [Mb: nat,Na: 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),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,Na)) ) ) ).

% fact_less_mono
tff(fact_8490_fact__less__mono,axiom,
    ! [Mb: nat,Na: 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),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,Na)) ) ) ).

% fact_less_mono
tff(fact_8491_fact__less__mono,axiom,
    ! [Mb: nat,Na: 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),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ) ).

% fact_less_mono
tff(fact_8492_fact__fact__dvd__fact,axiom,
    ! [K: nat,Na: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,Na))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))) ).

% fact_fact_dvd_fact
tff(fact_8493_fact__fact__dvd__fact,axiom,
    ! [K: nat,Na: nat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,Na))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))) ).

% fact_fact_dvd_fact
tff(fact_8494_fact__fact__dvd__fact,axiom,
    ! [K: nat,Na: 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,K)),aa_nat_int(semiri1406184849735516958ct_int,Na))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))) ).

% fact_fact_dvd_fact
tff(fact_8495_fact__fact__dvd__fact,axiom,
    ! [K: nat,Na: nat] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))) ).

% fact_fact_dvd_fact
tff(fact_8496_fact__fact__dvd__fact,axiom,
    ! [K: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,Na))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))) ).

% fact_fact_dvd_fact
tff(fact_8497_fact__mod,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_int_int(modulo_modulo_int(aa_nat_int(semiri1406184849735516958ct_int,Na)),aa_nat_int(semiri1406184849735516958ct_int,Mb)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_8498_fact__mod,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( modulo364778990260209775nteger(aa_nat_Code_integer(semiri3624122377584611663nteger,Na),aa_nat_Code_integer(semiri3624122377584611663nteger,Mb)) = zero_z3403309356797280102nteger ) ) ).

% fact_mod
tff(fact_8499_fact__mod,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(semiri1408675320244567234ct_nat,Mb)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_8500_fact__le__power,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Na)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),Na))) ).

% fact_le_power
tff(fact_8501_fact__le__power,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Na)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),Na))) ).

% fact_le_power
tff(fact_8502_fact__le__power,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),Na))) ).

% fact_le_power
tff(fact_8503_fact__le__power,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),Na))) ).

% fact_le_power
tff(fact_8504_norm__less__p1,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,aa_real_real(real_V7735802525324610683m_real,X))),one_one_real))) ).

% norm_less_p1
tff(fact_8505_norm__less__p1,axiom,
    ! [X: complex] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,X))),one_one_complex))) ).

% norm_less_p1
tff(fact_8506_choose__dvd,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_Code_integer(semiri3624122377584611663nteger,Na)) ) ).

% choose_dvd
tff(fact_8507_choose__dvd,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ) ).

% choose_dvd
tff(fact_8508_choose__dvd,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_int(semiri1406184849735516958ct_int,Na)) ) ).

% choose_dvd
tff(fact_8509_choose__dvd,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_real(semiri2265585572941072030t_real,Na)) ) ).

% choose_dvd
tff(fact_8510_choose__dvd,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ).

% choose_dvd
tff(fact_8511_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_complex(semiri5044797733671781792omplex,Mb) = $ite(Mb = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8512_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_rat(semiri773545260158071498ct_rat,Mb) = $ite(Mb = zero_zero_nat,one_one_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8513_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_int(semiri1406184849735516958ct_int,Mb) = $ite(Mb = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8514_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_real(semiri2265585572941072030t_real,Mb) = $ite(Mb = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8515_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_nat(semiri1408675320244567234ct_nat,Mb) = $ite(Mb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8516_fact__reduce,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_complex(semiri5044797733671781792omplex,Na) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8517_fact__reduce,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_rat(semiri773545260158071498ct_rat,Na) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8518_fact__reduce,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_int(semiri1406184849735516958ct_int,Na) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8519_fact__reduce,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_real(semiri2265585572941072030t_real,Na) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8520_fact__reduce,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Na) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8521_tan__gt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tan_real,X)) ) ) ).

% tan_gt_zero
tff(fact_8522_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(tan_real,X3)) ) ) ).

% lemma_tan_total
tff(fact_8523_lemma__tan__total1,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(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X3) = Y ) ) ).

% lemma_tan_total1
tff(fact_8524_tan__mono__lt__eq,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_8525_tan__monotone_H,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_8526_tan__monotone,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X)) ) ) ) ).

% tan_monotone
tff(fact_8527_tan__total,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(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X3) = Y )
      & ! [Y4: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(tan_real,Y4) = Y ) )
         => ( Y4 = X3 ) ) ) ).

% tan_total
tff(fact_8528_add__tan__eq,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(X),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y))) ) ) ) ).

% add_tan_eq
tff(fact_8529_add__tan__eq,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).

% add_tan_eq
tff(fact_8530_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ? [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(tan_real,X3) = Y ) ) ) ).

% tan_total_pos
tff(fact_8531_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,X)) ) ) ).

% tan_pos_pi2_le
tff(fact_8532_tan__less__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,X)),zero_zero_real) ) ) ).

% tan_less_zero
tff(fact_8533_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_8534_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)) ) ) ) ).

% tan_mono_le
tff(fact_8535_tan__bound__pi2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,X))),one_one_real) ) ).

% tan_bound_pi2
tff(fact_8536_arctan,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y)) = Y ) ) ).

% arctan
tff(fact_8537_arctan__tan,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,X)) = X ) ) ) ).

% arctan_tan
tff(fact_8538_arctan__unique,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( ( aa_real_real(tan_real,X) = Y )
         => ( aa_real_real(arctan,Y) = X ) ) ) ) ).

% arctan_unique
tff(fact_8539_Maclaurin__zero,axiom,
    ! [X: real,Na: nat,Diff: fun_na5723442673449079972x_real] :
      ( ( X = zero_zero_real )
     => ( ( Na != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun5746040925223427547t_real(aTP_Lamm_ts(X),Diff),set_ord_lessThan_nat(Na)) = aa_complex_real(aa_nat480766339200631653x_real(Diff,zero_zero_nat),zero_zero_complex) ) ) ) ).

% Maclaurin_zero
tff(fact_8540_Maclaurin__zero,axiom,
    ! [X: real,Na: nat,Diff: fun_na6724076748692969250l_real] :
      ( ( X = zero_zero_real )
     => ( ( Na != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_tt(X),Diff),set_ord_lessThan_nat(Na)) = aa_real_real(aa_nat_fun_real_real(Diff,zero_zero_nat),zero_zero_real) ) ) ) ).

% Maclaurin_zero
tff(fact_8541_Maclaurin__zero,axiom,
    ! [X: real,Na: nat,Diff: fun_nat_fun_rat_real] :
      ( ( X = zero_zero_real )
     => ( ( Na != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun507281639146497029t_real(aTP_Lamm_tu(X),Diff),set_ord_lessThan_nat(Na)) = aa_rat_real(aa_nat_fun_rat_real(Diff,zero_zero_nat),zero_zero_rat) ) ) ) ).

% Maclaurin_zero
tff(fact_8542_Maclaurin__zero,axiom,
    ! [X: real,Na: nat,Diff: fun_nat_fun_nat_real] :
      ( ( X = zero_zero_real )
     => ( ( Na != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun7039663905532832253t_real(aTP_Lamm_tv(X),Diff),set_ord_lessThan_nat(Na)) = aa_nat_real(aa_nat_fun_nat_real(Diff,zero_zero_nat),zero_zero_nat) ) ) ) ).

% Maclaurin_zero
tff(fact_8543_Maclaurin__zero,axiom,
    ! [X: real,Na: nat,Diff: fun_nat_fun_int_real] :
      ( ( X = zero_zero_real )
     => ( ( Na != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun241848180832411225t_real(aTP_Lamm_tw(X),Diff),set_ord_lessThan_nat(Na)) = aa_int_real(aa_nat_fun_int_real(Diff,zero_zero_nat),zero_zero_int) ) ) ) ).

% Maclaurin_zero
tff(fact_8544_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ? [B8: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_tx(H),J),set_ord_lessThan_nat(Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B8),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Na)),aa_nat_real(semiri2265585572941072030t_real,Na)))) ) ).

% Maclaurin_lemma
tff(fact_8545_tan__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).

% tan_add
tff(fact_8546_tan__add,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_add
tff(fact_8547_tan__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(X),Y)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(minus_minus_complex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_8548_tan__diff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_8549_lemma__tan__add1,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_8550_lemma__tan__add1,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_8551_minus__sin__cos__eq,axiom,
    ! [X: real] : aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_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_8552_minus__sin__cos__eq,axiom,
    ! [X: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(sin_complex,X)) = aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% minus_sin_cos_eq
tff(fact_8553_tan__total__pi4,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ? [Z2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))),Z2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))
          & ( aa_real_real(tan_real,Z2) = X ) ) ) ).

% tan_total_pi4
tff(fact_8554_tan__half,axiom,
    ! [X: complex] : aa_complex_complex(tan_complex,X) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X))),aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X))),one_one_complex)) ).

% tan_half
tff(fact_8555_tan__half,axiom,
    ! [X: real] : aa_real_real(tan_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X))),one_one_real)) ).

% tan_half
tff(fact_8556_arsinh__def,axiom,
    ! [X: real] : aa_real_real(arsinh_real,X) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% arsinh_def
tff(fact_8557_Maclaurin__sin__expansion3,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ? [T6: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T6)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T6),X)
            & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ty,X),set_ord_lessThan_nat(Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_8558_Maclaurin__sin__expansion4,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ? [T6: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T6)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),X)
          & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ty,X),set_ord_lessThan_nat(Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_8559_sin__coeff__def,axiom,
    ! [X2: nat] :
      aa_nat_real(sin_coeff,X2) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X2),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X2),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X2))) ).

% sin_coeff_def
tff(fact_8560_Maclaurin__exp__lt,axiom,
    ! [X: real,Na: nat] :
      ( ( X != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ? [T6: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T6))
            & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X))
            & ( aa_real_real(exp_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tz,X),set_ord_lessThan_nat(Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_8561_round__altdef,axiom,
    ! [X: real] :
      aa_real_int(archim8280529875227126926d_real,X) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(archim2898591450579166408c_real,X)),aa_real_int(archim7802044766580827645g_real,X),aa_real_int(archim6058952711729229775r_real,X)) ).

% round_altdef
tff(fact_8562_round__altdef,axiom,
    ! [X: rat] :
      aa_rat_int(archim7778729529865785530nd_rat,X) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(archimedean_frac_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,X),aa_rat_int(archim3151403230148437115or_rat,X)) ).

% round_altdef
tff(fact_8563_exp__less__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)) ) ).

% exp_less_mono
tff(fact_8564_exp__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ).

% exp_less_cancel_iff
tff(fact_8565_exp__zero,axiom,
    aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).

% exp_zero
tff(fact_8566_exp__zero,axiom,
    aa_real_real(exp_real,zero_zero_real) = one_one_real ).

% exp_zero
tff(fact_8567_frac__of__int,axiom,
    ! [Z: int] : aa_real_real(archim2898591450579166408c_real,aa_int_real(ring_1_of_int_real,Z)) = zero_zero_real ).

% frac_of_int
tff(fact_8568_frac__of__int,axiom,
    ! [Z: int] : aa_rat_rat(archimedean_frac_rat,aa_int_rat(ring_1_of_int_rat,Z)) = zero_zero_rat ).

% frac_of_int
tff(fact_8569_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_8570_one__less__exp__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X) ) ).

% one_less_exp_iff
tff(fact_8571_exp__less__one__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,X)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real) ) ).

% exp_less_one_iff
tff(fact_8572_exp__ln__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,X)) = X )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X) ) ).

% exp_ln_iff
tff(fact_8573_exp__ln,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(exp_real,aa_real_real(ln_ln_real,X)) = X ) ) ).

% exp_ln
tff(fact_8574_fact__mono__nat,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ).

% fact_mono_nat
tff(fact_8575_fact__ge__self,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_ge_self
tff(fact_8576_exp__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ).

% exp_less_cancel
tff(fact_8577_exp__not__eq__zero,axiom,
    ! [X: complex] : aa_complex_complex(exp_complex,X) != zero_zero_complex ).

% exp_not_eq_zero
tff(fact_8578_exp__not__eq__zero,axiom,
    ! [X: real] : aa_real_real(exp_real,X) != zero_zero_real ).

% exp_not_eq_zero
tff(fact_8579_fact__less__mono__nat,axiom,
    ! [Mb: nat,Na: 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),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ) ).

% fact_less_mono_nat
tff(fact_8580_exp__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X3: real] : aa_real_real(exp_real,X3) = Y ) ).

% exp_total
tff(fact_8581_exp__gt__zero,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(exp_real,X)) ).

% exp_gt_zero
tff(fact_8582_not__exp__less__zero,axiom,
    ! [X: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,X)),zero_zero_real) ).

% not_exp_less_zero
tff(fact_8583_mult__exp__exp,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,Y)) = aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(X),Y)) ).

% mult_exp_exp
tff(fact_8584_mult__exp__exp,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)) = aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) ).

% mult_exp_exp
tff(fact_8585_exp__add__commuting,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(times_times_complex(X),Y) = aa_complex_complex(times_times_complex(Y),X) )
     => ( aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,Y)) ) ) ).

% exp_add_commuting
tff(fact_8586_exp__add__commuting,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X) )
     => ( aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)) ) ) ).

% exp_add_commuting
tff(fact_8587_frac__ge__0,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,X)) ).

% frac_ge_0
tff(fact_8588_frac__ge__0,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,X)) ).

% frac_ge_0
tff(fact_8589_frac__lt__1,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(archim2898591450579166408c_real,X)),one_one_real) ).

% frac_lt_1
tff(fact_8590_frac__lt__1,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(archimedean_frac_rat,X)),one_one_rat) ).

% frac_lt_1
tff(fact_8591_frac__1__eq,axiom,
    ! [X: real] : aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) = aa_real_real(archim2898591450579166408c_real,X) ).

% frac_1_eq
tff(fact_8592_frac__1__eq,axiom,
    ! [X: rat] : aa_rat_rat(archimedean_frac_rat,aa_rat_rat(plus_plus_rat(X),one_one_rat)) = aa_rat_rat(archimedean_frac_rat,X) ).

% frac_1_eq
tff(fact_8593_fact__ge__Suc__0__nat,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_ge_Suc_0_nat
tff(fact_8594_exp__gt__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,X)) ) ).

% exp_gt_one
tff(fact_8595_dvd__fact,axiom,
    ! [Mb: nat,Na: 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),Na)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ) ).

% dvd_fact
tff(fact_8596_fact__diff__Suc,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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)),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Na)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na))) ) ) ).

% fact_diff_Suc
tff(fact_8597_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(ln_ln_real,X))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Y)),X) ) ) ).

% ln_ge_iff
tff(fact_8598_fact__div__fact__le__pow,axiom,
    ! [R2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),R2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),R2)) ) ).

% fact_div_fact_le_pow
tff(fact_8599_powr__def,axiom,
    ! [X: real,A: real] :
      aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A) = $ite(X = zero_zero_real,zero_zero_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(ln_ln_real,X)))) ).

% powr_def
tff(fact_8600_exp__divide__power__eq,axiom,
    ! [Na: nat,X: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,aa_complex_complex(divide1717551699836669952omplex(X),aa_nat_complex(semiri8010041392384452111omplex,Na)))),Na) = aa_complex_complex(exp_complex,X) ) ) ).

% exp_divide_power_eq
tff(fact_8601_exp__divide__power__eq,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_nat_real(semiri5074537144036343181t_real,Na)))),Na) = aa_real_real(exp_real,X) ) ) ).

% exp_divide_power_eq
tff(fact_8602_frac__eq,axiom,
    ! [X: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,X) = X )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ) ).

% frac_eq
tff(fact_8603_frac__eq,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,X) = X )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),one_one_rat) ) ) ).

% frac_eq
tff(fact_8604_frac__add,axiom,
    ! [X: real,Y: real] :
      aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real)) ).

% frac_add
tff(fact_8605_frac__add,axiom,
    ! [X: rat,Y: rat] :
      aa_rat_rat(archimedean_frac_rat,aa_rat_rat(plus_plus_rat(X),Y)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y)),aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat)) ).

% frac_add
tff(fact_8606_tanh__altdef,axiom,
    ! [X: real] : aa_real_real(tanh_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))) ).

% tanh_altdef
tff(fact_8607_sin__coeff__Suc,axiom,
    ! [Na: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(cos_coeff,Na)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na))) ).

% sin_coeff_Suc
tff(fact_8608_floor__add,axiom,
    ! [X: real,Y: real] :
      aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y))),one_one_int)) ).

% floor_add
tff(fact_8609_floor__add,axiom,
    ! [X: rat,Y: rat] :
      aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),Y)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))),one_one_int)) ).

% floor_add
tff(fact_8610_cos__coeff__Suc,axiom,
    ! [Na: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,Na))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na))) ).

% cos_coeff_Suc
tff(fact_8611_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Na: nat,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_nat_real(semiri5074537144036343181t_real,Na)))),Na)),aa_real_real(exp_real,X)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_8612_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_nat_real(semiri5074537144036343181t_real,Na)))),Na)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_8613_log__base__10__eq2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,X)) ) ) ).

% log_base_10_eq2
tff(fact_8614_log__base__10__eq1,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(ln_ln_real,X)) ) ) ).

% log_base_10_eq1
tff(fact_8615_sin__tan,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(tan_real,X)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_8616_cos__tan,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_8617_monoseq__def,axiom,
    ! [X5: fun_nat_real] :
      ( topolo6980174941875973593q_real(X5)
    <=> ( ! [M5: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,M5)),aa_nat_real(X5,N3)) )
        | ! [M5: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,N3)),aa_nat_real(X5,M5)) ) ) ) ).

% monoseq_def
tff(fact_8618_monoseq__def,axiom,
    ! [X5: fun_nat_set_int] :
      ( topolo3100542954746470799et_int(X5)
    <=> ( ! [M5: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N3)
           => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(X5,M5)),aa_nat_set_int(X5,N3)) )
        | ! [M5: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N3)
           => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(X5,N3)),aa_nat_set_int(X5,M5)) ) ) ) ).

% monoseq_def
tff(fact_8619_monoseq__def,axiom,
    ! [X5: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X5)
    <=> ( ! [M5: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,M5)),aa_nat_rat(X5,N3)) )
        | ! [M5: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,N3)),aa_nat_rat(X5,M5)) ) ) ) ).

% monoseq_def
tff(fact_8620_monoseq__def,axiom,
    ! [X5: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X5)
    <=> ( ! [M5: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N3)
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,M5)),aa_nat_num(X5,N3)) )
        | ! [M5: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N3)
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,N3)),aa_nat_num(X5,M5)) ) ) ) ).

% monoseq_def
tff(fact_8621_monoseq__def,axiom,
    ! [X5: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X5)
    <=> ( ! [M5: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,M5)),aa_nat_nat(X5,N3)) )
        | ! [M5: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,N3)),aa_nat_nat(X5,M5)) ) ) ) ).

% monoseq_def
tff(fact_8622_monoseq__def,axiom,
    ! [X5: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X5)
    <=> ( ! [M5: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,M5)),aa_nat_int(X5,N3)) )
        | ! [M5: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,N3)),aa_nat_int(X5,M5)) ) ) ) ).

% monoseq_def
tff(fact_8623_monoI2,axiom,
    ! [X5: 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(X5,N)),aa_nat_real(X5,M)) )
     => topolo6980174941875973593q_real(X5) ) ).

% monoI2
tff(fact_8624_monoI2,axiom,
    ! [X5: fun_nat_set_int] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(X5,N)),aa_nat_set_int(X5,M)) )
     => topolo3100542954746470799et_int(X5) ) ).

% monoI2
tff(fact_8625_monoI2,axiom,
    ! [X5: 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(X5,N)),aa_nat_rat(X5,M)) )
     => topolo4267028734544971653eq_rat(X5) ) ).

% monoI2
tff(fact_8626_monoI2,axiom,
    ! [X5: 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(X5,N)),aa_nat_num(X5,M)) )
     => topolo1459490580787246023eq_num(X5) ) ).

% monoI2
tff(fact_8627_monoI2,axiom,
    ! [X5: 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(X5,N)),aa_nat_nat(X5,M)) )
     => topolo4902158794631467389eq_nat(X5) ) ).

% monoI2
tff(fact_8628_monoI2,axiom,
    ! [X5: 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(X5,N)),aa_nat_int(X5,M)) )
     => topolo4899668324122417113eq_int(X5) ) ).

% monoI2
tff(fact_8629_monoI1,axiom,
    ! [X5: 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(X5,M)),aa_nat_real(X5,N)) )
     => topolo6980174941875973593q_real(X5) ) ).

% monoI1
tff(fact_8630_monoI1,axiom,
    ! [X5: fun_nat_set_int] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(X5,M)),aa_nat_set_int(X5,N)) )
     => topolo3100542954746470799et_int(X5) ) ).

% monoI1
tff(fact_8631_monoI1,axiom,
    ! [X5: 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(X5,M)),aa_nat_rat(X5,N)) )
     => topolo4267028734544971653eq_rat(X5) ) ).

% monoI1
tff(fact_8632_monoI1,axiom,
    ! [X5: 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(X5,M)),aa_nat_num(X5,N)) )
     => topolo1459490580787246023eq_num(X5) ) ).

% monoI1
tff(fact_8633_monoI1,axiom,
    ! [X5: 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(X5,M)),aa_nat_nat(X5,N)) )
     => topolo4902158794631467389eq_nat(X5) ) ).

% monoI1
tff(fact_8634_monoI1,axiom,
    ! [X5: 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(X5,M)),aa_nat_int(X5,N)) )
     => topolo4899668324122417113eq_int(X5) ) ).

% monoI1
tff(fact_8635_real__sqrt__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ).

% real_sqrt_less_iff
tff(fact_8636_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,X)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real) ) ).

% real_sqrt_lt_0_iff
tff(fact_8637_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ).

% real_sqrt_gt_0_iff
tff(fact_8638_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,X)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ).

% real_sqrt_lt_1_iff
tff(fact_8639_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ).

% real_sqrt_gt_1_iff
tff(fact_8640_real__sqrt__less__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)) ) ).

% real_sqrt_less_mono
tff(fact_8641_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,X)) ) ).

% real_sqrt_gt_zero
tff(fact_8642_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_8643_real__less__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_8644_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),U)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_8645_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( 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,X),aa_nat_real(aa_real_fun_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,X)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_8646_ln__sqrt,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,X)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_8647_arsinh__real__aux,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_aux
tff(fact_8648_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_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_8649_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
         => ( 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(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_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_8650_monoseq__Suc,axiom,
    ! [X5: fun_nat_real] :
      ( topolo6980174941875973593q_real(X5)
    <=> ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,N3)),aa_nat_real(X5,aa_nat_nat(suc,N3)))
        | ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,aa_nat_nat(suc,N3))),aa_nat_real(X5,N3)) ) ) ).

% monoseq_Suc
tff(fact_8651_monoseq__Suc,axiom,
    ! [X5: fun_nat_set_int] :
      ( topolo3100542954746470799et_int(X5)
    <=> ( ! [N3: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(X5,N3)),aa_nat_set_int(X5,aa_nat_nat(suc,N3)))
        | ! [N3: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(X5,aa_nat_nat(suc,N3))),aa_nat_set_int(X5,N3)) ) ) ).

% monoseq_Suc
tff(fact_8652_monoseq__Suc,axiom,
    ! [X5: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X5)
    <=> ( ! [N3: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,N3)),aa_nat_rat(X5,aa_nat_nat(suc,N3)))
        | ! [N3: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,aa_nat_nat(suc,N3))),aa_nat_rat(X5,N3)) ) ) ).

% monoseq_Suc
tff(fact_8653_monoseq__Suc,axiom,
    ! [X5: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X5)
    <=> ( ! [N3: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,N3)),aa_nat_num(X5,aa_nat_nat(suc,N3)))
        | ! [N3: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,aa_nat_nat(suc,N3))),aa_nat_num(X5,N3)) ) ) ).

% monoseq_Suc
tff(fact_8654_monoseq__Suc,axiom,
    ! [X5: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X5)
    <=> ( ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,N3)),aa_nat_nat(X5,aa_nat_nat(suc,N3)))
        | ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,aa_nat_nat(suc,N3))),aa_nat_nat(X5,N3)) ) ) ).

% monoseq_Suc
tff(fact_8655_monoseq__Suc,axiom,
    ! [X5: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X5)
    <=> ( ! [N3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,N3)),aa_nat_int(X5,aa_nat_nat(suc,N3)))
        | ! [N3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,aa_nat_nat(suc,N3))),aa_nat_int(X5,N3)) ) ) ).

% monoseq_Suc
tff(fact_8656_mono__SucI2,axiom,
    ! [X5: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,aa_nat_nat(suc,N))),aa_nat_real(X5,N))
     => topolo6980174941875973593q_real(X5) ) ).

% mono_SucI2
tff(fact_8657_mono__SucI2,axiom,
    ! [X5: fun_nat_set_int] :
      ( ! [N: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(X5,aa_nat_nat(suc,N))),aa_nat_set_int(X5,N))
     => topolo3100542954746470799et_int(X5) ) ).

% mono_SucI2
tff(fact_8658_mono__SucI2,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,aa_nat_nat(suc,N))),aa_nat_rat(X5,N))
     => topolo4267028734544971653eq_rat(X5) ) ).

% mono_SucI2
tff(fact_8659_mono__SucI2,axiom,
    ! [X5: fun_nat_num] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,aa_nat_nat(suc,N))),aa_nat_num(X5,N))
     => topolo1459490580787246023eq_num(X5) ) ).

% mono_SucI2
tff(fact_8660_mono__SucI2,axiom,
    ! [X5: fun_nat_nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,aa_nat_nat(suc,N))),aa_nat_nat(X5,N))
     => topolo4902158794631467389eq_nat(X5) ) ).

% mono_SucI2
tff(fact_8661_mono__SucI2,axiom,
    ! [X5: fun_nat_int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,aa_nat_nat(suc,N))),aa_nat_int(X5,N))
     => topolo4899668324122417113eq_int(X5) ) ).

% mono_SucI2
tff(fact_8662_mono__SucI1,axiom,
    ! [X5: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,N)),aa_nat_real(X5,aa_nat_nat(suc,N)))
     => topolo6980174941875973593q_real(X5) ) ).

% mono_SucI1
tff(fact_8663_mono__SucI1,axiom,
    ! [X5: fun_nat_set_int] :
      ( ! [N: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(X5,N)),aa_nat_set_int(X5,aa_nat_nat(suc,N)))
     => topolo3100542954746470799et_int(X5) ) ).

% mono_SucI1
tff(fact_8664_mono__SucI1,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,N)),aa_nat_rat(X5,aa_nat_nat(suc,N)))
     => topolo4267028734544971653eq_rat(X5) ) ).

% mono_SucI1
tff(fact_8665_mono__SucI1,axiom,
    ! [X5: fun_nat_num] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,N)),aa_nat_num(X5,aa_nat_nat(suc,N)))
     => topolo1459490580787246023eq_num(X5) ) ).

% mono_SucI1
tff(fact_8666_mono__SucI1,axiom,
    ! [X5: fun_nat_nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,N)),aa_nat_nat(X5,aa_nat_nat(suc,N)))
     => topolo4902158794631467389eq_nat(X5) ) ).

% mono_SucI1
tff(fact_8667_mono__SucI1,axiom,
    ! [X5: fun_nat_int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,N)),aa_nat_int(X5,aa_nat_nat(suc,N)))
     => topolo4899668324122417113eq_int(X5) ) ).

% mono_SucI1
tff(fact_8668_pochhammer__double,axiom,
    ! [Z: complex,Na: 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))),Na)) = 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))),Na)))),aa_nat_complex(comm_s2602460028002588243omplex(Z),Na))),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))))),Na)) ).

% pochhammer_double
tff(fact_8669_pochhammer__double,axiom,
    ! [Z: rat,Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Na))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),Na)) ).

% pochhammer_double
tff(fact_8670_pochhammer__double,axiom,
    ! [Z: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Na))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Na)) ).

% pochhammer_double
tff(fact_8671_of__nat__code,axiom,
    ! [Na: nat] : aa_nat_complex(semiri8010041392384452111omplex,Na) = aa_complex_complex(aa_nat3829984526574172007omplex(semiri2816024913162550771omplex(aTP_Lamm_ua),Na),zero_zero_complex) ).

% of_nat_code
tff(fact_8672_of__nat__code,axiom,
    ! [Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,Na) = aa_rat_rat(semiri7787848453975740701ux_rat(aTP_Lamm_ub,Na),zero_zero_rat) ).

% of_nat_code
tff(fact_8673_of__nat__code,axiom,
    ! [Na: nat] : aa_nat_int(semiri1314217659103216013at_int,Na) = aa_int_int(semiri8420488043553186161ux_int(aTP_Lamm_uc,Na),zero_zero_int) ).

% of_nat_code
tff(fact_8674_of__nat__code,axiom,
    ! [Na: nat] : aa_nat_real(semiri5074537144036343181t_real,Na) = aa_real_real(aa_nat_fun_real_real(semiri7260567687927622513x_real(aTP_Lamm_ud),Na),zero_zero_real) ).

% of_nat_code
tff(fact_8675_of__nat__code,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Na) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_ue),Na),zero_zero_nat) ).

% of_nat_code
tff(fact_8676_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,Mb: nat] : groups2073611262835488442omplex(aTP_Lamm_uf(R2),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_8677_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_ug(R2),set_or1269000886237332187st_nat(zero_zero_nat,Mb)) = aa_rat_rat(times_times_rat(aa_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_8678_gchoose__row__sum__weighted,axiom,
    ! [R2: real,Mb: nat] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_uh,R2),set_or1269000886237332187st_nat(zero_zero_nat,Mb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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(aa_real_fun_nat_real(gbinomial_real,R2),aa_nat_nat(suc,Mb))) ).

% gchoose_row_sum_weighted
tff(fact_8679_central__binomial__lower__bound,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),Na)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),Na))) ) ).

% central_binomial_lower_bound
tff(fact_8680_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T6: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T6)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( Z != aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(cos_real,T6)),aa_real_real(sin_real,T6)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_8681_binomial__Suc__n,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),Na) = aa_nat_nat(suc,Na) ).

% binomial_Suc_n
tff(fact_8682_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_complex(gbinomial_complex(zero_zero_complex),aa_nat_nat(suc,K)) = zero_zero_complex ).

% gbinomial_0(2)
tff(fact_8683_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,zero_zero_real),aa_nat_nat(suc,K)) = zero_zero_real ).

% gbinomial_0(2)
tff(fact_8684_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_rat(gbinomial_rat(zero_zero_rat),aa_nat_nat(suc,K)) = zero_zero_rat ).

% gbinomial_0(2)
tff(fact_8685_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% gbinomial_0(2)
tff(fact_8686_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_int(gbinomial_int(zero_zero_int),aa_nat_nat(suc,K)) = zero_zero_int ).

% gbinomial_0(2)
tff(fact_8687_binomial__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,zero_zero_nat)) = Na ).

% binomial_1
tff(fact_8688_binomial__0__Suc,axiom,
    ! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% binomial_0_Suc
tff(fact_8689_binomial__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K) ) ).

% binomial_eq_0_iff
tff(fact_8690_gbinomial__0_I1_J,axiom,
    ! [A: complex] : aa_nat_complex(gbinomial_complex(A),zero_zero_nat) = one_one_complex ).

% gbinomial_0(1)
tff(fact_8691_gbinomial__0_I1_J,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),zero_zero_nat) = one_one_real ).

% gbinomial_0(1)
tff(fact_8692_gbinomial__0_I1_J,axiom,
    ! [A: rat] : aa_nat_rat(gbinomial_rat(A),zero_zero_nat) = one_one_rat ).

% gbinomial_0(1)
tff(fact_8693_gbinomial__0_I1_J,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),zero_zero_nat) = one_one_nat ).

% gbinomial_0(1)
tff(fact_8694_gbinomial__0_I1_J,axiom,
    ! [A: int] : aa_nat_int(gbinomial_int(A),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_8695_binomial__Suc__Suc,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,K))) ).

% binomial_Suc_Suc
tff(fact_8696_pochhammer__0,axiom,
    ! [A: complex] : aa_nat_complex(comm_s2602460028002588243omplex(A),zero_zero_nat) = one_one_complex ).

% pochhammer_0
tff(fact_8697_pochhammer__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_8698_pochhammer__0,axiom,
    ! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).

% pochhammer_0
tff(fact_8699_pochhammer__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),zero_zero_nat) = one_one_nat ).

% pochhammer_0
tff(fact_8700_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_8701_binomial__n__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_8702_zero__less__binomial__iff,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ).

% zero_less_binomial_iff
tff(fact_8703_binomial__eq__0,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_8704_Suc__times__binomial,axiom,
    ! [K: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) ).

% Suc_times_binomial
tff(fact_8705_Suc__times__binomial__eq,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).

% Suc_times_binomial_eq
tff(fact_8706_binomial__symmetric,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% binomial_symmetric
tff(fact_8707_choose__mult__lemma,axiom,
    ! [Mb: nat,R2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),R2)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),R2)),K)),K)),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_8708_binomial__le__pow,axiom,
    ! [R2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),R2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),R2)) ) ).

% binomial_le_pow
tff(fact_8709_pochhammer__pos,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,X),Na)) ) ).

% pochhammer_pos
tff(fact_8710_pochhammer__pos,axiom,
    ! [X: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),Na)) ) ).

% pochhammer_pos
tff(fact_8711_pochhammer__pos,axiom,
    ! [X: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),Na)) ) ).

% pochhammer_pos
tff(fact_8712_pochhammer__pos,axiom,
    ! [X: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X),Na)) ) ).

% pochhammer_pos
tff(fact_8713_pochhammer__neq__0__mono,axiom,
    ! [A: complex,Mb: nat,Na: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Mb) != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),Na) != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_8714_pochhammer__neq__0__mono,axiom,
    ! [A: real,Mb: nat,Na: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Mb) != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Na) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_8715_pochhammer__neq__0__mono,axiom,
    ! [A: rat,Mb: nat,Na: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Mb) != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Na) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_8716_pochhammer__eq__0__mono,axiom,
    ! [A: complex,Na: nat,Mb: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Na) = zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),Mb) = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_8717_pochhammer__eq__0__mono,axiom,
    ! [A: real,Na: nat,Mb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Na) = zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Mb) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_8718_pochhammer__eq__0__mono,axiom,
    ! [A: rat,Na: nat,Mb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Na) = zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Mb) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_8719_zero__less__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) ) ).

% zero_less_binomial
tff(fact_8720_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_8721_choose__mult,axiom,
    ! [K: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Mb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K))) ) ) ) ).

% choose_mult
tff(fact_8722_binomial__Suc__Suc__eq__times,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))),aa_nat_nat(suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_8723_gbinomial__pochhammer_H,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)),aa_nat_complex(semiri5044797733671781792omplex,K)) ).

% gbinomial_pochhammer'
tff(fact_8724_gbinomial__pochhammer_H,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(divide_divide_rat(aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).

% gbinomial_pochhammer'
tff(fact_8725_gbinomial__pochhammer_H,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)),aa_nat_real(semiri2265585572941072030t_real,K)) ).

% gbinomial_pochhammer'
tff(fact_8726_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_8727_gbinomial__Suc__Suc,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_8728_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_8729_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Na)),K) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_8730_pochhammer__nonneg,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,X),Na)) ) ).

% pochhammer_nonneg
tff(fact_8731_pochhammer__nonneg,axiom,
    ! [X: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),Na)) ) ).

% pochhammer_nonneg
tff(fact_8732_pochhammer__nonneg,axiom,
    ! [X: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
     => 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,X),Na)) ) ).

% pochhammer_nonneg
tff(fact_8733_pochhammer__nonneg,axiom,
    ! [X: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X),Na)) ) ).

% pochhammer_nonneg
tff(fact_8734_pochhammer__0__left,axiom,
    ! [Na: nat] :
      aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),Na) = $ite(Na = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% pochhammer_0_left
tff(fact_8735_pochhammer__0__left,axiom,
    ! [Na: nat] :
      aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,zero_zero_real),Na) = $ite(Na = zero_zero_nat,one_one_real,zero_zero_real) ).

% pochhammer_0_left
tff(fact_8736_pochhammer__0__left,axiom,
    ! [Na: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),Na) = $ite(Na = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% pochhammer_0_left
tff(fact_8737_pochhammer__0__left,axiom,
    ! [Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),Na) = $ite(Na = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% pochhammer_0_left
tff(fact_8738_pochhammer__0__left,axiom,
    ! [Na: nat] :
      aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),Na) = $ite(Na = zero_zero_nat,one_one_int,zero_zero_int) ).

% pochhammer_0_left
tff(fact_8739_binomial__absorption,axiom,
    ! [K: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),K)) ).

% binomial_absorption
tff(fact_8740_gbinomial__addition__formula,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).

% gbinomial_addition_formula
tff(fact_8741_gbinomial__addition__formula,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).

% gbinomial_addition_formula
tff(fact_8742_gbinomial__addition__formula,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).

% gbinomial_addition_formula
tff(fact_8743_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K)),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_8744_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(gbinomial_rat(A),K)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_8745_gbinomial__mult__1_H,axiom,
    ! [A: complex,K: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),A) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,K)),aa_nat_complex(gbinomial_complex(A),K))),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_8746_gbinomial__mult__1_H,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),A) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_8747_gbinomial__mult__1_H,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_8748_gbinomial__mult__1,axiom,
    ! [A: complex,K: nat] : aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(A),K)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,K)),aa_nat_complex(gbinomial_complex(A),K))),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_8749_gbinomial__mult__1,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_8750_gbinomial__mult__1,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_8751_binomial__fact__lemma,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,Na) ) ) ).

% binomial_fact_lemma
tff(fact_8752_pochhammer__rec,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Na)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),Na)) ).

% pochhammer_rec
tff(fact_8753_pochhammer__rec,axiom,
    ! [A: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),Na)) ).

% pochhammer_rec
tff(fact_8754_pochhammer__rec,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Na)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),Na)) ).

% pochhammer_rec
tff(fact_8755_pochhammer__rec,axiom,
    ! [A: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Na)) = 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)),Na)) ).

% pochhammer_rec
tff(fact_8756_pochhammer__rec,axiom,
    ! [A: int,Na: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Na)) = 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)),Na)) ).

% pochhammer_rec
tff(fact_8757_pochhammer__rec_H,axiom,
    ! [Z: complex,Na: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,Na)) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,Na))),aa_nat_complex(comm_s2602460028002588243omplex(Z),Na)) ).

% pochhammer_rec'
tff(fact_8758_pochhammer__rec_H,axiom,
    ! [Z: rat,Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Na)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,Na))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Na)) ).

% pochhammer_rec'
tff(fact_8759_pochhammer__rec_H,axiom,
    ! [Z: int,Na: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(suc,Na)) = 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,Na))),aa_nat_int(comm_s4660882817536571857er_int(Z),Na)) ).

% pochhammer_rec'
tff(fact_8760_pochhammer__rec_H,axiom,
    ! [Z: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Na))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Na)) ).

% pochhammer_rec'
tff(fact_8761_pochhammer__rec_H,axiom,
    ! [Z: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,Na)) = 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,Na))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Na)) ).

% pochhammer_rec'
tff(fact_8762_pochhammer__Suc,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Na)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(A),Na)),aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,Na))) ).

% pochhammer_Suc
tff(fact_8763_pochhammer__Suc,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Na)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(A),Na)),aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,Na))) ).

% pochhammer_Suc
tff(fact_8764_pochhammer__Suc,axiom,
    ! [A: int,Na: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(A),Na)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,Na))) ).

% pochhammer_Suc
tff(fact_8765_pochhammer__Suc,axiom,
    ! [A: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,Na))) ).

% pochhammer_Suc
tff(fact_8766_pochhammer__Suc,axiom,
    ! [A: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Na))) ).

% pochhammer_Suc
tff(fact_8767_pochhammer__eq__0__iff,axiom,
    ! [A: complex,Na: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Na) = zero_zero_complex )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Na)
          & ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_8768_pochhammer__eq__0__iff,axiom,
    ! [A: rat,Na: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Na) = zero_zero_rat )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Na)
          & ( A = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_8769_pochhammer__eq__0__iff,axiom,
    ! [A: real,Na: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Na) = zero_zero_real )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Na)
          & ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_8770_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Na))),K) = zero_zero_complex )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8771_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Na))),K) = zero_zero_rat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8772_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na))),K) = zero_z3403309356797280102nteger )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8773_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),K) = zero_zero_int )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8774_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),K) = zero_zero_real )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8775_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Na))),K) = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8776_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Na))),K) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8777_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na))),K) = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8778_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K)
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),K) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8779_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),K) = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8780_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Na))),K) != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8781_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Na))),K) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8782_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na))),K) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8783_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),K) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8784_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),K) != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8785_pochhammer__product_H,axiom,
    ! [Z: complex,Na: nat,Mb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Mb)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),Na)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,Na))),Mb)) ).

% pochhammer_product'
tff(fact_8786_pochhammer__product_H,axiom,
    ! [Z: rat,Na: nat,Mb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Mb)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),Na)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,Na))),Mb)) ).

% pochhammer_product'
tff(fact_8787_pochhammer__product_H,axiom,
    ! [Z: int,Na: nat,Mb: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Mb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),Na)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Na))),Mb)) ).

% pochhammer_product'
tff(fact_8788_pochhammer__product_H,axiom,
    ! [Z: real,Na: nat,Mb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Mb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Na)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Na))),Mb)) ).

% pochhammer_product'
tff(fact_8789_pochhammer__product_H,axiom,
    ! [Z: nat,Na: 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,Na),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Na)),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,Na))),Mb)) ).

% pochhammer_product'
tff(fact_8790_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_8791_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_8792_binomial__maximum_H,axiom,
    ! [Na: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),Na)) ).

% binomial_maximum'
tff(fact_8793_binomial__mono,axiom,
    ! [K: nat,K6: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),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)),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K6)) ) ) ).

% binomial_mono
tff(fact_8794_binomial__antimono,axiom,
    ! [K: nat,K6: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) ) ) ) ).

% binomial_antimono
tff(fact_8795_binomial__maximum,axiom,
    ! [Na: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% binomial_maximum
tff(fact_8796_binomial__le__pow2,axiom,
    ! [Na: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% binomial_le_pow2
tff(fact_8797_choose__reduce__nat,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_8798_times__binomial__minus1__eq,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_8799_Suc__times__gbinomial,axiom,
    ! [K: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(gbinomial_complex(A),K)) ).

% Suc_times_gbinomial
tff(fact_8800_Suc__times__gbinomial,axiom,
    ! [K: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(gbinomial_rat(A),K)) ).

% Suc_times_gbinomial
tff(fact_8801_Suc__times__gbinomial,axiom,
    ! [K: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) ).

% Suc_times_gbinomial
tff(fact_8802_gbinomial__absorption,axiom,
    ! [K: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).

% gbinomial_absorption
tff(fact_8803_gbinomial__absorption,axiom,
    ! [K: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).

% gbinomial_absorption
tff(fact_8804_gbinomial__absorption,axiom,
    ! [K: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).

% gbinomial_absorption
tff(fact_8805_binomial__altdef__nat,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_8806_gbinomial__trinomial__revision,axiom,
    ! [K: nat,Mb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),Mb)),aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),K)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8807_gbinomial__trinomial__revision,axiom,
    ! [K: nat,Mb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),Mb)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8808_gbinomial__trinomial__revision,axiom,
    ! [K: nat,Mb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8809_pochhammer__product,axiom,
    ! [Mb: nat,Na: nat,Z: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(Z),Na) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),Mb)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Mb))) ) ) ).

% pochhammer_product
tff(fact_8810_pochhammer__product,axiom,
    ! [Mb: nat,Na: nat,Z: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),Na) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),Mb)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Mb))) ) ) ).

% pochhammer_product
tff(fact_8811_pochhammer__product,axiom,
    ! [Mb: nat,Na: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_nat_int(comm_s4660882817536571857er_int(Z),Na) = 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,Na),Mb))) ) ) ).

% pochhammer_product
tff(fact_8812_pochhammer__product,axiom,
    ! [Mb: nat,Na: nat,Z: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Na) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Mb)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Mb))) ) ) ).

% pochhammer_product
tff(fact_8813_pochhammer__product,axiom,
    ! [Mb: nat,Na: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Na) = 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,Na),Mb))) ) ) ).

% pochhammer_product
tff(fact_8814_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_8815_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),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)),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K6)) ) ) ).

% binomial_strict_mono
tff(fact_8816_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_8817_central__binomial__odd,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_8818_binomial__addition__formula,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_8819_binomial__fact,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,Na)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))) ) ) ).

% binomial_fact
tff(fact_8820_binomial__fact,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,Na)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))) ) ) ).

% binomial_fact
tff(fact_8821_binomial__fact,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri2265585572941072030t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))) ) ) ).

% binomial_fact
tff(fact_8822_fact__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,Na)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))) ) ) ).

% fact_binomial
tff(fact_8823_fact__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,Na)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))) ) ) ).

% fact_binomial
tff(fact_8824_fact__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri2265585572941072030t_real,Na)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))) ) ) ).

% fact_binomial
tff(fact_8825_gbinomial__rec,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8826_gbinomial__rec,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8827_gbinomial__rec,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8828_gbinomial__factors,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))),aa_nat_complex(gbinomial_complex(A),K)) ).

% gbinomial_factors
tff(fact_8829_gbinomial__factors,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))),aa_nat_rat(gbinomial_rat(A),K)) ).

% gbinomial_factors
tff(fact_8830_gbinomial__factors,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) ).

% gbinomial_factors
tff(fact_8831_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(R2),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,R2)),K)) = aa_complex_complex(times_times_complex(R2),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,R2)),one_one_complex)),K)) ).

% pochhammer_absorb_comp
tff(fact_8832_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(R2),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,R2)),K)) = aa_rat_rat(times_times_rat(R2),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,R2)),one_one_rat)),K)) ).

% pochhammer_absorb_comp
tff(fact_8833_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(R2,aa_nat_Code_integer(semiri4939895301339042750nteger,K))),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2)),K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2),one_one_Code_integer)),K)) ).

% pochhammer_absorb_comp
tff(fact_8834_pochhammer__absorb__comp,axiom,
    ! [R2: int,K: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_nat_int(semiri1314217659103216013at_int,K))),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,R2)),K)) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,R2)),one_one_int)),K)) ).

% pochhammer_absorb_comp
tff(fact_8835_pochhammer__absorb__comp,axiom,
    ! [R2: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,R2)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,R2)),one_one_real)),K)) ).

% pochhammer_absorb_comp
tff(fact_8836_gbinomial__minus,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% gbinomial_minus
tff(fact_8837_gbinomial__minus,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,A)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).

% gbinomial_minus
tff(fact_8838_gbinomial__minus,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(uminus_uminus_real,A)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% gbinomial_minus
tff(fact_8839_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8840_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8841_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8842_pochhammer__minus,axiom,
    ! [B: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% pochhammer_minus
tff(fact_8843_pochhammer__minus,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).

% pochhammer_minus
tff(fact_8844_pochhammer__minus,axiom,
    ! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K)),one_one_Code_integer)),K)) ).

% pochhammer_minus
tff(fact_8845_pochhammer__minus,axiom,
    ! [B: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K)) ).

% pochhammer_minus
tff(fact_8846_pochhammer__minus,axiom,
    ! [B: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% pochhammer_minus
tff(fact_8847_pochhammer__minus_H,axiom,
    ! [B: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K)) ).

% pochhammer_minus'
tff(fact_8848_pochhammer__minus_H,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K)) ).

% pochhammer_minus'
tff(fact_8849_pochhammer__minus_H,axiom,
    ! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K)),one_one_Code_integer)),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K)) ).

% pochhammer_minus'
tff(fact_8850_pochhammer__minus_H,axiom,
    ! [B: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K)) ).

% pochhammer_minus'
tff(fact_8851_pochhammer__minus_H,axiom,
    ! [B: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K)) ).

% pochhammer_minus'
tff(fact_8852_gbinomial__sum__up__index,axiom,
    ! [K: nat,Na: nat] : groups2073611262835488442omplex(aTP_Lamm_ui(K),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8853_gbinomial__sum__up__index,axiom,
    ! [K: nat,Na: nat] : groups2906978787729119204at_rat(aTP_Lamm_uj(K),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8854_gbinomial__sum__up__index,axiom,
    ! [K: nat,Na: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_uk,K),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8855_gbinomial__absorption_H,axiom,
    ! [K: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8856_gbinomial__absorption_H,axiom,
    ! [K: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8857_gbinomial__absorption_H,axiom,
    ! [K: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8858_binomial__code,axiom,
    ! [Na: nat,K: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K),
        zero_zero_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)),one_one_nat)),Na),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K))) ) ).

% binomial_code
tff(fact_8859_gbinomial__code,axiom,
    ! [A: complex,K: nat] :
      aa_nat_complex(gbinomial_complex(A),K) = $ite(K = zero_zero_nat,one_one_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_ul(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_complex)),aa_nat_complex(semiri5044797733671781792omplex,K))) ).

% gbinomial_code
tff(fact_8860_gbinomial__code,axiom,
    ! [A: rat,K: nat] :
      aa_nat_rat(gbinomial_rat(A),K) = $ite(K = zero_zero_nat,one_one_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_um(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_rat)),aa_nat_rat(semiri773545260158071498ct_rat,K))) ).

% gbinomial_code
tff(fact_8861_gbinomial__code,axiom,
    ! [A: real,K: nat] :
      aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = $ite(K = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_un(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_real)),aa_nat_real(semiri2265585572941072030t_real,K))) ).

% gbinomial_code
tff(fact_8862_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,Na: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,Na))),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,Na))) = groups6464643781859351333omplex(aTP_Lamm_uo(Z),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))),Na)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8863_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,Na: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Na))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_nat_nat(suc,Na))) = groups73079841787564623at_rat(aTP_Lamm_up(Z),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))),Na)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8864_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,Na))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_nat_nat(suc,Na))) = groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_uq,Z),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))),Na)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8865_pochhammer__code,axiom,
    ! [A: complex,Na: nat] :
      aa_nat_complex(comm_s2602460028002588243omplex(A),Na) = $ite(Na = zero_zero_nat,one_one_complex,aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_ur(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),one_one_complex)) ).

% pochhammer_code
tff(fact_8866_pochhammer__code,axiom,
    ! [A: rat,Na: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(A),Na) = $ite(Na = zero_zero_nat,one_one_rat,aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_us(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),one_one_rat)) ).

% pochhammer_code
tff(fact_8867_pochhammer__code,axiom,
    ! [A: int,Na: nat] :
      aa_nat_int(comm_s4660882817536571857er_int(A),Na) = $ite(Na = zero_zero_nat,one_one_int,aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_ut(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),one_one_int)) ).

% pochhammer_code
tff(fact_8868_pochhammer__code,axiom,
    ! [A: real,Na: nat] :
      aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Na) = $ite(Na = zero_zero_nat,one_one_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_uu(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),one_one_real)) ).

% pochhammer_code
tff(fact_8869_pochhammer__code,axiom,
    ! [A: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Na) = $ite(Na = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_uv(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),one_one_nat)) ).

% pochhammer_code
tff(fact_8870_gbinomial__partial__row__sum,axiom,
    ! [A: complex,Mb: nat] : groups2073611262835488442omplex(aTP_Lamm_uf(A),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_8871_gbinomial__partial__row__sum,axiom,
    ! [A: rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_ug(A),set_ord_atMost_nat(Mb)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_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_8872_gbinomial__partial__row__sum,axiom,
    ! [A: real,Mb: nat] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_uh,A),set_ord_atMost_nat(Mb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_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(aa_real_fun_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_8873_atMost__iff,axiom,
    ! [I: real,K: real] :
      ( aa_set_real_o(member_real(I),set_ord_atMost_real(K))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),K) ) ).

% atMost_iff
tff(fact_8874_atMost__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(I),set_or4236626031148496127et_nat(K))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,I),K) ) ).

% atMost_iff
tff(fact_8875_atMost__iff,axiom,
    ! [I: set_int,K: set_int] :
      ( member_set_int(I,set_or58775011639299419et_int(K))
    <=> aa_set_int_o(ord_less_eq_set_int(I),K) ) ).

% atMost_iff
tff(fact_8876_atMost__iff,axiom,
    ! [I: rat,K: rat] :
      ( aa_set_rat_o(member_rat(I),set_ord_atMost_rat(K))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),K) ) ).

% atMost_iff
tff(fact_8877_atMost__iff,axiom,
    ! [I: num,K: num] :
      ( aa_set_num_o(member_num(I),set_ord_atMost_num(K))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I),K) ) ).

% atMost_iff
tff(fact_8878_atMost__iff,axiom,
    ! [I: nat,K: nat] :
      ( aa_set_nat_o(member_nat(I),set_ord_atMost_nat(K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K) ) ).

% atMost_iff
tff(fact_8879_atMost__iff,axiom,
    ! [I: int,K: int] :
      ( aa_set_int_o(member_int(I),set_ord_atMost_int(K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K) ) ).

% atMost_iff
tff(fact_8880_finite__atMost,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,set_ord_atMost_nat(K)) ).

% finite_atMost
tff(fact_8881_of__nat__prod,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_uw(F),A2) ).

% of_nat_prod
tff(fact_8882_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = groups129246275422532515t_real(aTP_Lamm_pk(F),A2) ).

% of_nat_prod
tff(fact_8883_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_pj(F)),A2) ).

% of_nat_prod
tff(fact_8884_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = groups705719431365010083at_int(aTP_Lamm_pi(F),A2) ).

% of_nat_prod
tff(fact_8885_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_real(ring_1_of_int_real,groups705719431365010083at_int(F,A2)) = groups129246275422532515t_real(aTP_Lamm_jv(F),A2) ).

% of_int_prod
tff(fact_8886_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_rat(ring_1_of_int_rat,groups705719431365010083at_int(F,A2)) = groups73079841787564623at_rat(aTP_Lamm_ux(F),A2) ).

% of_int_prod
tff(fact_8887_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_int(ring_1_of_int_int,groups705719431365010083at_int(F,A2)) = groups705719431365010083at_int(aTP_Lamm_uy(F),A2) ).

% of_int_prod
tff(fact_8888_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_real(ring_1_of_int_real,groups1705073143266064639nt_int(F,A2)) = groups2316167850115554303t_real(aTP_Lamm_uz(F),A2) ).

% of_int_prod
tff(fact_8889_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_rat(ring_1_of_int_rat,groups1705073143266064639nt_int(F,A2)) = groups1072433553688619179nt_rat(aTP_Lamm_va(F),A2) ).

% of_int_prod
tff(fact_8890_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_int(ring_1_of_int_int,groups1705073143266064639nt_int(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_vb(F),A2) ).

% of_int_prod
tff(fact_8891_prod__zero__iff,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( groups6464643781859351333omplex(F,A2) = zero_zero_complex )
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A2)
            & ( aa_nat_complex(F,X4) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_8892_prod__zero__iff,axiom,
    ! [A2: set_int,F: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( groups7440179247065528705omplex(F,A2) = zero_zero_complex )
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A2)
            & ( aa_int_complex(F,X4) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_8893_prod__zero__iff,axiom,
    ! [A2: set_complex,F: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( groups3708469109370488835omplex(F,A2) = zero_zero_complex )
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A2)
            & ( aa_complex_complex(F,X4) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_8894_prod__zero__iff,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( groups129246275422532515t_real(F,A2) = zero_zero_real )
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A2)
            & ( aa_nat_real(F,X4) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_8895_prod__zero__iff,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( groups2316167850115554303t_real(F,A2) = zero_zero_real )
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A2)
            & ( aa_int_real(F,X4) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_8896_prod__zero__iff,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( groups766887009212190081x_real(F,A2) = zero_zero_real )
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A2)
            & ( aa_complex_real(F,X4) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_8897_prod__zero__iff,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( groups73079841787564623at_rat(F,A2) = zero_zero_rat )
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A2)
            & ( aa_nat_rat(F,X4) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_8898_prod__zero__iff,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( groups1072433553688619179nt_rat(F,A2) = zero_zero_rat )
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A2)
            & ( aa_int_rat(F,X4) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_8899_prod__zero__iff,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( groups225925009352817453ex_rat(F,A2) = zero_zero_rat )
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A2)
            & ( aa_complex_rat(F,X4) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_8900_prod__zero__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( groups1707563613775114915nt_nat(F,A2) = zero_zero_nat )
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A2)
            & ( aa_int_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_8901_prod_Oempty,axiom,
    ! [G: fun_real_complex] : groups713298508707869441omplex(G,bot_bot_set_real) = one_one_complex ).

% prod.empty
tff(fact_8902_prod_Oempty,axiom,
    ! [G: fun_real_real] : groups1681761925125756287l_real(G,bot_bot_set_real) = one_one_real ).

% prod.empty
tff(fact_8903_prod_Oempty,axiom,
    ! [G: fun_real_rat] : groups4061424788464935467al_rat(G,bot_bot_set_real) = one_one_rat ).

% prod.empty
tff(fact_8904_prod_Oempty,axiom,
    ! [G: fun_real_nat] : groups4696554848551431203al_nat(G,bot_bot_set_real) = one_one_nat ).

% prod.empty
tff(fact_8905_prod_Oempty,axiom,
    ! [G: fun_real_int] : groups4694064378042380927al_int(G,bot_bot_set_real) = one_one_int ).

% prod.empty
tff(fact_8906_prod_Oempty,axiom,
    ! [G: fun_nat_complex] : groups6464643781859351333omplex(G,bot_bot_set_nat) = one_one_complex ).

% prod.empty
tff(fact_8907_prod_Oempty,axiom,
    ! [G: fun_nat_real] : groups129246275422532515t_real(G,bot_bot_set_nat) = one_one_real ).

% prod.empty
tff(fact_8908_prod_Oempty,axiom,
    ! [G: fun_nat_rat] : groups73079841787564623at_rat(G,bot_bot_set_nat) = one_one_rat ).

% prod.empty
tff(fact_8909_prod_Oempty,axiom,
    ! [G: fun_int_complex] : groups7440179247065528705omplex(G,bot_bot_set_int) = one_one_complex ).

% prod.empty
tff(fact_8910_prod_Oempty,axiom,
    ! [G: fun_int_real] : groups2316167850115554303t_real(G,bot_bot_set_int) = one_one_real ).

% prod.empty
tff(fact_8911_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_complex] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( groups6464643781859351333omplex(G,A2) = one_one_complex ) ) ).

% prod.infinite
tff(fact_8912_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups7440179247065528705omplex(G,A2) = one_one_complex ) ) ).

% prod.infinite
tff(fact_8913_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups3708469109370488835omplex(G,A2) = one_one_complex ) ) ).

% prod.infinite
tff(fact_8914_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_real] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( groups129246275422532515t_real(G,A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_8915_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups2316167850115554303t_real(G,A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_8916_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups766887009212190081x_real(G,A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_8917_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_rat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( groups73079841787564623at_rat(G,A2) = one_one_rat ) ) ).

% prod.infinite
tff(fact_8918_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups1072433553688619179nt_rat(G,A2) = one_one_rat ) ) ).

% prod.infinite
tff(fact_8919_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups225925009352817453ex_rat(G,A2) = one_one_rat ) ) ).

% prod.infinite
tff(fact_8920_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups1707563613775114915nt_nat(G,A2) = one_one_nat ) ) ).

% prod.infinite
tff(fact_8921_atMost__subset__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ord_le4403425263959731960et_int(set_or58775011639299419et_int(X),set_or58775011639299419et_int(Y))
    <=> aa_set_int_o(ord_less_eq_set_int(X),Y) ) ).

% atMost_subset_iff
tff(fact_8922_atMost__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_ord_atMost_rat(X)),set_ord_atMost_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% atMost_subset_iff
tff(fact_8923_atMost__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_ord_atMost_num(X)),set_ord_atMost_num(Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y) ) ).

% atMost_subset_iff
tff(fact_8924_atMost__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_ord_atMost_nat(X)),set_ord_atMost_nat(Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ).

% atMost_subset_iff
tff(fact_8925_atMost__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_ord_atMost_int(X)),set_ord_atMost_int(Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ).

% atMost_subset_iff
tff(fact_8926_dvd__prod__eqI,axiom,
    ! [A2: set_real,A: real,B: nat,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(A),A2)
       => ( ( B = aa_real_nat(F,A) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),groups4696554848551431203al_nat(F,A2)) ) ) ) ).

% dvd_prod_eqI
tff(fact_8927_dvd__prod__eqI,axiom,
    ! [A2: set_int,A: int,B: nat,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(A),A2)
       => ( ( B = aa_int_nat(F,A) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),groups1707563613775114915nt_nat(F,A2)) ) ) ) ).

% dvd_prod_eqI
tff(fact_8928_dvd__prod__eqI,axiom,
    ! [A2: set_complex,A: complex,B: nat,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(A),A2)
       => ( ( B = aa_complex_nat(F,A) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),groups861055069439313189ex_nat(F,A2)) ) ) ) ).

% dvd_prod_eqI
tff(fact_8929_dvd__prod__eqI,axiom,
    ! [A2: set_real,A: real,B: int,F: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(A),A2)
       => ( ( B = aa_real_int(F,A) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),groups4694064378042380927al_int(F,A2)) ) ) ) ).

% dvd_prod_eqI
tff(fact_8930_dvd__prod__eqI,axiom,
    ! [A2: set_complex,A: complex,B: int,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(A),A2)
       => ( ( B = aa_complex_int(F,A) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),groups858564598930262913ex_int(F,A2)) ) ) ) ).

% dvd_prod_eqI
tff(fact_8931_dvd__prod__eqI,axiom,
    ! [A2: set_real,A: real,B: code_integer,F: fun_re6978948838062957830nteger] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(A),A2)
       => ( ( B = aa_real_Code_integer(F,A) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),groups6225526099057966256nteger(F,A2)) ) ) ) ).

% dvd_prod_eqI
tff(fact_8932_dvd__prod__eqI,axiom,
    ! [A2: set_nat,A: nat,B: code_integer,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(A),A2)
       => ( ( B = aa_nat_Code_integer(F,A) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),groups3455450783089532116nteger(F,A2)) ) ) ) ).

% dvd_prod_eqI
tff(fact_8933_dvd__prod__eqI,axiom,
    ! [A2: set_int,A: int,B: code_integer,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(A),A2)
       => ( ( B = aa_int_Code_integer(F,A) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),groups3827104343326376752nteger(F,A2)) ) ) ) ).

% dvd_prod_eqI
tff(fact_8934_dvd__prod__eqI,axiom,
    ! [A2: set_complex,A: complex,B: code_integer,F: fun_co8153928105989009288nteger] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(A),A2)
       => ( ( B = aa_com4217951221296830281nteger(F,A) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),groups8682486955453173170nteger(F,A2)) ) ) ) ).

% dvd_prod_eqI
tff(fact_8935_dvd__prod__eqI,axiom,
    ! [A2: set_nat,A: nat,B: nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(A),A2)
       => ( ( B = aa_nat_nat(F,A) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) ) ) ) ).

% dvd_prod_eqI
tff(fact_8936_dvd__prodI,axiom,
    ! [A2: set_real,A: real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(A),A2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_real_nat(F,A)),groups4696554848551431203al_nat(F,A2)) ) ) ).

% dvd_prodI
tff(fact_8937_dvd__prodI,axiom,
    ! [A2: set_int,A: int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(A),A2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(F,A)),groups1707563613775114915nt_nat(F,A2)) ) ) ).

% dvd_prodI
tff(fact_8938_dvd__prodI,axiom,
    ! [A2: set_complex,A: complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(A),A2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_complex_nat(F,A)),groups861055069439313189ex_nat(F,A2)) ) ) ).

% dvd_prodI
tff(fact_8939_dvd__prodI,axiom,
    ! [A2: set_real,A: real,F: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(A),A2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_real_int(F,A)),groups4694064378042380927al_int(F,A2)) ) ) ).

% dvd_prodI
tff(fact_8940_dvd__prodI,axiom,
    ! [A2: set_complex,A: complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(A),A2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_complex_int(F,A)),groups858564598930262913ex_int(F,A2)) ) ) ).

% dvd_prodI
tff(fact_8941_dvd__prodI,axiom,
    ! [A2: set_real,A: real,F: fun_re6978948838062957830nteger] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(A),A2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_real_Code_integer(F,A)),groups6225526099057966256nteger(F,A2)) ) ) ).

% dvd_prodI
tff(fact_8942_dvd__prodI,axiom,
    ! [A2: set_nat,A: nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(A),A2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(F,A)),groups3455450783089532116nteger(F,A2)) ) ) ).

% dvd_prodI
tff(fact_8943_dvd__prodI,axiom,
    ! [A2: set_int,A: int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(A),A2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_int_Code_integer(F,A)),groups3827104343326376752nteger(F,A2)) ) ) ).

% dvd_prodI
tff(fact_8944_dvd__prodI,axiom,
    ! [A2: set_complex,A: complex,F: fun_co8153928105989009288nteger] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(A),A2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_com4217951221296830281nteger(F,A)),groups8682486955453173170nteger(F,A2)) ) ) ).

% dvd_prodI
tff(fact_8945_dvd__prodI,axiom,
    ! [A2: set_nat,A: nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(A),A2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(F,A)),aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) ) ) ).

% dvd_prodI
tff(fact_8946_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_vc(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_complex(B,A),one_one_complex) ) ) ).

% prod.delta
tff(fact_8947_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_vd(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_complex(B,A),one_one_complex) ) ) ).

% prod.delta
tff(fact_8948_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_ve(A),B),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_int_complex(B,A),one_one_complex) ) ) ).

% prod.delta
tff(fact_8949_prod_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_vf(A),B),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_complex_complex(B,A),one_one_complex) ) ) ).

% prod.delta
tff(fact_8950_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_vg(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_8951_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_vh(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_8952_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_vi(A),B),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_int_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_8953_prod_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_vj(A),B),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_complex_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_8954_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_vk(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_rat(B,A),one_one_rat) ) ) ).

% prod.delta
tff(fact_8955_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_vl(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_rat(B,A),one_one_rat) ) ) ).

% prod.delta
tff(fact_8956_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_vm(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_complex(B,A),one_one_complex) ) ) ).

% prod.delta'
tff(fact_8957_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_vn(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_complex(B,A),one_one_complex) ) ) ).

% prod.delta'
tff(fact_8958_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_vo(A),B),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_int_complex(B,A),one_one_complex) ) ) ).

% prod.delta'
tff(fact_8959_prod_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_vp(A),B),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_complex_complex(B,A),one_one_complex) ) ) ).

% prod.delta'
tff(fact_8960_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_vq(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_8961_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_vr(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_8962_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_vs(A),B),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_int_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_8963_prod_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_vt(A),B),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_complex_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_8964_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_vu(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_rat(B,A),one_one_rat) ) ) ).

% prod.delta'
tff(fact_8965_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_vv(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_rat(B,A),one_one_rat) ) ) ).

% prod.delta'
tff(fact_8966_prod_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ~ aa_set_real_o(member_real(X),A2)
       => ( groups713298508707869441omplex(G,insert_real(X,A2)) = aa_complex_complex(times_times_complex(aa_real_complex(G,X)),groups713298508707869441omplex(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8967_prod_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(X),A2)
       => ( groups6464643781859351333omplex(G,insert_nat(X,A2)) = aa_complex_complex(times_times_complex(aa_nat_complex(G,X)),groups6464643781859351333omplex(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8968_prod_Oinsert,axiom,
    ! [A2: set_int,X: int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int(X),A2)
       => ( groups7440179247065528705omplex(G,insert_int(X,A2)) = aa_complex_complex(times_times_complex(aa_int_complex(G,X)),groups7440179247065528705omplex(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8969_prod_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ~ aa_set_complex_o(member_complex(X),A2)
       => ( groups3708469109370488835omplex(G,insert_complex(X,A2)) = aa_complex_complex(times_times_complex(aa_complex_complex(G,X)),groups3708469109370488835omplex(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8970_prod_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ~ aa_set_real_o(member_real(X),A2)
       => ( groups1681761925125756287l_real(G,insert_real(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,X)),groups1681761925125756287l_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8971_prod_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(X),A2)
       => ( groups129246275422532515t_real(G,insert_nat(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,X)),groups129246275422532515t_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8972_prod_Oinsert,axiom,
    ! [A2: set_int,X: int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int(X),A2)
       => ( groups2316167850115554303t_real(G,insert_int(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,X)),groups2316167850115554303t_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8973_prod_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ~ aa_set_complex_o(member_complex(X),A2)
       => ( groups766887009212190081x_real(G,insert_complex(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,X)),groups766887009212190081x_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8974_prod_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ~ aa_set_real_o(member_real(X),A2)
       => ( groups4061424788464935467al_rat(G,insert_real(X,A2)) = aa_rat_rat(times_times_rat(aa_real_rat(G,X)),groups4061424788464935467al_rat(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8975_prod_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(X),A2)
       => ( groups73079841787564623at_rat(G,insert_nat(X,A2)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,X)),groups73079841787564623at_rat(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8976_Icc__subset__Iic__iff,axiom,
    ! [L: set_int,H: set_int,H2: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(L,H),set_or58775011639299419et_int(H2))
    <=> ( ~ aa_set_int_o(ord_less_eq_set_int(L),H)
        | aa_set_int_o(ord_less_eq_set_int(H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8977_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_8978_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_8979_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H: nat,H2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_or1269000886237332187st_nat(L,H)),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_8980_Icc__subset__Iic__iff,axiom,
    ! [L: int,H: int,H2: int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_int_set_int(aa_int8639817059373041269et_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_8981_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_8982_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups2906978787729119204at_rat(G,set_ord_atMost_nat(aa_nat_nat(suc,Na))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_ord_atMost_nat(Na))),aa_nat_rat(G,aa_nat_nat(suc,Na))) ).

% sum.atMost_Suc
tff(fact_8983_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_int,Na: nat] : groups3539618377306564664at_int(G,set_ord_atMost_nat(aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_ord_atMost_nat(Na))),aa_nat_int(G,aa_nat_nat(suc,Na))) ).

% sum.atMost_Suc
tff(fact_8984_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_ord_atMost_nat(aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_ord_atMost_nat(Na))),aa_nat_nat(G,aa_nat_nat(suc,Na))) ).

% sum.atMost_Suc
tff(fact_8985_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_real,Na: nat] : groups6591440286371151544t_real(G,set_ord_atMost_nat(aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,set_ord_atMost_nat(Na))),aa_nat_real(G,aa_nat_nat(suc,Na))) ).

% sum.atMost_Suc
tff(fact_8986_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_complex,Na: nat] : groups6464643781859351333omplex(G,set_ord_lessThan_nat(aa_nat_nat(suc,Na))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,set_ord_lessThan_nat(Na))),aa_nat_complex(G,Na)) ).

% prod.lessThan_Suc
tff(fact_8987_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Na: nat] : groups129246275422532515t_real(G,set_ord_lessThan_nat(aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,set_ord_lessThan_nat(Na))),aa_nat_real(G,Na)) ).

% prod.lessThan_Suc
tff(fact_8988_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups73079841787564623at_rat(G,set_ord_lessThan_nat(aa_nat_nat(suc,Na))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_ord_lessThan_nat(Na))),aa_nat_rat(G,Na)) ).

% prod.lessThan_Suc
tff(fact_8989_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),set_ord_lessThan_nat(aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),set_ord_lessThan_nat(Na))),aa_nat_nat(G,Na)) ).

% prod.lessThan_Suc
tff(fact_8990_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Na: nat] : groups705719431365010083at_int(G,set_ord_lessThan_nat(aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_ord_lessThan_nat(Na))),aa_nat_int(G,Na)) ).

% prod.lessThan_Suc
tff(fact_8991_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_complex,Na: nat] : groups6464643781859351333omplex(G,set_ord_atMost_nat(aa_nat_nat(suc,Na))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,set_ord_atMost_nat(Na))),aa_nat_complex(G,aa_nat_nat(suc,Na))) ).

% prod.atMost_Suc
tff(fact_8992_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_real,Na: nat] : groups129246275422532515t_real(G,set_ord_atMost_nat(aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,set_ord_atMost_nat(Na))),aa_nat_real(G,aa_nat_nat(suc,Na))) ).

% prod.atMost_Suc
tff(fact_8993_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups73079841787564623at_rat(G,set_ord_atMost_nat(aa_nat_nat(suc,Na))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_ord_atMost_nat(Na))),aa_nat_rat(G,aa_nat_nat(suc,Na))) ).

% prod.atMost_Suc
tff(fact_8994_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),set_ord_atMost_nat(aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),set_ord_atMost_nat(Na))),aa_nat_nat(G,aa_nat_nat(suc,Na))) ).

% prod.atMost_Suc
tff(fact_8995_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_int,Na: nat] : groups705719431365010083at_int(G,set_ord_atMost_nat(aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_ord_atMost_nat(Na))),aa_nat_int(G,aa_nat_nat(suc,Na))) ).

% prod.atMost_Suc
tff(fact_8996_atMost__0,axiom,
    set_ord_atMost_nat(zero_zero_nat) = insert_nat(zero_zero_nat,bot_bot_set_nat) ).

% atMost_0
tff(fact_8997_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_complex,Mb: nat,Na: nat] :
      groups6464643781859351333omplex(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),Mb),one_one_complex,aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,set_or1269000886237332187st_nat(Mb,Na))),aa_nat_complex(G,aa_nat_nat(suc,Na)))) ).

% prod.cl_ivl_Suc
tff(fact_8998_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,Mb: nat,Na: nat] :
      groups129246275422532515t_real(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),Mb),one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,set_or1269000886237332187st_nat(Mb,Na))),aa_nat_real(G,aa_nat_nat(suc,Na)))) ).

% prod.cl_ivl_Suc
tff(fact_8999_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,Mb: nat,Na: nat] :
      groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),Mb),one_one_rat,aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(Mb,Na))),aa_nat_rat(G,aa_nat_nat(suc,Na)))) ).

% prod.cl_ivl_Suc
tff(fact_9000_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,Mb: nat,Na: nat] :
      aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),Mb),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(Mb,Na))),aa_nat_nat(G,aa_nat_nat(suc,Na)))) ).

% prod.cl_ivl_Suc
tff(fact_9001_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,Mb: nat,Na: nat] :
      groups705719431365010083at_int(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),Mb),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or1269000886237332187st_nat(Mb,Na))),aa_nat_int(G,aa_nat_nat(suc,Na)))) ).

% prod.cl_ivl_Suc
tff(fact_9002_atMost__atLeast0,axiom,
    ! [Na: nat] : set_ord_atMost_nat(Na) = set_or1269000886237332187st_nat(zero_zero_nat,Na) ).

% atMost_atLeast0
tff(fact_9003_lessThan__Suc__atMost,axiom,
    ! [K: nat] : set_ord_lessThan_nat(aa_nat_nat(suc,K)) = set_ord_atMost_nat(K) ).

% lessThan_Suc_atMost
tff(fact_9004_atMost__Suc,axiom,
    ! [K: nat] : set_ord_atMost_nat(aa_nat_nat(suc,K)) = insert_nat(aa_nat_nat(suc,K),set_ord_atMost_nat(K)) ).

% atMost_Suc
tff(fact_9005_finite__nat__iff__bounded__le,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
    <=> ? [K3: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),set_ord_atMost_nat(K3)) ) ).

% finite_nat_iff_bounded_le
tff(fact_9006_sum__choose__upper,axiom,
    ! [Mb: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vw,Mb)),set_ord_atMost_nat(Na)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),aa_nat_nat(suc,Mb)) ).

% sum_choose_upper
tff(fact_9007_sum__choose__lower,axiom,
    ! [R2: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vx,R2)),set_ord_atMost_nat(Na)) = 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),Na))),Na) ).

% sum_choose_lower
tff(fact_9008_choose__rising__sum_I1_J,axiom,
    ! [Na: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vy,Na)),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,Na),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_9009_choose__rising__sum_I2_J,axiom,
    ! [Na: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vy,Na)),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,Na),Mb)),one_one_nat)),Mb) ).

% choose_rising_sum(2)
tff(fact_9010_atLeast1__atMost__eq__remove0,axiom,
    ! [Na: nat] : set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na) = minus_minus_set_nat(set_ord_atMost_nat(Na),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_9011_fact__eq__fact__times,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_dr),set_or1269000886237332187st_nat(aa_nat_nat(suc,Na),Mb))) ) ) ).

% fact_eq_fact_times
tff(fact_9012_sum__choose__diagonal,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vz(Mb),Na)),set_ord_atMost_nat(Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),Mb) ) ) ).

% sum_choose_diagonal
tff(fact_9013_vandermonde,axiom,
    ! [Mb: nat,Na: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_wa(Mb),Na),R2)),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),Na)),R2) ).

% vandermonde
tff(fact_9014_fact__div__fact,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_dr),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat),Mb)) ) ) ).

% fact_div_fact
tff(fact_9015_binomial,axiom,
    ! [A: nat,B: nat,Na: 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)),Na) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_wb(A),B),Na)),set_ord_atMost_nat(Na)) ).

% binomial
tff(fact_9016_polynomial__product__nat,axiom,
    ! [Mb: nat,A: fun_nat_nat,Na: nat,B: fun_nat_nat,X: 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,Na),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_wc(A),X)),set_ord_atMost_nat(Mb))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_wc(B),X)),set_ord_atMost_nat(Na))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_we(A),B),X)),set_ord_atMost_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na))) ) ) ) ).

% polynomial_product_nat
tff(fact_9017_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))),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_9018_cot__less__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cot_real,X)),zero_zero_real) ) ) ).

% cot_less_zero
tff(fact_9019_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_wf(D))) ).

% int_ge_less_than2_def
tff(fact_9020_of__nat__id,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Na) = Na ).

% of_nat_id
tff(fact_9021_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J))) = groups1705073143266064639nt_int(aTP_Lamm_ds,aa_int_set_int(aa_int8639817059373041269et_int(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_9022_cot__gt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cot_real,X)) ) ) ).

% cot_gt_zero
tff(fact_9023_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_wg(D))) ).

% int_ge_less_than_def
tff(fact_9024_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_wh,Y)) ).

% arctan_def
tff(fact_9025_modulo__int__unfold,axiom,
    ! [K: int,Mb: nat,L: int,Na: nat] :
      aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Na))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Na = zero_zero_nat ) ),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,Mb)),
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Na))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),Mb))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Na))))) ) ).

% modulo_int_unfold
tff(fact_9026_powr__int,axiom,
    ! [X: real,I: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,I)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,I)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I))))) ) ) ).

% powr_int
tff(fact_9027_nat__int,axiom,
    ! [Na: nat] : aa_int_nat(nat2,aa_nat_int(semiri1314217659103216013at_int,Na)) = Na ).

% nat_int
tff(fact_9028_nat__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_num_int(numeral_numeral_int,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_numeral
tff(fact_9029_nat__of__bool,axiom,
    ! [P: $o] : aa_int_nat(nat2,aa_o_int(zero_n2684676970156552555ol_int,(P))) = aa_o_nat(zero_n2687167440665602831ol_nat,(P)) ).

% nat_of_bool
tff(fact_9030_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_9031_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa_int_nat(nat2,I) = zero_zero_nat )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int) ) ).

% nat_0_iff
tff(fact_9032_nat__le__0,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,Z) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_9033_zless__nat__conj,axiom,
    ! [W2: int,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ) ).

% zless_nat_conj
tff(fact_9034_nat__neg__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = zero_zero_nat ).

% nat_neg_numeral
tff(fact_9035_nat__zminus__int,axiom,
    ! [Na: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_9036_int__nat__eq,axiom,
    ! [Z: int] :
      aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),Z,zero_zero_int) ).

% int_nat_eq
tff(fact_9037_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),L)),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_9038_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),aa_int_int(sgn_sgn_int,R2))),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_9039_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),K))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_9040_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,R2)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_9041_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% zero_less_nat_eq
tff(fact_9042_diff__nat__numeral,axiom,
    ! [V: num,V3: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V3)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V3))) ).

% diff_nat_numeral
tff(fact_9043_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na) = aa_int_nat(nat2,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_9044_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_9045_nat__abs__dvd__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),Na)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% nat_abs_dvd_iff
tff(fact_9046_dvd__nat__abs__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),aa_int_nat(nat2,aa_int_int(abs_abs_int,K)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,Na)),K) ) ).

% dvd_nat_abs_iff
tff(fact_9047_nat__ceiling__le__eq,axiom,
    ! [X: real,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,A)) ) ).

% nat_ceiling_le_eq
tff(fact_9048_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_9049_nat__numeral__diff__1,axiom,
    ! [V: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_9050_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_9051_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)),A) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_9052_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_9053_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)),A) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_9054_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_9055_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ).

% nat_mono
tff(fact_9056_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N: nat,L3: int] : K != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L3)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% int_sgnE
tff(fact_9057_ex__nat,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_13: nat] : aa_nat_o(P,X_13)
    <=> ? [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
          & aa_nat_o(P,aa_int_nat(nat2,X4)) ) ) ).

% ex_nat
tff(fact_9058_all__nat,axiom,
    ! [P: fun_nat_o] :
      ( ! [X_13: nat] : aa_nat_o(P,X_13)
    <=> ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
         => aa_nat_o(P,aa_int_nat(nat2,X4)) ) ) ).

% all_nat
tff(fact_9059_eq__nat__nat__iff,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,Z) = aa_int_nat(nat2,Z5) )
        <=> ( Z = Z5 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_9060_nat__mono__iff,axiom,
    ! [Z: int,W2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ) ).

% nat_mono_iff
tff(fact_9061_zless__nat__eq__int__zless,axiom,
    ! [Mb: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_9062_nat__le__iff,axiom,
    ! [X: int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,X)),Na)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% nat_le_iff
tff(fact_9063_nat__0__le,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_9064_int__eq__iff,axiom,
    ! [Mb: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = Z )
    <=> ( ( Mb = aa_int_nat(nat2,Z) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ) ).

% int_eq_iff
tff(fact_9065_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_9066_nat__abs__mult__distrib,axiom,
    ! [W2: int,Z: int] : aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,W2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Z))) ).

% nat_abs_mult_distrib
tff(fact_9067_nat__plus__as__int,axiom,
    ! [X2: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_plus_as_int
tff(fact_9068_zsgn__def,axiom,
    ! [I: int] :
      aa_int_int(sgn_sgn_int,I) = $ite(
        I = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),I),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% zsgn_def
tff(fact_9069_powr__real__of__int,axiom,
    ! [X: real,Na: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,Na)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,Na)),aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Na))))) ) ) ).

% powr_real_of_int
tff(fact_9070_nat__less__eq__zless,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ) ).

% nat_less_eq_zless
tff(fact_9071_nat__le__eq__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),W2)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ) ).

% nat_le_eq_zle
tff(fact_9072_nat__eq__iff,axiom,
    ! [W2: int,Mb: nat] :
      ( ( aa_int_nat(nat2,W2) = Mb )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2),W2 = aa_nat_int(semiri1314217659103216013at_int,Mb),Mb = zero_zero_nat) ) ).

% nat_eq_iff
tff(fact_9073_nat__eq__iff2,axiom,
    ! [Mb: nat,W2: int] :
      ( ( Mb = aa_int_nat(nat2,W2) )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2),W2 = aa_nat_int(semiri1314217659103216013at_int,Mb),Mb = zero_zero_nat) ) ).

% nat_eq_iff2
tff(fact_9074_split__nat,axiom,
    ! [P: fun_nat_o,I: int] :
      ( aa_nat_o(P,aa_int_nat(nat2,I))
    <=> ( ! [N3: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N3) )
           => aa_nat_o(P,N3) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_nat
tff(fact_9075_le__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Na)),K) ) ) ).

% le_nat_iff
tff(fact_9076_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_9077_nat__mult__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ).

% nat_mult_distrib
tff(fact_9078_Suc__as__int,axiom,
    ! [X2: nat] : aa_nat_nat(suc,X2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X2)),one_one_int)) ).

% Suc_as_int
tff(fact_9079_forall__pos__mono__1,axiom,
    ! [P: fun_real_o,E2: real] :
      ( ! [D4: real,E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D4),E)
         => ( aa_real_o(P,D4)
           => aa_real_o(P,E) ) )
     => ( ! [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),E2)
         => aa_real_o(P,E2) ) ) ) ).

% forall_pos_mono_1
tff(fact_9080_forall__pos__mono,axiom,
    ! [P: fun_real_o,E2: real] :
      ( ! [D4: real,E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D4),E)
         => ( aa_real_o(P,D4)
           => aa_real_o(P,E) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero_nat )
           => aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N))) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(P,E2) ) ) ) ).

% forall_pos_mono
tff(fact_9081_real__arch__inverse,axiom,
    ! [E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
    <=> ? [N3: nat] :
          ( ( N3 != zero_zero_nat )
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N3)))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N3))),E2) ) ) ).

% real_arch_inverse
tff(fact_9082_nat__diff__distrib,axiom,
    ! [Z5: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z5),Z)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_diff_distrib
tff(fact_9083_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_9084_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% nat_abs_triangle_ineq
tff(fact_9085_nat__floor__neg,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_9086_nat__power__eq,axiom,
    ! [Z: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),Na) ) ) ).

% nat_power_eq
tff(fact_9087_ln__inverse,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(ln_ln_real,aa_real_real(inverse_inverse_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,X)) ) ) ).

% ln_inverse
tff(fact_9088_floor__eq3,axiom,
    ! [Na: nat,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = Na ) ) ) ).

% floor_eq3
tff(fact_9089_le__nat__floor,axiom,
    ! [X: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))) ) ).

% le_nat_floor
tff(fact_9090_divide__int__def,axiom,
    ! [K: int,L: int] :
      aa_int_int(divide_divide_int(K),L) = $ite(
        L = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))))) ) ).

% divide_int_def
tff(fact_9091_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_9092_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_9093_nat__less__iff,axiom,
    ! [W2: int,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),Mb)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),aa_nat_int(semiri1314217659103216013at_int,Mb)) ) ) ).

% nat_less_iff
tff(fact_9094_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z5))) ) ) ).

% nat_mult_distrib_neg
tff(fact_9095_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_9096_log__inverse,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(inverse_inverse_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)) ) ) ) ) ).

% log_inverse
tff(fact_9097_floor__eq4,axiom,
    ! [Na: nat,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Na)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = Na ) ) ) ).

% floor_eq4
tff(fact_9098_diff__nat__eq__if,axiom,
    ! [Z: int,Z5: int] :
      aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),zero_zero_int),
        aa_int_nat(nat2,Z),
        $let(
          d: int,
          d:= aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5),
          $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,d),zero_zero_int),zero_zero_nat,aa_int_nat(nat2,d)) ) ) ).

% diff_nat_eq_if
tff(fact_9099_nat__dvd__iff,axiom,
    ! [Z: int,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,Z)),Mb)
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,Mb)),Mb = zero_zero_nat) ) ).

% nat_dvd_iff
tff(fact_9100_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q4: int] :
      ( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,L))
       => ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L)),R2) )
         => aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_9101_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(inverse_inverse_real,X))) ) ).

% plus_inverse_ge_2
tff(fact_9102_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,X) ) ) ).

% real_inv_sqrt_pow2
tff(fact_9103_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) ) ) )
         => ~ ! [R4: int,Q2: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R4) )
               => ( ( aa_int_int(sgn_sgn_int,R4) = aa_int_int(sgn_sgn_int,A22) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R4)),aa_int_int(abs_abs_int,A22))
                   => ( A1 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q2),A22)),R4) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_9104_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,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q5),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),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,Q5),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_9105_divide__int__unfold,axiom,
    ! [K: int,Mb: nat,L: int,Na: nat] :
      aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Na))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Na = zero_zero_nat ) ),
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na)),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),Na)),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),Mb)))))) ) ).

% divide_int_unfold
tff(fact_9106_sinh__ln__real,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(inverse_inverse_real,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_9107_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_9108_sinh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,X)),aa_real_real(sinh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ).

% sinh_real_less_iff
tff(fact_9109_sinh__real__neg__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,X)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real) ) ).

% sinh_real_neg_iff
tff(fact_9110_sinh__real__pos__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sinh_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X) ) ).

% sinh_real_pos_iff
tff(fact_9111_sinh__less__cosh__real,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X)) ).

% sinh_less_cosh_real
tff(fact_9112_cosh__real__pos,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cosh_real,X)) ).

% cosh_real_pos
tff(fact_9113_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_9114_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( 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(cosh_real,X)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_9115_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),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_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_9116_sgn__real__def,axiom,
    ! [A: real] :
      aa_real_real(sgn_sgn_real,A) = $ite(
        A = zero_zero_real,
        zero_zero_real,
        $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% sgn_real_def
tff(fact_9117_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),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_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_9118_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_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_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_9119_sgn__power__injE,axiom,
    ! [A: real,Na: nat,X: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Na)) = X )
     => ( ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B)),Na)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_9120_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,X)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_9121_floor__real__def,axiom,
    ! [X: real] : aa_real_int(archim6058952711729229775r_real,X) = the_int(aa_real_fun_int_o(aTP_Lamm_wi,X)) ).

% floor_real_def
tff(fact_9122_cosh__ln__real,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(inverse_inverse_real,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_9123_floor__rat__def,axiom,
    ! [X: rat] : aa_rat_int(archim3151403230148437115or_rat,X) = the_int(aTP_Lamm_wj(X)) ).

% floor_rat_def
tff(fact_9124_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_9125_sum__power2,axiom,
    ! [K: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),set_or4665077453230672383an_nat(zero_zero_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),one_one_nat) ).

% sum_power2
tff(fact_9126_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,set_or4665077453230672383an_nat(L,U)) ).

% finite_atLeastLessThan
tff(fact_9127_atLeastLessThan__singleton,axiom,
    ! [Mb: nat] : set_or4665077453230672383an_nat(Mb,aa_nat_nat(suc,Mb)) = insert_nat(Mb,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_9128_sgn__rat__def,axiom,
    ! [A: rat] :
      aa_rat_rat(sgn_sgn_rat,A) = $ite(
        A = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% sgn_rat_def
tff(fact_9129_abs__rat__def,axiom,
    ! [A: rat] :
      aa_rat_rat(abs_abs_rat,A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A),A) ).

% abs_rat_def
tff(fact_9130_less__eq__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
        | ( X = Y ) ) ) ).

% less_eq_rat_def
tff(fact_9131_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
     => ~ ! [S2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),S2)
           => ! [T6: rat] :
                ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),T6)
               => ( R2 != aa_rat_rat(plus_plus_rat(S2),T6) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9132_ex__nat__less__eq,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),Na)
          & aa_nat_o(P,M5) )
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_or4665077453230672383an_nat(zero_zero_nat,Na))
          & aa_nat_o(P,X4) ) ) ).

% ex_nat_less_eq
tff(fact_9133_all__nat__less__eq,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),Na)
         => aa_nat_o(P,M5) )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_or4665077453230672383an_nat(zero_zero_nat,Na))
         => aa_nat_o(P,X4) ) ) ).

% all_nat_less_eq
tff(fact_9134_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : set_or4665077453230672383an_nat(L,aa_nat_nat(suc,U)) = set_or1269000886237332187st_nat(L,U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_9135_lessThan__atLeast0,axiom,
    ! [Na: nat] : set_ord_lessThan_nat(Na) = set_or4665077453230672383an_nat(zero_zero_nat,Na) ).

% lessThan_atLeast0
tff(fact_9136_atLeastLessThan0,axiom,
    ! [Mb: nat] : set_or4665077453230672383an_nat(Mb,zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_9137_atLeast0__lessThan__Suc,axiom,
    ! [Na: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,Na)) = insert_nat(Na,set_or4665077453230672383an_nat(zero_zero_nat,Na)) ).

% atLeast0_lessThan_Suc
tff(fact_9138_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N4: set_nat,Na: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N4),set_or4665077453230672383an_nat(zero_zero_nat,Na))
     => aa_set_nat_o(finite_finite_nat,N4) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_9139_atLeastLessThanSuc,axiom,
    ! [Mb: nat,Na: nat] :
      set_or4665077453230672383an_nat(Mb,aa_nat_nat(suc,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na),insert_nat(Na,set_or4665077453230672383an_nat(Mb,Na)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_9140_prod__Suc__Suc__fact,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),Na)) = aa_nat_nat(semiri1408675320244567234ct_nat,Na) ).

% prod_Suc_Suc_fact
tff(fact_9141_prod__Suc__fact,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(zero_zero_nat,Na)) = aa_nat_nat(semiri1408675320244567234ct_nat,Na) ).

% prod_Suc_fact
tff(fact_9142_atLeastLessThan__nat__numeral,axiom,
    ! [Mb: nat,K: num] :
      set_or4665077453230672383an_nat(Mb,aa_num_nat(numeral_numeral_nat,K)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_num_nat(pred_numeral,K)),insert_nat(aa_num_nat(pred_numeral,K),set_or4665077453230672383an_nat(Mb,aa_num_nat(pred_numeral,K))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_9143_atLeast1__lessThan__eq__remove0,axiom,
    ! [Na: nat] : set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),Na) = minus_minus_set_nat(set_ord_lessThan_nat(Na),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_9144_Chebyshev__sum__upper__nat,axiom,
    ! [Na: 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),Na)
           => 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),Na)
             => 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,Na),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_wk(A),B)),set_or4665077453230672383an_nat(zero_zero_nat,Na)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),set_or4665077453230672383an_nat(zero_zero_nat,Na))),aa_set_nat_nat(groups3542108847815614940at_nat(B),set_or4665077453230672383an_nat(zero_zero_nat,Na)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_9145_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_9146_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : aa_set_int_o(finite_finite_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,L),U)) ).

% finite_atLeastLessThan_int
tff(fact_9147_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : aa_set_int_o(finite_finite_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,zero_zero_int),U)) ).

% finite_atLeastZeroLessThan_int
tff(fact_9148_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_9149_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_wl),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_9150_normalize__negative,axiom,
    ! [Q4: int,P2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Q4),zero_zero_int)
     => ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4)) = 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,Q4))) ) ) ).

% normalize_negative
tff(fact_9151_Cauchy__iff2,axiom,
    ! [X5: fun_nat_real] :
      ( topolo4055970368930404560y_real(X5)
    <=> ! [J3: nat] :
        ? [M7: nat] :
        ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),M5)
         => ! [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(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(X5,M5)),aa_nat_real(X5,N3)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_9152_vebt__mint_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(X) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),X)
       => ( ! [A3: $o,B3: $o] :
              ( ( X = 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))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = 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_9153_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(aa_num_rat(numeral_numeral_rat,K)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),one_one_int) ).

% quotient_of_number(3)
tff(fact_9154_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_9155_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_9156_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_9157_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int) ).

% quotient_of_number(5)
tff(fact_9158_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_9159_rat__divide__code,axiom,
    ! [P2: rat,Q4: rat] : quotient_of(aa_rat_rat(divide_divide_rat(P2),Q4)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_wn(Q4)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_9160_rat__times__code,axiom,
    ! [P2: rat,Q4: rat] : quotient_of(aa_rat_rat(times_times_rat(P2),Q4)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_wp(Q4)),quotient_of(P2)) ).

% rat_times_code
tff(fact_9161_quotient__of__div,axiom,
    ! [R2: rat,Na: int,D: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Na),D) )
     => ( R2 = aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,Na)),aa_int_rat(ring_1_of_int_rat,D)) ) ) ).

% quotient_of_div
tff(fact_9162_rat__minus__code,axiom,
    ! [P2: rat,Q4: rat] : quotient_of(aa_rat_rat(minus_minus_rat(P2),Q4)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_wr(Q4)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_9163_quotient__of__denom__pos,axiom,
    ! [R2: rat,P2: int,Q4: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q4) ) ).

% quotient_of_denom_pos
tff(fact_9164_rat__plus__code,axiom,
    ! [P2: rat,Q4: rat] : quotient_of(aa_rat_rat(plus_plus_rat(P2),Q4)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_wt(Q4)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_9165_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P2: int,Q4: int] :
      ( ( aa_Pro6970504127699319548nt_int(normalize,R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q4) ) ).

% normalize_denom_pos
tff(fact_9166_normalize__crossproduct,axiom,
    ! [Q4: int,S: int,P2: int,R2: int] :
      ( ( Q4 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R2),S)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P2),S) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),Q4) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9167_rat__uminus__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(uminus_uminus_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_wu),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_9168_rat__less__code,axiom,
    ! [P2: rat,Q4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),Q4)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_ww(Q4)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_9169_rat__abs__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(abs_abs_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_wx),quotient_of(P2)) ).

% rat_abs_code
tff(fact_9170_vebt__maxt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(X) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),X)
       => ( ! [A3: $o,B3: $o] :
              ( ( X = 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))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = 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_9171_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_9172_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,X)
      <=> (Y) )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X)
       => ( ( ( X = vEBT_Leaf($false,$false) )
           => ( (Y)
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( X = vEBT_Leaf($true,(Uv2)) )
               => ( ~ (Y)
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( X = vEBT_Leaf((Uu2),$true) )
                 => ( ~ (Y)
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
                   => ( (Y)
                     => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
                     => ( ~ (Y)
                       => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_9173_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,X)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X)
       => ( ( ( X = vEBT_Leaf($false,$false) )
           => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_9174_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,X)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X)
       => ( ! [Uv2: $o] :
              ( ( X = vEBT_Leaf($true,(Uv2)) )
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) )
         => ( ! [Uu2: $o] :
                ( ( X = vEBT_Leaf((Uu2),$true) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_9175_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),aa_num_int(numeral_numeral_int,K))) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,K)) ).

% Frct_code_post(5)
tff(fact_9176_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_jq(L),Qr) ).

% divmod_step_integer_def
tff(fact_9177_divmod__integer_H__def,axiom,
    ! [Mb: num,Na: num] : unique3479559517661332726nteger(Mb,Na) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Na)),modulo364778990260209775nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Na))) ).

% divmod_integer'_def
tff(fact_9178_sgn__integer__code,axiom,
    ! [K: code_integer] :
      aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = $ite(
        K = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) ) ).

% sgn_integer_code
tff(fact_9179_exhaustive__integer_H_Ocases,axiom,
    ! [X: produc7509114972854644512nteger] :
      ~ ! [F2: fun_Co3460408610320954864e_term,D4: code_integer,I3: code_integer] : X != aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,F2),produc1086072967326762835nteger(D4,I3)) ).

% exhaustive_integer'.cases
tff(fact_9180_full__exhaustive__integer_H_Ocases,axiom,
    ! [X: produc2453742332425399446nteger] :
      ~ ! [F2: fun_Pr5098669352780829690e_term,D4: code_integer,I3: code_integer] : X != aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,F2),produc1086072967326762835nteger(D4,I3)) ).

% full_exhaustive_integer'.cases
tff(fact_9181_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_9182_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_9183_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_9184_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_9185_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_9186_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_9187_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),one_one_int)) = aa_num_rat(numeral_numeral_rat,K) ).

% Frct_code_post(4)
tff(fact_9188_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),aa_num_int(numeral_numeral_int,L))) = aa_rat_rat(divide_divide_rat(aa_num_rat(numeral_numeral_rat,K)),aa_num_rat(numeral_numeral_rat,L)) ).

% Frct_code_post(6)
tff(fact_9189_integer__of__int__code,axiom,
    ! [K: int] :
      aa_int_Code_integer(code_integer_of_int,K) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
        aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K))),
        $ite(
          K = zero_zero_int,
          zero_z3403309356797280102nteger,
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% integer_of_int_code
tff(fact_9190_pair__leqI2,axiom,
    ! [A: nat,B: nat,S: nat,T: 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,S),T)
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_9191_card__Collect__less__nat,axiom,
    ! [Na: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_by,Na))) = Na ).

% card_Collect_less_nat
tff(fact_9192_card__atMost,axiom,
    ! [U: nat] : aa_set_nat_nat(finite_card_nat,set_ord_atMost_nat(U)) = aa_nat_nat(suc,U) ).

% card_atMost
tff(fact_9193_card__Collect__le__nat,axiom,
    ! [Na: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bz,Na))) = aa_nat_nat(suc,Na) ).

% card_Collect_le_nat
tff(fact_9194_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_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_9195_abs__integer__code,axiom,
    ! [K: code_integer] :
      abs_abs_Code_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),K) ).

% abs_integer_code
tff(fact_9196_less__integer__code_I1_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_integer_code(1)
tff(fact_9197_less__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa2),X) ) ).

% less_integer.abs_eq
tff(fact_9198_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_wy(M6),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_9199_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_wz(M6),I)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_wy(M6),I))) ) ) ).

% card_less_Suc
tff(fact_9200_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_wz(M6),I))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_wy(M6),I))) ) ) ).

% card_less_Suc2
tff(fact_9201_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),set_or4665077453230672383an_nat(K,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A2))))
     => ( A2 = set_or4665077453230672383an_nat(K,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A2))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_9202_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N4: set_nat,Na: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N4),set_or4665077453230672383an_nat(zero_zero_nat,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N4)),Na) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_9203_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_dr),set_or4665077453230672383an_nat(zero_zero_nat,aa_set_nat_nat(finite_card_nat,S3)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_dr),S3)) ).

% card_sum_le_nat_sum
tff(fact_9204_card__nth__roots,axiom,
    ! [C: complex,Na: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_xa(C),Na))) = Na ) ) ) ).

% card_nth_roots
tff(fact_9205_card__roots__unity__eq,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_he,Na))) = Na ) ) ).

% card_roots_unity_eq
tff(fact_9206_pair__leqI1,axiom,
    ! [A: nat,B: nat,S: nat,T: 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),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_9207_int__of__integer__code,axiom,
    ! [K: code_integer] :
      code_int_of_integer(K) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),
        aa_int_int(uminus_uminus_int,code_int_of_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))),
        $ite(K = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_xb,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ) ).

% int_of_integer_code
tff(fact_9208_less__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Xa2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,code_int_of_integer(X)),code_int_of_integer(Xa2)) ) ).

% less_integer.rep_eq
tff(fact_9209_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),L)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,code_int_of_integer(K)),code_int_of_integer(L)) ) ).

% integer_less_iff
tff(fact_9210_divmod__integer__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = produc1086072967326762835nteger(divide6298287555418463151nteger(K,L),modulo364778990260209775nteger(K,L)) ).

% divmod_integer_def
tff(fact_9211_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      code_nat_of_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_xc,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% nat_of_integer_code
tff(fact_9212_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = produc6677183202524767010eger_o(divide6298287555418463151nteger(K,numera6620942414471956472nteger(aa_num_num(bit0,one))),~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),K)) ).

% bit_cut_integer_def
tff(fact_9213_divmod__abs__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_abs(K,L) = produc1086072967326762835nteger(divide6298287555418463151nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L)),modulo364778990260209775nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L))) ).

% divmod_abs_def
tff(fact_9214_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger)
     => ( code_nat_of_integer(K) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_9215_nat__of__integer__code__post_I1_J,axiom,
    code_nat_of_integer(zero_z3403309356797280102nteger) = zero_zero_nat ).

% nat_of_integer_code_post(1)
tff(fact_9216_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_9217_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_9218_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      code_bit_cut_integer(K) = $ite(K = zero_z3403309356797280102nteger,produc6677183202524767010eger_o(zero_z3403309356797280102nteger,$false),produc9125791028180074456eger_o(aTP_Lamm_xd(K),code_divmod_abs(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% bit_cut_integer_code
tff(fact_9219_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L),
          $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_xe(L),code_divmod_abs(K,L))),
          $ite(
            L = zero_z3403309356797280102nteger,
            produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
            aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),
              $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_xf(L),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_9220_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_xg),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_9221_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_ci),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_9222_less__eq__nat_Osimps_I2_J,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Na)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,Mb)),Na) ) ).

% less_eq_nat.simps(2)
tff(fact_9223_diff__Suc,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Na)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_dr),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)) ).

% diff_Suc
tff(fact_9224_binomial__def,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_xh(Na),K))) ).

% binomial_def
tff(fact_9225_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_dr),Nat) ).

% pred_def
tff(fact_9226_wmin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),XS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,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_9227_wmax__insertI,axiom,
    ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y),YS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,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(X,XS),YS)),fun_max_weak) ) ) ) ).

% wmax_insertI
tff(fact_9228_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero_nat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ).

% bezw_0
tff(fact_9229_push__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se545348938243370406it_int(Na),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% push_bit_negative_int_iff
tff(fact_9230_push__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ).

% push_bit_of_Suc_0
tff(fact_9231_wmin__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] : aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,bot_bo2099793752762293965at_nat)),fun_min_weak) ).

% wmin_emptyI
tff(fact_9232_wmax__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,X5)
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,X5)),fun_max_weak) ) ).

% wmax_emptyI
tff(fact_9233_bit__push__bit__iff__int,axiom,
    ! [Mb: nat,K: int,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_se545348938243370406it_int(Mb),K)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Mb)) ) ) ).

% bit_push_bit_iff_int
tff(fact_9234_bit__push__bit__iff__nat,axiom,
    ! [Mb: nat,Q4: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),Q4)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Mb)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9235_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y6: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X5: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y),Y6)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_less)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,Y6)),fun_max_strict)
         => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X,X5),Y6)),fun_max_strict) ) ) ) ).

% smax_insertI
tff(fact_9236_smin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),XS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,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_9237_prod__decode__aux_Osimps,axiom,
    ! [K: nat,Mb: nat] :
      nat_prod_decode_aux(K,Mb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),K),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),Mb)),nat_prod_decode_aux(aa_nat_nat(suc,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,K)))) ).

% prod_decode_aux.simps
tff(fact_9238_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(X,Xa2) = Y )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa2),X),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa2)),nat_prod_decode_aux(aa_nat_nat(suc,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X)))) ) ) ).

% prod_decode_aux.elims
tff(fact_9239_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_9240_smin__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] :
      ( ( X5 != bot_bo2099793752762293965at_nat )
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,bot_bo2099793752762293965at_nat)),fun_min_strict) ) ).

% smin_emptyI
tff(fact_9241_Suc__0__div__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_div_numeral
tff(fact_9242_Suc__0__mod__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_mod_numeral
tff(fact_9243_finite__enumerate,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ? [R4: fun_nat_nat] :
          ( aa_set_nat_o(strict1292158309912662752at_nat(R4),set_ord_lessThan_nat(aa_set_nat_nat(finite_card_nat,S3)))
          & ! [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),aa_set_nat_nat(finite_card_nat,S3))
             => aa_set_nat_o(member_nat(aa_nat_nat(R4,N5)),S3) ) ) ) ).

% finite_enumerate
tff(fact_9244_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_9245_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_9246_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(quotient_of(R2))) ).

% quotient_of_denom_pos'
tff(fact_9247_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Y)
     => ( bezw(X,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,X),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ).

% bezw_non_0
tff(fact_9248_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( bezw(X,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,X),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,X),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,X),Xa2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa2)))))) ) ) ).

% bezw.elims
tff(fact_9249_bezw_Osimps,axiom,
    ! [X: nat,Y: nat] :
      bezw(X,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,X),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)))))) ).

% bezw.simps
tff(fact_9250_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_9251_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa2) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),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,X),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,X),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,X),Xa2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa2)))))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)) ) ) ) ).

% bezw.pelims
tff(fact_9252_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(X,Xa2) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))
       => ~ ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa2),X),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa2)),nat_prod_decode_aux(aa_nat_nat(suc,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X)))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_9253_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(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(divide_divide_int(product_fst_int_int(P2)),a)),aa_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(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(divide_divide_int(product_fst_int_int(P2)),a)),aa_int_int(divide_divide_int(product_snd_int_int(P2)),a)) ) ) ) ).

% normalize_def
tff(fact_9254_gcd__1__int,axiom,
    ! [Mb: int] : aa_int_int(gcd_gcd_int(Mb),one_one_int) = one_one_int ).

% gcd_1_int
tff(fact_9255_gcd__neg2__int,axiom,
    ! [X: int,Y: int] : aa_int_int(gcd_gcd_int(X),aa_int_int(uminus_uminus_int,Y)) = aa_int_int(gcd_gcd_int(X),Y) ).

% gcd_neg2_int
tff(fact_9256_gcd__neg1__int,axiom,
    ! [X: int,Y: int] : aa_int_int(gcd_gcd_int(aa_int_int(uminus_uminus_int,X)),Y) = aa_int_int(gcd_gcd_int(X),Y) ).

% gcd_neg1_int
tff(fact_9257_abs__gcd__int,axiom,
    ! [X: int,Y: int] : aa_int_int(abs_abs_int,aa_int_int(gcd_gcd_int(X),Y)) = aa_int_int(gcd_gcd_int(X),Y) ).

% abs_gcd_int
tff(fact_9258_gcd__abs1__int,axiom,
    ! [X: int,Y: int] : aa_int_int(gcd_gcd_int(aa_int_int(abs_abs_int,X)),Y) = aa_int_int(gcd_gcd_int(X),Y) ).

% gcd_abs1_int
tff(fact_9259_gcd__abs2__int,axiom,
    ! [X: int,Y: int] : aa_int_int(gcd_gcd_int(X),aa_int_int(abs_abs_int,Y)) = aa_int_int(gcd_gcd_int(X),Y) ).

% gcd_abs2_int
tff(fact_9260_gcd__pos__int,axiom,
    ! [Mb: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(gcd_gcd_int(Mb),Na))
    <=> ( ( Mb != zero_zero_int )
        | ( Na != zero_zero_int ) ) ) ).

% gcd_pos_int
tff(fact_9261_gcd__neg__numeral__1__int,axiom,
    ! [Na: num,X: int] : aa_int_int(gcd_gcd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))),X) = aa_int_int(gcd_gcd_int(aa_num_int(numeral_numeral_int,Na)),X) ).

% gcd_neg_numeral_1_int
tff(fact_9262_gcd__neg__numeral__2__int,axiom,
    ! [X: int,Na: num] : aa_int_int(gcd_gcd_int(X),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_int_int(gcd_gcd_int(X),aa_num_int(numeral_numeral_int,Na)) ).

% gcd_neg_numeral_2_int
tff(fact_9263_gcd__0__int,axiom,
    ! [X: int] : aa_int_int(gcd_gcd_int(X),zero_zero_int) = aa_int_int(abs_abs_int,X) ).

% gcd_0_int
tff(fact_9264_gcd__0__left__int,axiom,
    ! [X: int] : aa_int_int(gcd_gcd_int(zero_zero_int),X) = aa_int_int(abs_abs_int,X) ).

% gcd_0_left_int
tff(fact_9265_gcd__proj1__if__dvd__int,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),Y)
     => ( aa_int_int(gcd_gcd_int(X),Y) = aa_int_int(abs_abs_int,X) ) ) ).

% gcd_proj1_if_dvd_int
tff(fact_9266_gcd__proj2__if__dvd__int,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y),X)
     => ( aa_int_int(gcd_gcd_int(X),Y) = aa_int_int(abs_abs_int,Y) ) ) ).

% gcd_proj2_if_dvd_int
tff(fact_9267_gcd__idem__int,axiom,
    ! [X: int] : aa_int_int(gcd_gcd_int(X),X) = aa_int_int(abs_abs_int,X) ).

% gcd_idem_int
tff(fact_9268_gcd__red__int,axiom,
    ! [X: int,Y: int] : aa_int_int(gcd_gcd_int(X),Y) = aa_int_int(gcd_gcd_int(Y),aa_int_int(modulo_modulo_int(X),Y)) ).

% gcd_red_int
tff(fact_9269_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(gcd_gcd_int(X),Y)) ).

% gcd_ge_0_int
tff(fact_9270_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U2: int,V2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U2),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V2),Y)) = aa_int_int(gcd_gcd_int(X),Y) ).

% bezout_int
tff(fact_9271_gcd__mult__distrib__int,axiom,
    ! [K: int,Mb: int,Na: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,K)),aa_int_int(gcd_gcd_int(Mb),Na)) = aa_int_int(gcd_gcd_int(aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Na)) ).

% gcd_mult_distrib_int
tff(fact_9272_gcd__le2__int,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,aa_int_int(gcd_gcd_int(A),B)),B) ) ).

% gcd_le2_int
tff(fact_9273_gcd__le1__int,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,aa_int_int(gcd_gcd_int(A),B)),A) ) ).

% gcd_le1_int
tff(fact_9274_gcd__cases__int,axiom,
    ! [X: int,Y: int,P: fun_int_o] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => aa_int_o(P,aa_int_int(gcd_gcd_int(X),Y)) ) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
           => aa_int_o(P,aa_int_int(gcd_gcd_int(X),aa_int_int(uminus_uminus_int,Y))) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),zero_zero_int)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
             => aa_int_o(P,aa_int_int(gcd_gcd_int(aa_int_int(uminus_uminus_int,X)),Y)) ) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),zero_zero_int)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
               => aa_int_o(P,aa_int_int(gcd_gcd_int(aa_int_int(uminus_uminus_int,X)),aa_int_int(uminus_uminus_int,Y))) ) )
           => aa_int_o(P,aa_int_int(gcd_gcd_int(X),Y)) ) ) ) ) ).

% gcd_cases_int
tff(fact_9275_gcd__unique__int,axiom,
    ! [D: int,A: int,B: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),A)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B)
        & ! [E3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),A)
              & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),B) )
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),D) ) )
    <=> ( D = aa_int_int(gcd_gcd_int(A),B) ) ) ).

% gcd_unique_int
tff(fact_9276_gcd__non__0__int,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Y)
     => ( aa_int_int(gcd_gcd_int(X),Y) = aa_int_int(gcd_gcd_int(Y),aa_int_int(modulo_modulo_int(X),Y)) ) ) ).

% gcd_non_0_int
tff(fact_9277_gcd__code__int,axiom,
    ! [K: int,L: int] :
      aa_int_int(gcd_gcd_int(K),L) = aa_int_int(abs_abs_int,
        $ite(L = zero_zero_int,K,aa_int_int(gcd_gcd_int(L),aa_int_int(modulo_modulo_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))))) ).

% gcd_code_int
tff(fact_9278_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_9279_gcd__nat_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),A) = A ).

% gcd_nat.left_neutral
tff(fact_9280_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_9281_gcd__nat_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),zero_zero_nat) = A ).

% gcd_nat.right_neutral
tff(fact_9282_gcd__0__nat,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),zero_zero_nat) = X ).

% gcd_0_nat
tff(fact_9283_gcd__0__left__nat,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),X) = X ).

% gcd_0_left_nat
tff(fact_9284_gcd__1__nat,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),one_one_nat) = one_one_nat ).

% gcd_1_nat
tff(fact_9285_gcd__proj2__if__dvd__nat,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Y),X)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = Y ) ) ).

% gcd_proj2_if_dvd_nat
tff(fact_9286_gcd__proj1__if__dvd__nat,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X ) ) ).

% gcd_proj1_if_dvd_nat
tff(fact_9287_gcd__nat_Obounded__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(gcd_gcd_nat,B),C))
    <=> ( 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) ) ) ).

% gcd_nat.bounded_iff
tff(fact_9288_gcd__nat_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = B ) ) ).

% gcd_nat.absorb2
tff(fact_9289_gcd__nat_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = A ) ) ).

% gcd_nat.absorb1
tff(fact_9290_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_9291_gcd__pos__nat,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Na))
    <=> ( ( Mb != zero_zero_nat )
        | ( Na != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9292_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(K),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% or_negative_int_iff
tff(fact_9293_gcd__int__int__eq,axiom,
    ! [Mb: nat,Na: nat] : aa_int_int(gcd_gcd_int(aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Na)) ).

% gcd_int_int_eq
tff(fact_9294_or__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% or_nat_numerals(4)
tff(fact_9295_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_9296_gcd__nat__abs__right__eq,axiom,
    ! [Na: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Na),aa_int_nat(nat2,aa_int_int(abs_abs_int,K))) = aa_int_nat(nat2,aa_int_int(gcd_gcd_int(aa_nat_int(semiri1314217659103216013at_int,Na)),K)) ).

% gcd_nat_abs_right_eq
tff(fact_9297_gcd__nat__abs__left__eq,axiom,
    ! [K: int,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),Na) = aa_int_nat(nat2,aa_int_int(gcd_gcd_int(K),aa_nat_int(semiri1314217659103216013at_int,Na))) ).

% gcd_nat_abs_left_eq
tff(fact_9298_or__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% or_nat_numerals(3)
tff(fact_9299_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_9300_gcd__mult__distrib__nat,axiom,
    ! [K: nat,Mb: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) ).

% gcd_mult_distrib_nat
tff(fact_9301_gcd__red__nat,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ).

% gcd_red_nat
tff(fact_9302_gcd__unique__nat,axiom,
    ! [D: nat,A: nat,B: 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)
        & ! [E3: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,E3),A)
              & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,E3),B) )
           => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,E3),D) ) )
    <=> ( D = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ).

% gcd_unique_nat
tff(fact_9303_gcd__nat_Ostrict__coboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
        & ( B != C ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C)
        & ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) != C ) ) ) ).

% gcd_nat.strict_coboundedI2
tff(fact_9304_gcd__nat_Ostrict__coboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
        & ( A != C ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C)
        & ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) != C ) ) ) ).

% gcd_nat.strict_coboundedI1
tff(fact_9305_gcd__nat_Ostrict__order__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) )
        & ( A != B ) ) ) ).

% gcd_nat.strict_order_iff
tff(fact_9306_gcd__nat_Ostrict__boundedE,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(gcd_gcd_nat,B),C))
        & ( A != aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C) ) )
     => ~ ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
            & ( A != B ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
              & ( A != C ) ) ) ) ).

% gcd_nat.strict_boundedE
tff(fact_9307_gcd__nat_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C) ) ).

% gcd_nat.coboundedI2
tff(fact_9308_gcd__nat_OcoboundedI1,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,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C) ) ).

% gcd_nat.coboundedI1
tff(fact_9309_gcd__nat_Oabsorb__iff2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = B ) ) ).

% gcd_nat.absorb_iff2
tff(fact_9310_gcd__nat_Oabsorb__iff1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = A ) ) ).

% gcd_nat.absorb_iff1
tff(fact_9311_gcd__nat_Ocobounded2,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B) ).

% gcd_nat.cobounded2
tff(fact_9312_gcd__nat_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A) ).

% gcd_nat.cobounded1
tff(fact_9313_gcd__nat_Oorder__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ).

% gcd_nat.order_iff
tff(fact_9314_gcd__nat_OboundedI,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(gcd_gcd_nat,B),C)) ) ) ).

% gcd_nat.boundedI
tff(fact_9315_gcd__nat_OboundedE,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(gcd_gcd_nat,B),C))
     => ~ ( 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) ) ) ).

% gcd_nat.boundedE
tff(fact_9316_gcd__nat_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
        & ( B != A ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = B ) ) ).

% gcd_nat.absorb4
tff(fact_9317_gcd__nat_Oabsorb3,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = A ) ) ).

% gcd_nat.absorb3
tff(fact_9318_gcd__nat_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% gcd_nat.orderI
tff(fact_9319_gcd__nat_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ).

% gcd_nat.orderE
tff(fact_9320_gcd__nat_Omono,axiom,
    ! [A: nat,C: nat,B: nat,D: 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,B),D)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,C),D)) ) ) ).

% gcd_nat.mono
tff(fact_9321_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_9322_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_9323_gcd__diff1__nat,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Na)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Na) ) ) ).

% gcd_diff1_nat
tff(fact_9324_gcd__diff2__nat,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Mb)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Na) ) ) ).

% gcd_diff2_nat
tff(fact_9325_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa2) = Y )
     => ( Y = $ite(Xa2 = zero_zero_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2))) ) ) ).

% gcd_nat.elims
tff(fact_9326_gcd__nat_Osimps,axiom,
    ! [X: nat,Y: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = $ite(Y = zero_zero_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y))) ).

% gcd_nat.simps
tff(fact_9327_gcd__non__0__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ) ) ).

% gcd_non_0_nat
tff(fact_9328_gcd__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] : code_int_of_integer(gcd_gcd_Code_integer(X,Xa2)) = aa_int_int(gcd_gcd_int(code_int_of_integer(X)),code_int_of_integer(Xa2)) ).

% gcd_integer.rep_eq
tff(fact_9329_gcd__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : gcd_gcd_Code_integer(aa_int_Code_integer(code_integer_of_int,Xa2),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(gcd_gcd_int(Xa2),X)) ).

% gcd_integer.abs_eq
tff(fact_9330_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X3: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) ) ).

% bezout_nat
tff(fact_9331_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X3: 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),X3))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) )
      | ( 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),X3))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9332_gcd__code__integer,axiom,
    ! [K: code_integer,L: code_integer] :
      gcd_gcd_Code_integer(K,L) = abs_abs_Code_integer($ite(L = zero_z3403309356797280102nteger,K,gcd_gcd_Code_integer(L,modulo364778990260209775nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L))))) ).

% gcd_code_integer
tff(fact_9333_gcd__int__def,axiom,
    ! [X: int,Y: int] : aa_int_int(gcd_gcd_int(X),Y) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,X))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Y)))) ).

% gcd_int_def
tff(fact_9334_bezw__aux,axiom,
    ! [X: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(bezw(X,Y))),aa_nat_int(semiri1314217659103216013at_int,X))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(X,Y))),aa_nat_int(semiri1314217659103216013at_int,Y))) ).

% bezw_aux
tff(fact_9335_OR__upper,axiom,
    ! [X: int,Na: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(X),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ) ).

% OR_upper
tff(fact_9336_or__Suc__0__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% or_Suc_0_eq
tff(fact_9337_Suc__0__or__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% Suc_0_or_eq
tff(fact_9338_or__nat__rec,axiom,
    ! [Mb: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Na) = 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))),Na) ))),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_9339_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa2) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))
       => ~ ( ( Y = $ite(Xa2 = zero_zero_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)) ) ) ) ).

% gcd_nat.pelims
tff(fact_9340_gcd__nat_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),A) = A ).

% gcd_nat.idem
tff(fact_9341_gcd__nat_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ).

% gcd_nat.left_idem
tff(fact_9342_gcd__nat_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ).

% gcd_nat.right_idem
tff(fact_9343_gcd__idem__nat,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),X) = X ).

% gcd_idem_nat
tff(fact_9344_or__nat__unfold,axiom,
    ! [Mb: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Na) = $ite(
        Mb = zero_zero_nat,
        Na,
        $ite(Na = 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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% or_nat_unfold
tff(fact_9345_max__Suc__Suc,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Na)) ).

% max_Suc_Suc
tff(fact_9346_max__0R,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),zero_zero_nat) = Na ).

% max_0R
tff(fact_9347_max__0L,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Na) = Na ).

% max_0L
tff(fact_9348_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_9349_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_9350_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_9351_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_9352_max__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),aa_num_nat(pred_numeral,K))) ).

% max_Suc_numeral
tff(fact_9353_max__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),Na)) ).

% max_numeral_Suc
tff(fact_9354_nat__add__max__right,axiom,
    ! [Mb: nat,Na: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q4)) ).

% nat_add_max_right
tff(fact_9355_nat__add__max__left,axiom,
    ! [Mb: nat,Na: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Na)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Q4)) ).

% nat_add_max_left
tff(fact_9356_nat__mult__max__left,axiom,
    ! [Mb: nat,Na: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Na)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q4)) ).

% nat_mult_max_left
tff(fact_9357_nat__mult__max__right,axiom,
    ! [Mb: nat,Na: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q4)) ).

% nat_mult_max_right
tff(fact_9358_nat__minus__add__max,axiom,
    ! [Na: 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,Na),Mb)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),Mb) ).

% nat_minus_add_max
tff(fact_9359_max__Suc2,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),aa_nat_nat(suc,Na)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Na),aa_nat_fun_nat_nat(aTP_Lamm_xi,Na)),Mb) ).

% max_Suc2
tff(fact_9360_max__Suc1,axiom,
    ! [Na: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Na)),Mb) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Na),aa_nat_fun_nat_nat(aTP_Lamm_xj,Na)),Mb) ).

% max_Suc1
tff(fact_9361_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          L = zero_z3403309356797280102nteger,
          produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
          aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),
            $ite(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_xk(L),code_divmod_abs(K,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_9362_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_9363_root__powr__inverse,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_real(aa_nat_fun_real_real(root,Na),X) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Na))) ) ) ) ).

% root_powr_inverse
tff(fact_9364_drop__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se8568078237143864401it_int(Na),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% drop_bit_negative_int_iff
tff(fact_9365_real__root__Suc__0,axiom,
    ! [X: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),X) = X ).

% real_root_Suc_0
tff(fact_9366_root__0,axiom,
    ! [X: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),X) = zero_zero_real ).

% root_0
tff(fact_9367_real__root__eq__iff,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Na),X) = aa_real_real(aa_nat_fun_real_real(root,Na),Y) )
      <=> ( X = Y ) ) ) ).

% real_root_eq_iff
tff(fact_9368_drop__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,Na = zero_zero_nat) ).

% drop_bit_of_Suc_0
tff(fact_9369_drop__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(bit_se8568078237143864401it_int(Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_9370_real__root__eq__0__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Na),X) = zero_zero_real )
      <=> ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_9371_real__root__less__iff,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ).

% real_root_less_iff
tff(fact_9372_real__root__le__iff,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ).

% real_root_le_iff
tff(fact_9373_real__root__eq__1__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Na),X) = one_one_real )
      <=> ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_9374_real__root__one,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_9375_real__root__lt__0__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real) ) ) ).

% real_root_lt_0_iff
tff(fact_9376_real__root__gt__0__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ) ).

% real_root_gt_0_iff
tff(fact_9377_real__root__le__0__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ) ).

% real_root_le_0_iff
tff(fact_9378_real__root__ge__0__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_9379_real__root__lt__1__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ) ).

% real_root_lt_1_iff
tff(fact_9380_real__root__gt__1__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ) ).

% real_root_gt_1_iff
tff(fact_9381_real__root__le__1__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ) ).

% real_root_le_1_iff
tff(fact_9382_real__root__ge__1__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_9383_drop__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(bit_se8568078237143864401it_int(Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9384_real__root__pow__pos2,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),Na) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_9385_real__root__less__mono,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_real_real(aa_nat_fun_real_real(root,Na),Y)) ) ) ).

% real_root_less_mono
tff(fact_9386_real__root__le__mono,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_real_real(aa_nat_fun_real_real(root,Na),Y)) ) ) ).

% real_root_le_mono
tff(fact_9387_real__root__power,axiom,
    ! [Na: nat,X: real,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),K) ) ) ).

% real_root_power
tff(fact_9388_real__root__abs,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_real_real(abs_abs_real,X)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)) ) ) ).

% real_root_abs
tff(fact_9389_sgn__root,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)) = aa_real_real(sgn_sgn_real,X) ) ) ).

% sgn_root
tff(fact_9390_real__root__gt__zero,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),X)) ) ) ).

% real_root_gt_zero
tff(fact_9391_real__root__strict__decreasing,axiom,
    ! [Na: nat,N4: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N4)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N4),X)),aa_real_real(aa_nat_fun_real_real(root,Na),X)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_9392_root__abs__power,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Na))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_9393_real__root__pos__pos,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),X)) ) ) ).

% real_root_pos_pos
tff(fact_9394_real__root__strict__increasing,axiom,
    ! [Na: nat,N4: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N4)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_real_real(aa_nat_fun_real_real(root,N4),X)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_9395_real__root__decreasing,axiom,
    ! [Na: nat,N4: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N4),X)),aa_real_real(aa_nat_fun_real_real(root,Na),X)) ) ) ) ).

% real_root_decreasing
tff(fact_9396_real__root__pow__pos,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),Na) = X ) ) ) ).

% real_root_pow_pos
tff(fact_9397_real__root__pos__unique,axiom,
    ! [Na: nat,Y: real,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Na) = X )
         => ( aa_real_real(aa_nat_fun_real_real(root,Na),X) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_9398_real__root__power__cancel,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_9399_real__root__increasing,axiom,
    ! [Na: nat,N4: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_real_real(aa_nat_fun_real_real(root,N4),X)) ) ) ) ) ).

% real_root_increasing
tff(fact_9400_root__sgn__power,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y)),Na))) = Y ) ) ).

% root_sgn_power
tff(fact_9401_sgn__power__root,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Na),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Na),X))),Na)) = X ) ) ).

% sgn_power_root
tff(fact_9402_ln__root,axiom,
    ! [Na: nat,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,Na),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% ln_root
tff(fact_9403_log__root,axiom,
    ! [Na: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_nat_fun_real_real(root,Na),A)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),A)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_root
tff(fact_9404_log__base__root,axiom,
    ! [Na: nat,B: real,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_nat_fun_real_real(root,Na),B)),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ) ).

% log_base_root
tff(fact_9405_split__root,axiom,
    ! [P: fun_real_o,Na: nat,X: real] :
      ( aa_real_o(P,aa_real_real(aa_nat_fun_real_real(root,Na),X))
    <=> ( ( ( Na = zero_zero_nat )
         => aa_real_o(P,zero_zero_real) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ! [Y5: real] :
              ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y5)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y5)),Na)) = X )
             => aa_real_o(P,Y5) ) ) ) ) ).

% split_root
tff(fact_9406_Suc__funpow,axiom,
    ! [Na: nat] : compow_fun_nat_nat(Na,suc) = aa_nat_fun_nat_nat(plus_plus_nat,Na) ).

% Suc_funpow
tff(fact_9407_min__Suc__Suc,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Na)) ).

% min_Suc_Suc
tff(fact_9408_min__0R,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Na),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_9409_min__0L,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),Na) = zero_zero_nat ).

% min_0L
tff(fact_9410_min__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Na),aa_num_nat(pred_numeral,K))) ).

% min_Suc_numeral
tff(fact_9411_min__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),Na)) ).

% min_numeral_Suc
tff(fact_9412_min__diff,axiom,
    ! [Mb: nat,I: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Na)),I) ).

% min_diff
tff(fact_9413_nat__mult__min__left,axiom,
    ! [Mb: nat,Na: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Na)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q4)) ).

% nat_mult_min_left
tff(fact_9414_nat__mult__min__right,axiom,
    ! [Mb: nat,Na: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Na),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q4)) ).

% nat_mult_min_right
tff(fact_9415_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_9416_min__Suc2,axiom,
    ! [Mb: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),aa_nat_nat(suc,Na)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_xl,Na)),Mb) ).

% min_Suc2
tff(fact_9417_min__Suc1,axiom,
    ! [Na: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Na)),Mb) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_xm,Na)),Mb) ).

% min_Suc1
tff(fact_9418_inf__int__def,axiom,
    inf_inf_int = ord_min_int ).

% inf_int_def
tff(fact_9419_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_bz,aTP_Lamm_by) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9420_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_xn) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_9421_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_xp),Xa2),X)) ).

% times_int.abs_eq
tff(fact_9422_int_Oabs__induct,axiom,
    ! [P: fun_int_o,X: int] :
      ( ! [Y3: product_prod_nat_nat] : aa_int_o(P,aa_Pro4240683299535347697at_int(abs_Integ,Y3))
     => aa_int_o(P,X) ) ).

% int.abs_induct
tff(fact_9423_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X3: nat,Y3: nat] : Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3)) ).

% eq_Abs_Integ
tff(fact_9424_nat_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] : aa_int_nat(nat2,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),X) ).

% nat.abs_eq
tff(fact_9425_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_9426_int__def,axiom,
    ! [Na: nat] : aa_nat_int(semiri1314217659103216013at_int,Na) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Na),zero_zero_nat)) ).

% int_def
tff(fact_9427_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_xq),X)) ).

% uminus_int.abs_eq
tff(fact_9428_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_9429_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: 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,X))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_xs),Xa2),X) ) ).

% less_int.abs_eq
tff(fact_9430_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: 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,X))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_xu),Xa2),X) ) ).

% less_eq_int.abs_eq
tff(fact_9431_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_xw),Xa2),X)) ).

% plus_int.abs_eq
tff(fact_9432_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_xy),Xa2),X)) ).

% minus_int.abs_eq
tff(fact_9433_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_max_strict,fun_max_weak)) ).

% max_rpair_set
tff(fact_9434_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_min_strict,fun_min_weak)) ).

% min_rpair_set
tff(fact_9435_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa2)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_xu),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa2)) ) ).

% less_eq_int.rep_eq
tff(fact_9436_less__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Xa2)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_xs),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa2)) ) ).

% less_int.rep_eq
tff(fact_9437_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_xz) ).

% prod_encode_def
tff(fact_9438_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_9439_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_9440_nat_Orep__eq,axiom,
    ! [X: int] : aa_int_nat(nat2,X) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),aa_int1417691210696471063at_nat(rep_Integ,X)) ).

% nat.rep_eq
tff(fact_9441_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,Mb: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,nat_prod_decode_aux(K,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K)),Mb) ).

% prod_encode_prod_decode_aux
tff(fact_9442_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_xq)) ).

% uminus_int_def
tff(fact_9443_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_xp)) ).

% times_int_def
tff(fact_9444_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_xy)) ).

% minus_int_def
tff(fact_9445_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_xw)) ).

% plus_int_def
tff(fact_9446_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,minus_minus_set_nat(M6,insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_9447_Gcd__dvd__nat,axiom,
    ! [A: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),A2)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_set_nat_nat(gcd_Gcd_nat,A2)),A) ) ).

% Gcd_dvd_nat
tff(fact_9448_Gcd__greatest__nat,axiom,
    ! [A2: set_nat,A: nat] :
      ( ! [B3: nat] :
          ( aa_set_nat_o(member_nat(B3),A2)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B3) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_set_nat_nat(gcd_Gcd_nat,A2)) ) ).

% Gcd_greatest_nat
tff(fact_9449_Gcd__nat__eq__one,axiom,
    ! [N4: set_nat] :
      ( aa_set_nat_o(member_nat(one_one_nat),N4)
     => ( aa_set_nat_nat(gcd_Gcd_nat,N4) = one_one_nat ) ) ).

% Gcd_nat_eq_one
tff(fact_9450_Gcd__in,axiom,
    ! [A2: set_nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_set_nat_o(member_nat(A3),A2)
         => ( aa_set_nat_o(member_nat(B3),A2)
           => aa_set_nat_o(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A3),B3)),A2) ) )
     => ( ( A2 != bot_bot_set_nat )
       => aa_set_nat_o(member_nat(aa_set_nat_nat(gcd_Gcd_nat,A2)),A2) ) ) ).

% Gcd_in
tff(fact_9451_sub__BitM__One__eq,axiom,
    ! [Na: num] : aa_num_int(neg_numeral_sub_int(aa_num_num(bitM,Na)),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(Na),one)) ).

% sub_BitM_One_eq
tff(fact_9452_num__of__nat_Osimps_I2_J,axiom,
    ! [Na: nat] :
      aa_nat_num(num_of_nat,aa_nat_nat(suc,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na),aa_num_num(inc,aa_nat_num(num_of_nat,Na)),one) ).

% num_of_nat.simps(2)
tff(fact_9453_pred__nat__def,axiom,
    pred_nat = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ya)) ).

% pred_nat_def
tff(fact_9454_abs__Gcd__eq,axiom,
    ! [K5: set_int] : aa_int_int(abs_abs_int,gcd_Gcd_int(K5)) = gcd_Gcd_int(K5) ).

% abs_Gcd_eq
tff(fact_9455_Gcd__dvd__int,axiom,
    ! [A: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,gcd_Gcd_int(A2)),A) ) ).

% Gcd_dvd_int
tff(fact_9456_Gcd__greatest__int,axiom,
    ! [A2: set_int,A: int] :
      ( ! [B3: int] :
          ( aa_set_int_o(member_int(B3),A2)
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B3) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),gcd_Gcd_int(A2)) ) ).

% Gcd_greatest_int
tff(fact_9457_Gcd__int__greater__eq__0,axiom,
    ! [K5: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),gcd_Gcd_int(K5)) ).

% Gcd_int_greater_eq_0
tff(fact_9458_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9459_numeral__num__of__nat,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,Na)) = Na ) ) ).

% numeral_num_of_nat
tff(fact_9460_num__of__nat__One,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),one_one_nat)
     => ( aa_nat_num(num_of_nat,Na) = one ) ) ).

% num_of_nat_One
tff(fact_9461_num__of__nat__double,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Na)) = aa_num_num(bit0,aa_nat_num(num_of_nat,Na)) ) ) ).

% num_of_nat_double
tff(fact_9462_num__of__nat__plus__distrib,axiom,
    ! [Mb: nat,Na: 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),Na)
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_nat_num(num_of_nat,Mb)),aa_nat_num(num_of_nat,Na)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9463_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(X),Xa2)
     => ( ( ? [Uu2: $o,Uv2: $o] : X = vEBT_Leaf((Uu2),(Uv2))
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( ( Deg = Xa2 )
                & $let(
                    n2: nat,
                    n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                    $let(
                      m2: nat,
                      m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n2),
                      ( ! [X4: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(vEBT_VEBT_valid(X4),n2) )
                      & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                      & ( aa_lis7044684121217598542BT_nat(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(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X6)
                            & ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                               => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_yb(Deg),TreeList),Summary),n2),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_9464_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Deg3: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(vEBT_Node(Mima2,Dega,TreeLista,Summarya)),Deg3)
    <=> ( ( Dega = Deg3 )
        & $let(
            n2: nat,
            n2:= 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),n2),
              ( ! [X4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
                 => aa_nat_o(vEBT_VEBT_valid(X4),n2) )
              & aa_nat_o(vEBT_VEBT_valid(Summarya),m2)
              & ( aa_lis7044684121217598542BT_nat(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(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X6)
                    & ! [X4: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_yb(Dega),TreeLista),Summarya),n2),m2))),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_9465_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(X),Xa2)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : X = vEBT_Leaf((Uu2),(Uv2))
         => ( (Y)
          <=> ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( (Y)
              <=> ~ ( ( Deg = Xa2 )
                    & $let(
                        n2: nat,
                        n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n2),
                          ( ! [X4: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(vEBT_VEBT_valid(X4),n2) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                          & ( aa_lis7044684121217598542BT_nat(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(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X6)
                                & ! [X4: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_yb(Deg),TreeList),Summary),n2),m2))),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_9466_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(X),Xa2)
     => ( ( ? [Uu2: $o,Uv2: $o] : X = vEBT_Leaf((Uu2),(Uv2))
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ~ ( ( Deg = Xa2 )
                  & $let(
                      n2: nat,
                      n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                      $let(
                        m2: nat,
                        m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n2),
                        ( ! [X4: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(vEBT_VEBT_valid(X4),n2) )
                        & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                        & ( aa_lis7044684121217598542BT_nat(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(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X6)
                              & ! [X4: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_yb(Deg),TreeList),Summary),n2),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_9467_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(X),Xa2)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( (Y)
                <=> ( Xa2 = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xa2)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( ( (Y)
                  <=> ( ( Deg = Xa2 )
                      & $let(
                          n2: nat,
                          n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n2),
                            ( ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(vEBT_VEBT_valid(X4),n2) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                            & ( aa_lis7044684121217598542BT_nat(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(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X6)
                                  & ! [X4: vEBT_VEBT] :
                                      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                     => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_yb(Deg),TreeList),Summary),n2),m2))),Mima) ) ) ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xa2)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_9468_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(X),Xa2)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xa2))
               => ( Xa2 != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xa2))
                 => ~ ( ( Deg = Xa2 )
                      & $let(
                          n2: nat,
                          n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n2),
                            ( ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(vEBT_VEBT_valid(X4),n2) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                            & ( aa_lis7044684121217598542BT_nat(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(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X6)
                                  & ! [X4: vEBT_VEBT] :
                                      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                     => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_yb(Deg),TreeList),Summary),n2),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_9469_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(X),Xa2)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xa2))
               => ( Xa2 = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xa2))
                 => ( ( Deg = Xa2 )
                    & $let(
                        n2: nat,
                        n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n2),
                          ( ! [X4: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(vEBT_VEBT_valid(X4),n2) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                          & ( aa_lis7044684121217598542BT_nat(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(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X6)
                                & ! [X4: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_yb(Deg),TreeList),Summary),n2),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_9470_GreatestI__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [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_9471_Greatest__le__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [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,K),order_Greatest_nat(P)) ) ) ).

% Greatest_le_nat
tff(fact_9472_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_o,B: nat] :
      ( ? [X_12: nat] : aa_nat_o(P,X_12)
     => ( ! [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_9473_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_9474_take__bit__num__simps_I2_J,axiom,
    ! [Na: nat] : bit_take_bit_num(aa_nat_nat(suc,Na),one) = some_num(one) ).

% take_bit_num_simps(2)
tff(fact_9475_take__bit__num__simps_I3_J,axiom,
    ! [Na: nat,Mb: num] : bit_take_bit_num(aa_nat_nat(suc,Na),aa_num_num(bit0,Mb)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_yc,bit_take_bit_num(Na,Mb)) ).

% take_bit_num_simps(3)
tff(fact_9476_take__bit__num__simps_I4_J,axiom,
    ! [Na: nat,Mb: num] : bit_take_bit_num(aa_nat_nat(suc,Na),aa_num_num(bit1,Mb)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Na,Mb))) ).

% take_bit_num_simps(4)
tff(fact_9477_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_yc,bit_take_bit_num(aa_num_nat(pred_numeral,R2),Mb)) ).

% take_bit_num_simps(6)
tff(fact_9478_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [Na: nat,Mb: num] : bit_take_bit_num(Na,aa_num_num(bit0,Mb)) = case_nat_option_num(none_num,aTP_Lamm_yd(Mb),Na) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_9479_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [Na: nat] : bit_take_bit_num(Na,one) = case_nat_option_num(none_num,aTP_Lamm_ye,Na) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_9480_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [Na: nat,Mb: num] : bit_take_bit_num(Na,aa_num_num(bit1,Mb)) = case_nat_option_num(none_num,aTP_Lamm_yf(Mb),Na) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_9481_take__bit__num__def,axiom,
    ! [Na: nat,Mb: num] :
      bit_take_bit_num(Na,Mb) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),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,Na),aa_num_nat(numeral_numeral_nat,Mb))))) ).

% take_bit_num_def
tff(fact_9482_Bit__Operations_Otake__bit__num__code,axiom,
    ! [Na: nat,Mb: num] : bit_take_bit_num(Na,Mb) = produc478579273971653890on_num(aTP_Lamm_yj,product_Pair_nat_num(Na,Mb)) ).

% Bit_Operations.take_bit_num_code
tff(fact_9483_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_yk) ).

% Rats_eq_int_div_nat
tff(fact_9484_Rats__dense__in__real,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => ? [X3: real] :
          ( aa_set_real_o(member_real(X3),field_5140801741446780682s_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X),X3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y) ) ) ).

% Rats_dense_in_real
tff(fact_9485_Rats__no__bot__less,axiom,
    ! [X: real] :
    ? [X3: real] :
      ( aa_set_real_o(member_real(X3),field_5140801741446780682s_real)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),X) ) ).

% Rats_no_bot_less
tff(fact_9486_rat__floor__lemma,axiom,
    ! [A: int,B: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(A),B))),aa_int_rat(fract(A),B))
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B)),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),B)),one_one_int))) ) ).

% rat_floor_lemma
tff(fact_9487_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,X: nat,Y: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yl,C),set_or4665077453230672383an_nat(X,Y)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Y),
        set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9488_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,set_or1269000886237332187st_nat(I,J)) = set_or1269000886237332187st_nat(aa_nat_nat(suc,I),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_9489_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,set_or4665077453230672383an_nat(I,J)) = set_or4665077453230672383an_nat(aa_nat_nat(suc,I),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_9490_less__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B)),aa_int_rat(fract(C),D))
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ).

% less_rat
tff(fact_9491_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] : ~ aa_set_nat_o(member_nat(zero_zero_nat),image_nat_nat(suc,A2)) ).

% zero_notin_Suc_image
tff(fact_9492_Rat__induct__pos,axiom,
    ! [P: fun_rat_o,Q4: rat] :
      ( ! [A3: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
         => aa_rat_o(P,aa_int_rat(fract(A3),B3)) )
     => aa_rat_o(P,Q4) ) ).

% Rat_induct_pos
tff(fact_9493_quotient__of__eq,axiom,
    ! [A: int,B: int,P2: int,Q4: int] :
      ( ( quotient_of(aa_int_rat(fract(A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4) )
     => ( aa_int_rat(fract(P2),Q4) = aa_int_rat(fract(A),B) ) ) ).

% quotient_of_eq
tff(fact_9494_normalize__eq,axiom,
    ! [A: int,B: int,P2: int,Q4: 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),Q4) )
     => ( aa_int_rat(fract(P2),Q4) = aa_int_rat(fract(A),B) ) ) ).

% normalize_eq
tff(fact_9495_quotient__of__Fract,axiom,
    ! [A: int,B: int] : quotient_of(aa_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_9496_image__Suc__lessThan,axiom,
    ! [Na: nat] : image_nat_nat(suc,set_ord_lessThan_nat(Na)) = set_or1269000886237332187st_nat(one_one_nat,Na) ).

% image_Suc_lessThan
tff(fact_9497_image__Suc__atMost,axiom,
    ! [Na: nat] : image_nat_nat(suc,set_ord_atMost_nat(Na)) = set_or1269000886237332187st_nat(one_one_nat,aa_nat_nat(suc,Na)) ).

% image_Suc_atMost
tff(fact_9498_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Na)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_or1269000886237332187st_nat(zero_zero_nat,Na))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9499_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,Na)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_or4665077453230672383an_nat(zero_zero_nat,Na))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9500_lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_ord_lessThan_nat(aa_nat_nat(suc,Na)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_ord_lessThan_nat(Na))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9501_atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_ord_atMost_nat(aa_nat_nat(suc,Na)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_ord_atMost_nat(Na))) ).

% atMost_Suc_eq_insert_0
tff(fact_9502_Fract__less__zero__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B)),zero_zero_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% Fract_less_zero_iff
tff(fact_9503_zero__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(fract(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ).

% zero_less_Fract_iff
tff(fact_9504_one__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(fract(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% one_less_Fract_iff
tff(fact_9505_Fract__less__one__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B)),one_one_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% Fract_less_one_iff
tff(fact_9506_zero__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(fract(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_Fract_iff
tff(fact_9507_Fract__le__zero__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(A),B)),zero_zero_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% Fract_le_zero_iff
tff(fact_9508_Fract__le__one__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(A),B)),one_one_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% Fract_le_one_iff
tff(fact_9509_one__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(fract(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% one_le_Fract_iff
tff(fact_9510_Gcd__abs__eq,axiom,
    ! [K5: set_int] : gcd_Gcd_int(image_int_int(abs_abs_int,K5)) = gcd_Gcd_int(K5) ).

% Gcd_abs_eq
tff(fact_9511_Gcd__int__eq,axiom,
    ! [N4: set_nat] : gcd_Gcd_int(image_nat_int(semiri1314217659103216013at_int,N4)) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,N4)) ).

% Gcd_int_eq
tff(fact_9512_Gcd__nat__abs__eq,axiom,
    ! [K5: set_int] : aa_set_nat_nat(gcd_Gcd_nat,image_int_nat(aTP_Lamm_ym,K5)) = aa_int_nat(nat2,gcd_Gcd_int(K5)) ).

% Gcd_nat_abs_eq
tff(fact_9513_finite__int__iff__bounded,axiom,
    ! [S3: set_int] :
      ( aa_set_int_o(finite_finite_int,S3)
    <=> ? [K3: int] : aa_set_int_o(ord_less_eq_set_int(image_int_int(abs_abs_int,S3)),set_ord_lessThan_int(K3)) ) ).

% finite_int_iff_bounded
tff(fact_9514_finite__int__iff__bounded__le,axiom,
    ! [S3: set_int] :
      ( aa_set_int_o(finite_finite_int,S3)
    <=> ? [K3: int] : aa_set_int_o(ord_less_eq_set_int(image_int_int(abs_abs_int,S3)),set_ord_atMost_int(K3)) ) ).

% finite_int_iff_bounded_le
tff(fact_9515_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S3: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S3)
    <=> ~ aa_set_nat_o(finite_finite_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),S3)) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_9516_Gcd__int__def,axiom,
    ! [K5: set_int] : gcd_Gcd_int(K5) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),K5))) ).

% Gcd_int_def
tff(fact_9517_range__mult,axiom,
    ! [A: real] :
      image_real_real(aa_rea7340605660300375871l_real(times_times_real,A),top_top_set_real) = $ite(A = zero_zero_real,insert_real(zero_zero_real,bot_bot_set_real),top_top_set_real) ).

% range_mult
tff(fact_9518_range__abs__Nats,axiom,
    image_int_int(abs_abs_int,top_top_set_int) = semiring_1_Nats_int ).

% range_abs_Nats
tff(fact_9519_infinite__UNIV__int,axiom,
    ~ aa_set_int_o(finite_finite_int,top_top_set_int) ).

% infinite_UNIV_int
tff(fact_9520_nat__not__finite,axiom,
    ~ aa_set_nat_o(finite_finite_nat,top_top_set_nat) ).

% nat_not_finite
tff(fact_9521_infinite__UNIV__nat,axiom,
    ~ aa_set_nat_o(finite_finite_nat,top_top_set_nat) ).

% infinite_UNIV_nat
tff(fact_9522_int__in__range__abs,axiom,
    ! [Na: nat] : aa_set_int_o(member_int(aa_nat_int(semiri1314217659103216013at_int,Na)),image_int_int(abs_abs_int,top_top_set_int)) ).

% int_in_range_abs
tff(fact_9523_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_9524_range__mod,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yn,Na),top_top_set_nat) = set_or4665077453230672383an_nat(zero_zero_nat,Na) ) ) ).

% range_mod
tff(fact_9525_root__def,axiom,
    ! [Na: nat,X: real] :
      aa_real_real(aa_nat_fun_real_real(root,Na),X) = $ite(Na = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_yo,Na)),X)) ).

% root_def
tff(fact_9526_DERIV__even__real__root,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_9527_DERIV__real__root__generic,axiom,
    ! [Na: nat,X: real,D5: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( X != zero_zero_real )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
             => ( D5 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
               => ( D5 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
               => ( D5 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),D5,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9528_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3))),aa_real_real(F,X)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_9529_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_9530_DERIV__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3))),aa_real_real(F,X)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_9531_DERIV__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_9532_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X3: real] :
            ( 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)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_9533_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X3: real] :
            ( 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)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_9534_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)),S3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D4)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_9535_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)),S3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D4)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3))),aa_real_real(F,X)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_9536_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,X: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)),S3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D4)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_9537_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,X: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)),S3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D4)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3))),aa_real_real(F,X)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_9538_MVT2,axiom,
    ! [A: real,B: real,F: fun_real_real,F5: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X3: real] :
            ( 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)
             => has_fi5821293074295781190e_real(F,aa_real_real(F5,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
       => ? [Z2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
            & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),aa_real_real(F5,Z2)) ) ) ) ) ).

% MVT2
tff(fact_9539_DERIV__local__const,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y3))),D)
             => ( aa_real_real(F,X) = aa_real_real(F,Y3) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_9540_DERIV__ln,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(inverse_inverse_real,X),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_ln
tff(fact_9541_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y3))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Y3)),aa_real_real(F,X)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_9542_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y3))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X)),aa_real_real(F,Y3)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_9543_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_9544_DERIV__pow,axiom,
    ! [Na: nat,X: real,S: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_yp,Na),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(X,S)) ).

% DERIV_pow
tff(fact_9545_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_yq,R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Z),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_9546_DERIV__log,axiom,
    ! [X: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(ln_ln_real,B)),X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_log
tff(fact_9547_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,Mb: real,X: real,R2: real] :
      ( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,X))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_yr(G),R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),Mb),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_9548_DERIV__powr,axiom,
    ! [G: fun_real_real,Mb: real,X: real,F: fun_real_real,R2: real] :
      ( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,X))
       => ( has_fi5821293074295781190e_real(F,R2,topolo2177554685111907308n_real(X,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_ys(G),F),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,X)),aa_real_real(F,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(ln_ln_real,aa_real_real(G,X)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Mb),aa_real_real(F,X))),aa_real_real(G,X)))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_9549_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => has_fi5821293074295781190e_real(arcosh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X,A2)) ) ).

% arcosh_real_has_field_derivative
tff(fact_9550_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_9551_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D5: real] :
      ( ( X != zero_zero_real )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( D5 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
           => ( D5 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D5,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_9552_artanh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => has_fi5821293074295781190e_real(artanh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X,A2)) ) ).

% artanh_real_has_field_derivative
tff(fact_9553_DERIV__real__root,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_9554_DERIV__arcsin,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => has_fi5821293074295781190e_real(arcsin,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_9555_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,Na: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M: nat,X3: 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)),X3),topolo2177554685111907308n_real(X3,top_top_set_real)) )
     => ? [T6: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X))
          & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_yt(Diff),X),set_ord_lessThan_nat(Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Na),T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_9556_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,Na: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M: nat,X3: 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)),X3),topolo2177554685111907308n_real(X3,top_top_set_real))
       => ? [T6: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X))
            & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_yt(Diff),X),set_ord_lessThan_nat(Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Na),T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_9557_DERIV__odd__real__root,axiom,
    ! [Na: nat,X: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( ( X != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_9558_Maclaurin__minus,axiom,
    ! [H: real,Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),zero_zero_real)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M: nat,T6: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),T6)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),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)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
           => ? [T6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),T6)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T6),zero_zero_real)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_tt(H),Diff),set_ord_lessThan_nat(Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Na),T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Na))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_9559_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T6: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T6)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),H) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
         => ? [T6: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T6)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),H)
              & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_tt(H),Diff),set_ord_lessThan_nat(Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Na),T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Na))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_9560_Maclaurin,axiom,
    ! [H: real,Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M: nat,T6: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T6)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),H) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
           => ? [T6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T6)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T6),H)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_tt(H),Diff),set_ord_lessThan_nat(Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Na),T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Na))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_9561_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( X != zero_zero_real )
         => ( ! [M: nat,X3: 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)),X3),topolo2177554685111907308n_real(X3,top_top_set_real))
           => ? [T6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T6))
                & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X))
                & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_yt(Diff),X),set_ord_lessThan_nat(Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Na),T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_9562_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M: nat,T6: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
       => ? [T6: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X))
            & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_yt(Diff),X),set_ord_lessThan_nat(Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Na),T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_9563_Taylor,axiom,
    ! [Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T6: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T6)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),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)),T6),topolo2177554685111907308n_real(T6,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),X)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
                 => ( ( X != C )
                   => ? [T6: real] :
                        ( $ite(
                            aa_real_o(aa_real_fun_real_o(ord_less_real,X),C),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),T6)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T6),C) ),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T6)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T6),X) ) )
                        & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_yu(Diff),C),X),set_ord_lessThan_nat(Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Na),T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),C)),Na))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_9564_Taylor__up,axiom,
    ! [Na: 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),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T6: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T6)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),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)),T6),topolo2177554685111907308n_real(T6,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)
             => ? [T6: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T6)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T6),B)
                  & ( aa_real_real(F,B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_yv(Diff),B),C),set_ord_lessThan_nat(Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Na),T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),Na))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_9565_Taylor__down,axiom,
    ! [Na: 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),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T6: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T6)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),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)),T6),topolo2177554685111907308n_real(T6,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)
             => ? [T6: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),T6)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T6),C)
                  & ( aa_real_real(F,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_yv(Diff),A),C),set_ord_lessThan_nat(Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Na),T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),Na))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_9566_Maclaurin__lemma2,axiom,
    ! [Na: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B2: real] :
      ( ! [M: nat,T6: real] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T6)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),H) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
     => ( ( Na = aa_nat_nat(suc,K) )
       => ! [M4: nat,T7: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T7)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T7),H) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_yx(Na),Diff),B2),M4),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T7)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_yy(Diff),M4),T7),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,M4))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,T7),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,M4)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,M4))))))),topolo2177554685111907308n_real(T7,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_9567_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => has_fi5821293074295781190e_real(aTP_Lamm_yz,suminf_real(aa_real_fun_nat_real(aTP_Lamm_za,X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_9568_DERIV__arccos,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => has_fi5821293074295781190e_real(arccos,aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_9569_DERIV__power__series_H,axiom,
    ! [R: real,F: fun_nat_real,X0: real] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_zb(F),X3)) )
     => ( aa_set_real_o(member_real(X0),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
         => has_fi5821293074295781190e_real(aTP_Lamm_zd(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_zb(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_9570_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),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_real,aa_real_real(arccos,Y)),aa_real_real(arccos,X)) ) ) ) ).

% arccos_less_arccos
tff(fact_9571_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_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_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ) ) ) ).

% arccos_less_mono
tff(fact_9572_DERIV__isconst3,axiom,
    ! [A: real,B: real,X: real,Y: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_set_real_o(member_real(X),set_or1633881224788618240n_real(A,B))
       => ( aa_set_real_o(member_real(Y),set_or1633881224788618240n_real(A,B))
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),set_or1633881224788618240n_real(A,B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real)) )
           => ( aa_real_real(F,X) = aa_real_real(F,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_9573_arccos__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,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_9574_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,X)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_9575_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,set_or5834768355832116004an_nat(L,U)) ).

% finite_greaterThanLessThan
tff(fact_9576_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : aa_set_int_o(finite_finite_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or5832277885323065728an_int,L),U)) ).

% finite_greaterThanLessThan_int
tff(fact_9577_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_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_9578_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : set_or4665077453230672383an_nat(aa_nat_nat(suc,L),U) = set_or5834768355832116004an_nat(L,U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_9579_LIM__fun__less__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [R4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
            & ! [X2: real] :
                ( ( ( X2 != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X2))),R4) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),zero_zero_real) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_9580_LIM__fun__not__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( ( L != zero_zero_real )
       => ? [R4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
            & ! [X2: real] :
                ( ( ( X2 != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X2))),R4) )
               => ( aa_real_real(F,X2) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_9581_LIM__fun__gt__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [R4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
            & ! [X2: real] :
                ( ( ( X2 != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X2))),R4) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,X2)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_9582_isCont__inverse__function2,axiom,
    ! [A: real,X: real,B: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),B)
       => ( ! [Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B)
               => ( aa_real_real(G,aa_real_real(F,Z2)) = Z2 ) ) )
         => ( ! [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B)
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_9583_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Na: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I,J))),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Na)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9584_isCont__arcosh,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arcosh_real) ) ).

% isCont_arcosh
tff(fact_9585_DERIV__inverse__function,axiom,
    ! [F: fun_real_real,D5: real,G: fun_real_real,X: real,A: real,B: real] :
      ( has_fi5821293074295781190e_real(F,D5,topolo2177554685111907308n_real(aa_real_real(G,X),top_top_set_real))
     => ( ( D5 != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),B)
           => ( ! [Y3: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),B)
                   => ( aa_real_real(F,aa_real_real(G,Y3)) = Y3 ) ) )
             => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),G)
               => has_fi5821293074295781190e_real(G,aa_real_real(inverse_inverse_real,D5),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_9586_isCont__arccos,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arccos) ) ) ).

% isCont_arccos
tff(fact_9587_isCont__arcsin,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arcsin) ) ) ).

% isCont_arcsin
tff(fact_9588_LIM__less__bound,axiom,
    ! [B: real,X: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),X)
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),set_or1633881224788618240n_real(B,X))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3)) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X)) ) ) ) ).

% LIM_less_bound
tff(fact_9589_isCont__artanh,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),artanh_real) ) ) ).

% isCont_artanh
tff(fact_9590_isCont__inverse__function,axiom,
    ! [D: real,X: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
     => ( ! [Z2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z2),X))),D)
           => ( aa_real_real(G,aa_real_real(F,Z2)) = Z2 ) )
       => ( ! [Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z2),X))),D)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_9591_GMVT_H,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [Z2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) ) )
       => ( ! [Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B)
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),G) ) )
         => ( ! [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z2),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) )
           => ( ! [Z2: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
                   => has_fi5821293074295781190e_real(F,aa_real_real(F5,Z2),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) )
             => ? [C2: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C2),B)
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(G2,C2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B)),aa_real_real(G,A))),aa_real_real(F5,C2)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_9592_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))
         => ! [N5: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_ze(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_ze(A),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))),N5))),groups6591440286371151544t_real(aTP_Lamm_ze(A),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))),N5)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_9593_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)
         => ! [N5: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_ze(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_ze(A),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))),N5)),one_one_nat))),groups6591440286371151544t_real(aTP_Lamm_ze(A),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))),N5))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_9594_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_zf(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ze(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_9595_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
tff(fact_9596_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_9597_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => filterlim_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),at_top_nat,at_top_nat) ) ).

% mult_nat_left_at_top
tff(fact_9598_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zg,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_9599_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_zh(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L3: real] :
                ( ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N5)),L3)
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L3),at_top_nat)
                & ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),aa_nat_real(G,N5))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_9600_LIMSEQ__inverse__zero,axiom,
    ! [X5: fun_nat_real] :
      ( ! [R4: real] :
        ? [N6: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,R4),aa_nat_real(X5,N)) )
     => filterlim_nat_real(aTP_Lamm_zi(X5),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_9601_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zj,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_9602_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_zk,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_9603_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zl,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_9604_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)
       => ( ! [E: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
             => ? [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,N5)),E)) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_9605_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_9606_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zm(X),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_9607_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(aa_real_fun_nat_real(power_power_real,C),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_9608_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(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,C)),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_9609_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zn,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_9610_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zo,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_9611_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zp,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_9612_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_ze(A)) ) ) ) ).

% summable
tff(fact_9613_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_pz,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_9614_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_zq(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ze(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_9615_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,Na: 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_ze(A),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),suminf_real(aTP_Lamm_ze(A))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_9616_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] :
              ( ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_ze(A),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))),N5)))),L3)
              & filterlim_nat_real(aTP_Lamm_zq(A),topolo2815343760600316023s_real(L3),at_top_nat)
              & ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),groups6591440286371151544t_real(aTP_Lamm_ze(A),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))),N5)),one_one_nat))))
              & filterlim_nat_real(aTP_Lamm_zf(A),topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_9617_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_zf(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ze(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_9618_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,Na: 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_ze(A))),groups6591440286371151544t_real(aTP_Lamm_ze(A),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),one_one_nat)))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_9619_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_zr(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_9620_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_zs(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_9621_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X3)
         => aa_nat_o(P,X3) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_9622_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_9623_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ord_le2510731241096832064er_nat(F3,at_top_nat)
    <=> ! [N8: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N8),F3) ) ).

% le_sequentially
tff(fact_9624_sequentially__offset,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_zs(P),K),at_top_nat) ) ).

% sequentially_offset
tff(fact_9625_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),X3)
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X3,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),zero_zero_real) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_top_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_9626_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,set_or6659071591806873216st_nat(L,U)) ).

% finite_greaterThanAtMost
tff(fact_9627_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : set_or1269000886237332187st_nat(aa_nat_nat(suc,L),U) = set_or6659071591806873216st_nat(L,U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_9628_eventually__at__left__real,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_zt(B),A),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ).

% eventually_at_left_real
tff(fact_9629_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Na: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I,J))),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Na)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9630_filterlim__pow__at__bot__even,axiom,
    ! [Na: nat,F: fun_real_real,F3: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zu(Na),F),at_top_real,F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_9631_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : aa_set_int_o(finite_finite_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or6656581121297822940st_int,L),U)) ).

% finite_greaterThanAtMost_int
tff(fact_9632_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B)
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X3,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_bot_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_9633_filterlim__pow__at__bot__odd,axiom,
    ! [Na: nat,F: fun_real_real,F3: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zu(Na),F),at_bot_real,F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_9634_GMVT,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X3: real] :
            ( ( 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) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),F) )
       => ( ! [X3: real] :
              ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X3,top_top_set_real)) )
         => ( ! [X3: real] :
                ( ( 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) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),G) )
           => ( ! [X3: real] :
                  ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
                    & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X3,top_top_set_real)) )
             => ? [G_c: real,F_c: real,C2: real] :
                  ( has_fi5821293074295781190e_real(G,G_c,topolo2177554685111907308n_real(C2,top_top_set_real))
                  & has_fi5821293074295781190e_real(F,F_c,topolo2177554685111907308n_real(C2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C2),B)
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),G_c) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B)),aa_real_real(G,A))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_9635_MVT,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
         => ? [L3: real,Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
              & has_fi5821293074295781190e_real(F,L3,topolo2177554685111907308n_real(Z2,top_top_set_real))
              & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),L3) ) ) ) ) ) ).

% MVT
tff(fact_9636_Rolle__deriv,axiom,
    ! [A: real,B: real,F: fun_real_real,F5: fun_re7681423839302404990l_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_real(F,A) = aa_real_real(F,B) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => ( ! [X3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
                 => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F5,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
           => ? [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
                & ! [X2: real] : aa_real_real(aa_rea7340605660300375871l_real(F5,Z2),X2) = zero_zero_real ) ) ) ) ) ).

% Rolle_deriv
tff(fact_9637_mvt,axiom,
    ! [A: real,B: real,F: fun_real_real,F5: fun_re7681423839302404990l_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
               => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F5,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
         => ~ ! [Xi: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xi)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xi),B)
                 => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) != aa_real_real(aa_rea7340605660300375871l_real(F5,Xi),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ) ) ) ) ) ) ).

% mvt
tff(fact_9638_DERIV__pos__imp__increasing__open,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_9639_DERIV__neg__imp__decreasing__open,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),zero_zero_real) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_9640_DERIV__isconst__end,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
         => ( aa_real_real(F,B) = aa_real_real(F,A) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_9641_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: fun_real_real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
             => ( aa_real_real(F,X) = aa_real_real(F,A) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_9642_Rolle,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_real(F,A) = aa_real_real(F,B) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => ( ! [X3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
                 => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
           => ? [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
                & has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(Z2,top_top_set_real)) ) ) ) ) ) ).

% Rolle
tff(fact_9643_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_9644_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_9645_eventually__at__right__real,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_zt(A),B),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ).

% eventually_at_right_real
tff(fact_9646_greaterThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_or1210151606488870762an_nat,K),insert_nat(aa_nat_nat(suc,K),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_9647_atLeast__0,axiom,
    set_ord_atLeast_nat(zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_9648_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] : set_ord_atLeast_nat(aa_nat_nat(suc,K)) = aa_nat_set_nat(set_or1210151606488870762an_nat,K) ).

% atLeast_Suc_greaterThan
tff(fact_9649_atLeast__Suc,axiom,
    ! [K: nat] : set_ord_atLeast_nat(aa_nat_nat(suc,K)) = minus_minus_set_nat(set_ord_atLeast_nat(K),insert_nat(K,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_9650_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_zv,M6))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_9651_Max__divisors__self__nat,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_df,Na))) = Na ) ) ).

% Max_divisors_self_nat
tff(fact_9652_bdd__above__nat,axiom,
    ! [X5: set_nat] :
      ( aa_set_nat_o(condit2214826472909112428ve_nat,X5)
    <=> aa_set_nat_o(finite_finite_nat,X5) ) ).

% bdd_above_nat
tff(fact_9653_card__le__Suc__Max,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,S3)),aa_nat_nat(suc,aa_set_nat_nat(lattic8265883725875713057ax_nat,S3))) ) ).

% card_le_Suc_Max
tff(fact_9654_Sup__nat__def,axiom,
    ! [X5: set_nat] :
      aa_set_nat_nat(complete_Sup_Sup_nat,X5) = $ite(X5 = bot_bot_set_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,X5)) ).

% Sup_nat_def
tff(fact_9655_divide__nat__def,axiom,
    ! [Mb: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Na) = $ite(Na = zero_zero_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_zw(Mb),Na)))) ).

% divide_nat_def
tff(fact_9656_gcd__is__Max__divisors__nat,axiom,
    ! [Na: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Na) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_zx(Na),Mb))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_9657_Max__divisors__self__int,axiom,
    ! [Na: int] :
      ( ( Na != zero_zero_int )
     => ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_dg,Na))) = aa_int_int(abs_abs_int,Na) ) ) ).

% Max_divisors_self_int
tff(fact_9658_gcd__is__Max__divisors__int,axiom,
    ! [Na: int,Mb: int] :
      ( ( Na != zero_zero_int )
     => ( aa_int_int(gcd_gcd_int(Mb),Na) = lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_zy(Na),Mb))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_9659_uniformity__complex__def,axiom,
    topolo896644834953643431omplex = comple8358262395181532106omplex(image_5971271580939081552omplex(aTP_Lamm_aaa,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_complex_def
tff(fact_9660_uniformity__real__def,axiom,
    topolo1511823702728130853y_real = comple2936214249959783750l_real(image_2178119161166701260l_real(aTP_Lamm_aac,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_real_def
tff(fact_9661_eventually__prod__sequentially,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] :
      ( eventu1038000079068216329at_nat(P,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N8: nat] :
        ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),M5)
         => ! [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),M5)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_9662_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_9663_mono__times__nat,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na)) ) ).

% mono_times_nat
tff(fact_9664_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aad,K)) ) ).

% mono_ge2_power_minus_self
tff(fact_9665_less__eq,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),Na)),transi6264000038957366511cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Na) ) ).

% less_eq
tff(fact_9666_inj__sgn__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_set_real_o(inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_yo,Na)),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_9667_log__inj,axiom,
    ! [B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => aa_set_real_o(inj_on_real_real(aa_rea7340605660300375871l_real(log,B)),set_or5849166863359141190n_real(zero_zero_real)) ) ).

% log_inj
tff(fact_9668_inj__Suc,axiom,
    ! [N4: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N4) ).

% inj_Suc
tff(fact_9669_inj__on__diff__nat,axiom,
    ! [N4: set_nat,K: nat] :
      ( ! [N: nat] :
          ( aa_set_nat_o(member_nat(N),N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N) )
     => aa_set_nat_o(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yl,K)),N4) ) ).

% inj_on_diff_nat
tff(fact_9670_inj__on__set__encode,axiom,
    aa_set_set_nat_o(inj_on_set_nat_nat(nat_set_encode),collect_set_nat(finite_finite_nat)) ).

% inj_on_set_encode
tff(fact_9671_pred__nat__trancl__eq__le,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),Na)),transi2905341329935302413cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na) ) ).

% pred_nat_trancl_eq_le
tff(fact_9672_measure__function__int,axiom,
    fun_is_measure_int(comp_int_nat_int(nat2,abs_abs_int)) ).

% measure_function_int
tff(fact_9673_powr__real__of__int_H,axiom,
    ! [X: real,Na: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( ( ( X != zero_zero_real )
          | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,Na)) = aa_int_real(aa_real_fun_int_real(power_int_real,X),Na) ) ) ) ).

% powr_real_of_int'
tff(fact_9674_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( set_or4665077453230672383an_nat(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = sup_sup_set_nat(set_or4665077453230672383an_nat(I,J),set_or4665077453230672383an_nat(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_9675_min__weak__def,axiom,
    fun_min_weak = 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_9676_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
tff(fact_9677_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_9678_max__weak__def,axiom,
    fun_max_weak = 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_9679_pos__deriv__imp__strict__mono,axiom,
    ! [F: fun_real_real,F5: fun_real_real] :
      ( ! [X3: real] : has_fi5821293074295781190e_real(F,aa_real_real(F5,X3),topolo2177554685111907308n_real(X3,top_top_set_real))
     => ( ! [X3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F5,X3))
       => order_7092887310737990675l_real(F) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_9680_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
tff(fact_9681_infinite__enumerate,axiom,
    ! [S3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
     => ? [R4: fun_nat_nat] :
          ( order_5726023648592871131at_nat(R4)
          & ! [N5: nat] : aa_set_nat_o(member_nat(aa_nat_nat(R4,N5)),S3) ) ) ).

% infinite_enumerate
tff(fact_9682_strict__mono__imp__increasing,axiom,
    ! [F: fun_nat_nat,Na: nat] :
      ( order_5726023648592871131at_nat(F)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(F,Na)) ) ).

% strict_mono_imp_increasing
tff(fact_9683_less__int__def,axiom,
    ord_less_int = map_fu3191548434056052140_int_o(rep_Integ,map_fu4826362097070443709at_o_o(rep_Integ,id_o),produc6395135316500328296_nat_o(aTP_Lamm_xs)) ).

% less_int_def
tff(fact_9684_nat__def,axiom,
    nat2 = map_fu2345160673673942751at_nat(rep_Integ,id_nat,produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat_def
tff(fact_9685_less__eq__int__def,axiom,
    ord_less_eq_int = map_fu3191548434056052140_int_o(rep_Integ,map_fu4826362097070443709at_o_o(rep_Integ,id_o),produc6395135316500328296_nat_o(aTP_Lamm_xu)) ).

% less_eq_int_def
tff(fact_9686_positive__rat,axiom,
    ! [A: int,B: int] :
      ( aa_rat_o(positive,aa_int_rat(fract(A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ).

% positive_rat
tff(fact_9687_less__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
    <=> aa_rat_o(positive,aa_rat_rat(minus_minus_rat(Y),X)) ) ).

% less_rat_def
tff(fact_9688_Rat_Opositive_Orep__eq,axiom,
    ! [X: rat] :
      ( aa_rat_o(positive,X)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(aa_rat7754553326175492643nt_int(rep_Rat,X))),product_snd_int_int(aa_rat7754553326175492643nt_int(rep_Rat,X)))) ) ).

% Rat.positive.rep_eq
tff(fact_9689_Rat_Opositive__def,axiom,
    positive = map_fu898904425404107465nt_o_o(rep_Rat,id_o,aTP_Lamm_aae) ).

% Rat.positive_def
tff(fact_9690_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,I)),J)
     => ( linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I,J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9691_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),J)
     => ( linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I,J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9692_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(set_ord_lessThan_nat(aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(set_ord_lessThan_nat(K)),cons_nat(K,nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_9693_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(set_ord_atMost_nat(aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(set_ord_atMost_nat(K)),cons_nat(aa_nat_nat(suc,K),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_9694_list__encode_Oelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( nat_list_encode(X) = Y )
     => ( ( ( X = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X3: nat,Xs2: list_nat] :
              ( ( X = cons_nat(X3,Xs2) )
             => ( Y != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),nat_list_encode(Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_9695_list__encode_Osimps_I1_J,axiom,
    nat_list_encode(nil_nat) = zero_zero_nat ).

% list_encode.simps(1)
tff(fact_9696_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] : nat_list_encode(cons_nat(X,Xs)) = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),nat_list_encode(Xs)))) ).

% list_encode.simps(2)
tff(fact_9697_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list_int] :
      upto_aux(I,J,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),Js,upto_aux(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js))) ).

% upto_aux_rec
tff(fact_9698_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_9699_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( upto(X,Xa2) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa2))
       => ~ ( ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa2),cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),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,X),Xa2)) ) ) ) ).

% upto.pelims
tff(fact_9700_upto__empty,axiom,
    ! [J: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,J),I)
     => ( upto(I,J) = nil_int ) ) ).

% upto_empty
tff(fact_9701_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int = upto(I,J) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil2
tff(fact_9702_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil
tff(fact_9703_upto__single,axiom,
    ! [I: int] : upto(I,I) = cons_int(I,nil_int) ).

% upto_single
tff(fact_9704_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K))),J)
     => ( aa_nat_int(nth_int(upto(I,J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_9705_length__upto,axiom,
    ! [I: int,J: int] : aa_list_int_nat(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_9706_upto__rec__numeral_I1_J,axiom,
    ! [Mb: num,Na: num] :
      upto(aa_num_int(numeral_numeral_int,Mb),aa_num_int(numeral_numeral_int,Na)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Na)),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,Na))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_9707_upto__rec__numeral_I2_J,axiom,
    ! [Mb: num,Na: num] :
      upto(aa_num_int(numeral_numeral_int,Mb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))),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,Na)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_9708_upto__rec__numeral_I3_J,axiom,
    ! [Mb: num,Na: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Na)) = $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,Na)),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,Na))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_9709_upto__rec__numeral_I4_J,axiom,
    ! [Mb: num,Na: 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,Na))) = $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,Na))),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,Na)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_9710_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,I),J) = set_int2(upto(I,J)) ).

% atLeastAtMost_upto
tff(fact_9711_distinct__upto,axiom,
    ! [I: int,J: int] : aa_list_int_o(distinct_int,upto(I,J)) ).

% distinct_upto
tff(fact_9712_upto__aux__def,axiom,
    ! [I: int,J: int,Js: list_int] : upto_aux(I,J,Js) = append_int(upto(I,J),Js) ).

% upto_aux_def
tff(fact_9713_upto__code,axiom,
    ! [I: int,J: int] : upto(I,J) = upto_aux(I,J,nil_int) ).

% upto_code
tff(fact_9714_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_9715_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_9716_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,I),J) = set_int2(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_9717_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or6656581121297822940st_int,I),J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_9718_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_9719_upto_Osimps,axiom,
    ! [I: int,J: int] :
      upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ).

% upto.simps
tff(fact_9720_upto_Oelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( upto(X,Xa2) = Y )
     => ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa2),cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa2)),nil_int) ) ) ).

% upto.elims
tff(fact_9721_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( upto(I,J) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_9722_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or5832277885323065728an_int,I),J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_9723_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K))) ) ) ) ).

% upto_split3
tff(fact_9724_list__encode_Opelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( nat_list_encode(X) = Y )
     => ( aa_list_nat_o(accp_list_nat(nat_list_encode_rel),X)
       => ( ( ( X = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),nil_nat) ) )
         => ~ ! [X3: nat,Xs2: list_nat] :
                ( ( X = cons_nat(X3,Xs2) )
               => ( ( Y = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),nat_list_encode(Xs2)))) )
                 => ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),cons_nat(X3,Xs2)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_9725_remdups__upt,axiom,
    ! [Mb: nat,Na: nat] : remdups_nat(upt(Mb,Na)) = upt(Mb,Na) ).

% remdups_upt
tff(fact_9726_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( hd_nat(upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_9727_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_9728_length__upt,axiom,
    ! [I: nat,J: nat] : aa_list_nat_nat(size_size_list_nat,upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) ).

% length_upt
tff(fact_9729_take__upt,axiom,
    ! [I: nat,Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb)),Na)
     => ( take_nat(Mb,upt(I,Na)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb)) ) ) ).

% take_upt
tff(fact_9730_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_9731_sorted__list__of__set__range,axiom,
    ! [Mb: nat,Na: nat] : linord2614967742042102400et_nat(set_or4665077453230672383an_nat(Mb,Na)) = upt(Mb,Na) ).

% sorted_list_of_set_range
tff(fact_9732_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_9733_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)
     => ( aa_nat_nat(nth_nat(upt(I,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).

% nth_upt
tff(fact_9734_upt__rec__numeral,axiom,
    ! [Mb: num,Na: num] :
      upt(aa_num_nat(numeral_numeral_nat,Mb),aa_num_nat(numeral_numeral_nat,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Na)),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,Na))),nil_nat) ).

% upt_rec_numeral
tff(fact_9735_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_9736_upt__conv__Cons__Cons,axiom,
    ! [Mb: nat,Na: nat,Ns: list_nat,Q4: nat] :
      ( ( cons_nat(Mb,cons_nat(Na,Ns)) = upt(Mb,Q4) )
    <=> ( cons_nat(Na,Ns) = upt(aa_nat_nat(suc,Mb),Q4) ) ) ).

% upt_conv_Cons_Cons
tff(fact_9737_map__add__upt,axiom,
    ! [Na: nat,Mb: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aaf,Na),upt(zero_zero_nat,Mb)) = upt(Na,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Na)) ).

% map_add_upt
tff(fact_9738_map__Suc__upt,axiom,
    ! [Mb: nat,Na: nat] : map_nat_nat(suc,upt(Mb,Na)) = upt(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Na)) ).

% map_Suc_upt
tff(fact_9739_atMost__upto,axiom,
    ! [Na: nat] : set_ord_atMost_nat(Na) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,Na))) ).

% atMost_upto
tff(fact_9740_greaterThanAtMost__upt,axiom,
    ! [Na: nat,Mb: nat] : set_or6659071591806873216st_nat(Na,Mb) = set_nat2(upt(aa_nat_nat(suc,Na),aa_nat_nat(suc,Mb))) ).

% greaterThanAtMost_upt
tff(fact_9741_atLeast__upt,axiom,
    ! [Na: nat] : set_ord_lessThan_nat(Na) = set_nat2(upt(zero_zero_nat,Na)) ).

% atLeast_upt
tff(fact_9742_atLeastAtMost__upt,axiom,
    ! [Na: nat,Mb: nat] : set_or1269000886237332187st_nat(Na,Mb) = set_nat2(upt(Na,aa_nat_nat(suc,Mb))) ).

% atLeastAtMost_upt
tff(fact_9743_atLeastLessThan__upt,axiom,
    ! [I: nat,J: nat] : set_or4665077453230672383an_nat(I,J) = set_nat2(upt(I,J)) ).

% atLeastLessThan_upt
tff(fact_9744_greaterThanLessThan__upt,axiom,
    ! [Na: nat,Mb: nat] : set_or5834768355832116004an_nat(Na,Mb) = set_nat2(upt(aa_nat_nat(suc,Na),Mb)) ).

% greaterThanLessThan_upt
tff(fact_9745_distinct__upt,axiom,
    ! [I: nat,J: nat] : aa_list_nat_o(distinct_nat,upt(I,J)) ).

% distinct_upt
tff(fact_9746_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_9747_map__decr__upt,axiom,
    ! [Mb: nat,Na: nat] : map_nat_nat(aTP_Lamm_aag,upt(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Na))) = upt(Mb,Na) ).

% map_decr_upt
tff(fact_9748_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_9749_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(X,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( I = X )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_9750_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_9751_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_9752_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_9753_sum__list__upt,axiom,
    ! [Mb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Na)
     => ( groups4561878855575611511st_nat(upt(Mb,Na)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_dr),set_or4665077453230672383an_nat(Mb,Na)) ) ) ).

% sum_list_upt
tff(fact_9754_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_aah(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_aai(Mb),N4)))),finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aaj(Mb),N4)))) ) ) ).

% card_length_sum_list_rec
tff(fact_9755_card__length__sum__list,axiom,
    ! [Mb: nat,N4: nat] : finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aah(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_9756_tl__upt,axiom,
    ! [Mb: nat,Na: nat] : tl_nat(upt(Mb,Na)) = upt(aa_nat_nat(suc,Mb),Na) ).

% tl_upt
tff(fact_9757_sorted__upt,axiom,
    ! [Mb: nat,Na: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),upt(Mb,Na)) ).

% sorted_upt
tff(fact_9758_sorted__wrt__upt,axiom,
    ! [Mb: nat,Na: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_nat),upt(Mb,Na)) ).

% sorted_wrt_upt
tff(fact_9759_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),aa_list_nat_nat(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_9760_sorted__upto,axiom,
    ! [Mb: int,Na: int] : aa_list_int_o(sorted_wrt_int(ord_less_eq_int),upto(Mb,Na)) ).

% sorted_upto
tff(fact_9761_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : aa_list_int_o(sorted_wrt_int(ord_less_int),upto(I,J)) ).

% sorted_wrt_upto
tff(fact_9762_pairs__le__eq__Sigma,axiom,
    ! [Mb: nat] : collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aak(Mb))) = produc457027306803732586at_nat(set_ord_atMost_nat(Mb),aTP_Lamm_aal(Mb)) ).

% pairs_le_eq_Sigma
tff(fact_9763_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_nat)) ).

% natLess_def
tff(fact_9764_vimage__Suc__insert__Suc,axiom,
    ! [Na: nat,A2: set_nat] : vimage_nat_nat(suc,insert_nat(aa_nat_nat(suc,Na),A2)) = insert_nat(Na,vimage_nat_nat(suc,A2)) ).

% vimage_Suc_insert_Suc
tff(fact_9765_vimage__Suc__insert__0,axiom,
    ! [A2: set_nat] : vimage_nat_nat(suc,insert_nat(zero_zero_nat,A2)) = vimage_nat_nat(suc,A2) ).

% vimage_Suc_insert_0
tff(fact_9766_finite__vimage__Suc__iff,axiom,
    ! [F3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,vimage_nat_nat(suc,F3))
    <=> aa_set_nat_o(finite_finite_nat,F3) ) ).

% finite_vimage_Suc_iff
tff(fact_9767_set__decode__div__2,axiom,
    ! [X: nat] : nat_set_decode(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = vimage_nat_nat(suc,nat_set_decode(X)) ).

% set_decode_div_2
tff(fact_9768_set__encode__vimage__Suc,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(nat_set_encode,vimage_nat_nat(suc,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(nat_set_encode,A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% set_encode_vimage_Suc
tff(fact_9769_Restr__natLeq,axiom,
    ! [Na: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_by,Na)),aTP_Lamm_aam(Na))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aan(Na))) ).

% Restr_natLeq
tff(fact_9770_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_eq_nat)) ).

% natLeq_def
tff(fact_9771_Restr__natLeq2,axiom,
    ! [Na: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq,Na),aTP_Lamm_aao(Na))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aan(Na))) ).

% Restr_natLeq2
tff(fact_9772_natLeq__underS__less,axiom,
    ! [Na: nat] : order_underS_nat(bNF_Ca8665028551170535155natLeq,Na) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_by,Na)) ).

% natLeq_underS_less
tff(fact_9773_Arg__bounded,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ).

% Arg_bounded
tff(fact_9774_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,aa_complex_real(arg,Z)) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ) ).

% Arg_correct
tff(fact_9775_Field__natLeq__on,axiom,
    ! [Na: nat] : field_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aan(Na)))) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_by,Na)) ).

% Field_natLeq_on
tff(fact_9776_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,X) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
         => ( aa_complex_real(arg,Z) = X ) ) ) ) ).

% cis_Arg_unique
tff(fact_9777_bij__betw__roots__unity,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_set_complex_o(bij_betw_nat_complex(aTP_Lamm_aap(Na),set_ord_lessThan_nat(Na)),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_he,Na))) ) ).

% bij_betw_roots_unity
tff(fact_9778_bij__betw__nth__root__unity,axiom,
    ! [C: complex,Na: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_set_complex_o(bij_be1856998921033663316omplex(times_times_complex(aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_nat_fun_real_real(root,Na),aa_complex_real(real_V1022390504157884413omplex,C)))),aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,Na))))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_he,Na))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_xa(C),Na))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9779_bij__betw__Suc,axiom,
    ! [M6: set_nat,N4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(bij_betw_nat_nat(suc),M6),N4)
    <=> ( image_nat_nat(suc,M6) = N4 ) ) ).

% bij_betw_Suc
tff(fact_9780_Arg__def,axiom,
    ! [Z: complex] :
      aa_complex_real(arg,Z) = $ite(Z = zero_zero_complex,zero_zero_real,fChoice_real(aa_com1951324360330529881real_o(aTP_Lamm_aaq,Z))) ).

% Arg_def
tff(fact_9781_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% not_negative_int_iff
tff(fact_9782_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% not_nonnegative_int_iff
tff(fact_9783_wf__int__ge__less__than2,axiom,
    ! [D: int] : wf_int(int_ge_less_than2(D)) ).

% wf_int_ge_less_than2
tff(fact_9784_wf__int__ge__less__than,axiom,
    ! [D: int] : wf_int(int_ge_less_than(D)) ).

% wf_int_ge_less_than
tff(fact_9785_wf__less,axiom,
    aa_set3702741264012733747_nat_o(wf_nat,collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_nat))) ).

% wf_less
tff(fact_9786_and__not__num__eq__None__iff,axiom,
    ! [Mb: num,Na: num] :
      ( ( bit_and_not_num(Mb,Na) = none_num )
    <=> ( aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Na))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_9787_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one,one) = none_num ).

% and_not_num.simps(1)
tff(fact_9788_and__not__num_Osimps_I3_J,axiom,
    ! [Na: num] : bit_and_not_num(one,aa_num_num(bit1,Na)) = none_num ).

% and_not_num.simps(3)
tff(fact_9789_and__not__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( bit_and_not_num(X,Xa2) = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != none_num ) ) )
       => ( ( ( X = one )
           => ( ? [N: num] : Xa2 = aa_num_num(bit0,N)
             => ( Y != some_num(one) ) ) )
         => ( ( ( X = one )
             => ( ? [N: num] : Xa2 = aa_num_num(bit1,N)
               => ( Y != none_num ) ) )
           => ( ! [M: num] :
                  ( ( X = aa_num_num(bit0,M) )
                 => ( ( Xa2 = one )
                   => ( Y != some_num(aa_num_num(bit0,M)) ) ) )
             => ( ! [M: num] :
                    ( ( X = 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] :
                      ( ( X = 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] :
                        ( ( X = aa_num_num(bit1,M) )
                       => ( ( Xa2 = one )
                         => ( Y != some_num(aa_num_num(bit0,M)) ) ) )
                   => ( ! [M: num] :
                          ( ( X = aa_num_num(bit1,M) )
                         => ! [N: num] :
                              ( ( Xa2 = aa_num_num(bit0,N) )
                             => ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_aar,bit_and_not_num(M,N)) ) ) )
                     => ~ ! [M: num] :
                            ( ( X = 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_9790_and__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( bit_un7362597486090784418nd_num(X,Xa2) = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != some_num(one) ) ) )
       => ( ( ( X = one )
           => ( ? [N: num] : Xa2 = aa_num_num(bit0,N)
             => ( Y != none_num ) ) )
         => ( ( ( X = one )
             => ( ? [N: num] : Xa2 = aa_num_num(bit1,N)
               => ( Y != some_num(one) ) ) )
           => ( ( ? [M: num] : X = aa_num_num(bit0,M)
               => ( ( Xa2 = one )
                 => ( Y != none_num ) ) )
             => ( ! [M: num] :
                    ( ( X = 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] :
                      ( ( X = 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] : X = aa_num_num(bit1,M)
                     => ( ( Xa2 = one )
                       => ( Y != some_num(one) ) ) )
                   => ( ! [M: num] :
                          ( ( X = 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] :
                            ( ( X = aa_num_num(bit1,M) )
                           => ! [N: num] :
                                ( ( Xa2 = aa_num_num(bit1,N) )
                               => ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_aar,bit_un7362597486090784418nd_num(M,N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_9791_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_9792_and__num_Osimps_I2_J,axiom,
    ! [Na: num] : bit_un7362597486090784418nd_num(one,aa_num_num(bit0,Na)) = none_num ).

% and_num.simps(2)
tff(fact_9793_xor__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( bit_un2480387367778600638or_num(X,Xa2) = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != none_num ) ) )
       => ( ( ( X = one )
           => ! [N: num] :
                ( ( Xa2 = aa_num_num(bit0,N) )
               => ( Y != some_num(aa_num_num(bit1,N)) ) ) )
         => ( ( ( X = one )
             => ! [N: num] :
                  ( ( Xa2 = aa_num_num(bit1,N) )
                 => ( Y != some_num(aa_num_num(bit0,N)) ) ) )
           => ( ! [M: num] :
                  ( ( X = aa_num_num(bit0,M) )
                 => ( ( Xa2 = one )
                   => ( Y != some_num(aa_num_num(bit1,M)) ) ) )
             => ( ! [M: num] :
                    ( ( X = 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] :
                      ( ( X = 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] :
                        ( ( X = aa_num_num(bit1,M) )
                       => ( ( Xa2 = one )
                         => ( Y != some_num(aa_num_num(bit0,M)) ) ) )
                   => ( ! [M: num] :
                          ( ( X = 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] :
                            ( ( X = 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_9794_xor__num_Osimps_I1_J,axiom,
    bit_un2480387367778600638or_num(one,one) = none_num ).

% xor_num.simps(1)
tff(fact_9795_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_rs,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uu)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_1
tff(fact_9796_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_tj,Uu) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu)) ).

% ATP.lambda_2
tff(fact_9797_ATP_Olambda__3,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_rr,Uu) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uu)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu)) ).

% ATP.lambda_3
tff(fact_9798_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_aag,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_4
tff(fact_9799_ATP_Olambda__5,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_ua,Uu) = aa_complex_complex(plus_plus_complex(Uu),one_one_complex) ).

% ATP.lambda_5
tff(fact_9800_ATP_Olambda__6,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ud,Uu) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),one_one_real) ).

% ATP.lambda_6
tff(fact_9801_ATP_Olambda__7,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_ub,Uu) = aa_rat_rat(plus_plus_rat(Uu),one_one_rat) ).

% ATP.lambda_7
tff(fact_9802_ATP_Olambda__8,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ue,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_8
tff(fact_9803_ATP_Olambda__9,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_uc,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).

% ATP.lambda_9
tff(fact_9804_ATP_Olambda__10,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_aae,Uu)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu)),product_snd_int_int(Uu))) ) ).

% ATP.lambda_10
tff(fact_9805_ATP_Olambda__11,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_yz,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_qh,Uu)) ).

% ATP.lambda_11
tff(fact_9806_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_zv,Uu) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_df,Uu)) ).

% ATP.lambda_12
tff(fact_9807_ATP_Olambda__13,axiom,
    ! [Uu: real] : aa_rea5428163979925208277l_real(aTP_Lamm_aac,Uu) = princi6114159922880469582l_real(collec3799799289383736868l_real(produc5414030515140494994real_o(aTP_Lamm_aab(Uu)))) ).

% ATP.lambda_13
tff(fact_9808_ATP_Olambda__14,axiom,
    ! [Uu: real] : aa_rea5117059003799598681omplex(aTP_Lamm_aaa,Uu) = princi3496590319149328850omplex(collec8663557070575231912omplex(produc6771430404735790350plex_o(aTP_Lamm_zz(Uu)))) ).

% ATP.lambda_14
tff(fact_9809_ATP_Olambda__15,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_zk,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).

% ATP.lambda_15
tff(fact_9810_ATP_Olambda__16,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_aar,Uu) = some_num(aa_num_num(bit1,Uu)) ).

% ATP.lambda_16
tff(fact_9811_ATP_Olambda__17,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_yc,Uu) = some_num(aa_num_num(bit0,Uu)) ).

% ATP.lambda_17
tff(fact_9812_ATP_Olambda__18,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_wu,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,Uu)) ).

% ATP.lambda_18
tff(fact_9813_ATP_Olambda__19,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_wx,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_19
tff(fact_9814_ATP_Olambda__20,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_jr,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_20
tff(fact_9815_ATP_Olambda__21,axiom,
    ! [Uu: int] : aa_int_nat(aTP_Lamm_ym,Uu) = aa_int_nat(nat2,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_21
tff(fact_9816_ATP_Olambda__22,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_yk,Uu)
    <=> ? [I2: int,N3: nat] :
          ( ( Uu = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I2)),aa_nat_real(semiri5074537144036343181t_real,N3)) )
          & ( N3 != zero_zero_nat ) ) ) ).

% ATP.lambda_22
tff(fact_9817_ATP_Olambda__23,axiom,
    ! [Uu: nat] : aa_nat_option_num(aTP_Lamm_ye,Uu) = some_num(one) ).

% ATP.lambda_23
tff(fact_9818_ATP_Olambda__24,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_yi(Uu),Uua) = case_num_option_num(some_num(one),aTP_Lamm_yg(Uua),aTP_Lamm_yh(Uua),Uu) ).

% ATP.lambda_24
tff(fact_9819_ATP_Olambda__25,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_wl,Uu),Uua) = $ite(Uu = 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,Uu)),Uua)),aa_int_int(abs_abs_int,Uu))) ).

% ATP.lambda_25
tff(fact_9820_ATP_Olambda__26,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_yj,Uu),Uua) = case_nat_option_num(none_num,aTP_Lamm_yi(Uua),Uu) ).

% ATP.lambda_26
tff(fact_9821_ATP_Olambda__27,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_yg(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_yc,bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_27
tff(fact_9822_ATP_Olambda__28,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_yd(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_yc,bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_28
tff(fact_9823_ATP_Olambda__29,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qa,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_29
tff(fact_9824_ATP_Olambda__30,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tr,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_30
tff(fact_9825_ATP_Olambda__31,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_wh,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(tan_real,Uua) = Uu ) ) ) ).

% ATP.lambda_31
tff(fact_9826_ATP_Olambda__32,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_xb,Uu),Uua) = $let(
        l2: int,
        l2:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Uu)),
        $ite(Uua = zero_z3403309356797280102nteger,l2,aa_int_int(aa_int_fun_int_int(plus_plus_int,l2),one_one_int)) ) ).

% ATP.lambda_32
tff(fact_9827_ATP_Olambda__33,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Extended_enat(aa_nat2025370463774033377d_enat(aTP_Lamm_pu,Uu),Uua) = $let(
        m3: extended_enat,
        m3:= times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uu)),
        $ite(Uua = zero_zero_nat,m3,plus_p3455044024723400733d_enat(m3,one_on7984719198319812577d_enat)) ) ).

% ATP.lambda_33
tff(fact_9828_ATP_Olambda__34,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_pt,Uu),Uua) = $let(
        m3: complex,
        m3:= aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_complex_complex(plus_plus_complex(m3),one_one_complex)) ) ).

% ATP.lambda_34
tff(fact_9829_ATP_Olambda__35,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_px,Uu),Uua) = $let(
        m3: real,
        m3:= aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m3),one_one_real)) ) ).

% ATP.lambda_35
tff(fact_9830_ATP_Olambda__36,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_pv,Uu),Uua) = $let(
        m3: rat,
        m3:= aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_rat_rat(plus_plus_rat(m3),one_one_rat)) ) ).

% ATP.lambda_36
tff(fact_9831_ATP_Olambda__37,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_py,Uu),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,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m3),one_one_nat)) ) ).

% ATP.lambda_37
tff(fact_9832_ATP_Olambda__38,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_pw,Uu),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,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_int_int(aa_int_fun_int_int(plus_plus_int,m3),one_one_int)) ) ).

% ATP.lambda_38
tff(fact_9833_ATP_Olambda__39,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa_real_o(aa_com1951324360330529881real_o(aTP_Lamm_aaq,Uu),Uua)
    <=> ( ( aa_complex_complex(sgn_sgn_complex,Uu) = aa_real_complex(cis,Uua) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi) ) ) ).

% ATP.lambda_39
tff(fact_9834_ATP_Olambda__40,axiom,
    ! [Uu: real,Uua: int] :
      ( aa_int_o(aa_real_fun_int_o(aTP_Lamm_wi,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uu),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int))) ) ) ).

% ATP.lambda_40
tff(fact_9835_ATP_Olambda__41,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_wj(Uu),Uua)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uu),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int))) ) ) ).

% ATP.lambda_41
tff(fact_9836_ATP_Olambda__42,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qh,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))) ).

% ATP.lambda_42
tff(fact_9837_ATP_Olambda__43,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_za,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_43
tff(fact_9838_ATP_Olambda__44,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ze(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_44
tff(fact_9839_ATP_Olambda__45,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_uf(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(Uu),Uua)),aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Uu),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_45
tff(fact_9840_ATP_Olambda__46,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uh,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Uu),Uua)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_46
tff(fact_9841_ATP_Olambda__47,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ug(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Uu),Uua)),aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Uu),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_47
tff(fact_9842_ATP_Olambda__48,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_xn,Uu),Uua)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_48
tff(fact_9843_ATP_Olambda__49,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tz,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_49
tff(fact_9844_ATP_Olambda__50,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aad,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uua)),Uua) ).

% ATP.lambda_50
tff(fact_9845_ATP_Olambda__51,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vy,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uu) ).

% ATP.lambda_51
tff(fact_9846_ATP_Olambda__52,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vx,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uua) ).

% ATP.lambda_52
tff(fact_9847_ATP_Olambda__53,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_he,Uu),Uua)
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_53
tff(fact_9848_ATP_Olambda__54,axiom,
    ! [Uu: nat,Uua: real] :
      ( aa_real_o(aa_nat_fun_real_o(aTP_Lamm_hd,Uu),Uua)
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_54
tff(fact_9849_ATP_Olambda__55,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pz,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_55
tff(fact_9850_ATP_Olambda__56,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_pf(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_56
tff(fact_9851_ATP_Olambda__57,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_pc(Uu),Uua) = aa_rat_rat(plus_plus_rat(aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_57
tff(fact_9852_ATP_Olambda__58,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_pe(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_58
tff(fact_9853_ATP_Olambda__59,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_pd(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_59
tff(fact_9854_ATP_Olambda__60,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ok(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,aa_nat_nat(suc,Uua))),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_60
tff(fact_9855_ATP_Olambda__61,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_oi(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,aa_nat_nat(suc,Uua))),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_61
tff(fact_9856_ATP_Olambda__62,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_oj(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,aa_nat_nat(suc,Uua))),aa_nat_int(Uu,Uua)) ).

% ATP.lambda_62
tff(fact_9857_ATP_Olambda__63,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_kr(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uua)),aa_nat_complex(power_power_complex(zero_zero_complex),Uua)) ).

% ATP.lambda_63
tff(fact_9858_ATP_Olambda__64,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_kt(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uua)) ).

% ATP.lambda_64
tff(fact_9859_ATP_Olambda__65,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ks(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ).

% ATP.lambda_65
tff(fact_9860_ATP_Olambda__66,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_sw(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)),aa_nat_int(power_power_int(zero_zero_int),Uua)) ).

% ATP.lambda_66
tff(fact_9861_ATP_Olambda__67,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_pb(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_67
tff(fact_9862_ATP_Olambda__68,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_oz(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_68
tff(fact_9863_ATP_Olambda__69,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_pa(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_69
tff(fact_9864_ATP_Olambda__70,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ov(Uu),Uua) = aa_complex_complex(minus_minus_complex(aa_nat_complex(Uu,Uua)),aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_70
tff(fact_9865_ATP_Olambda__71,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_oy(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_71
tff(fact_9866_ATP_Olambda__72,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ow(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_72
tff(fact_9867_ATP_Olambda__73,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ox(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_73
tff(fact_9868_ATP_Olambda__74,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_qr(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_74
tff(fact_9869_ATP_Olambda__75,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qp(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_75
tff(fact_9870_ATP_Olambda__76,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_qq(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_76
tff(fact_9871_ATP_Olambda__77,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_nu(Uu),Uua)
    <=> ( aa_complex_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_77
tff(fact_9872_ATP_Olambda__78,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_nx(Uu),Uua)
    <=> ( aa_complex_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_78
tff(fact_9873_ATP_Olambda__79,axiom,
    ! [Uu: fun_real_complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_nq(Uu),Uua)
    <=> ( aa_real_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_79
tff(fact_9874_ATP_Olambda__80,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_ns(Uu),Uua)
    <=> ( aa_real_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_80
tff(fact_9875_ATP_Olambda__81,axiom,
    ! [Uu: fun_real_rat,Uua: real] :
      ( aa_real_o(aTP_Lamm_nv(Uu),Uua)
    <=> ( aa_real_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_81
tff(fact_9876_ATP_Olambda__82,axiom,
    ! [Uu: fun_real_nat,Uua: real] :
      ( aa_real_o(aTP_Lamm_ny(Uu),Uua)
    <=> ( aa_real_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_82
tff(fact_9877_ATP_Olambda__83,axiom,
    ! [Uu: fun_int_complex,Uua: int] :
      ( aa_int_o(aTP_Lamm_nr(Uu),Uua)
    <=> ( aa_int_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_83
tff(fact_9878_ATP_Olambda__84,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( aa_int_o(aTP_Lamm_nt(Uu),Uua)
    <=> ( aa_int_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_84
tff(fact_9879_ATP_Olambda__85,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_nw(Uu),Uua)
    <=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_85
tff(fact_9880_ATP_Olambda__86,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_nz(Uu),Uua)
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_86
tff(fact_9881_ATP_Olambda__87,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_zf(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_ze(Uu),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_87
tff(fact_9882_ATP_Olambda__88,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_zq(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_ze(Uu),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_88
tff(fact_9883_ATP_Olambda__89,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xz,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uu) ).

% ATP.lambda_89
tff(fact_9884_ATP_Olambda__90,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ru(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Uua))),aa_nat_complex(power_power_complex(Uu),Uua)) ).

% ATP.lambda_90
tff(fact_9885_ATP_Olambda__91,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rt,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_91
tff(fact_9886_ATP_Olambda__92,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_yo,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_92
tff(fact_9887_ATP_Olambda__93,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ty,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_93
tff(fact_9888_ATP_Olambda__94,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tq,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_94
tff(fact_9889_ATP_Olambda__95,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_xc,Uu),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu),
        $let(
          l3: nat,
          l3:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l3),one_one_nat)) ) ) ).

% ATP.lambda_95
tff(fact_9890_ATP_Olambda__96,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ui(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),Uu) ).

% ATP.lambda_96
tff(fact_9891_ATP_Olambda__97,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_uk,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),Uu) ).

% ATP.lambda_97
tff(fact_9892_ATP_Olambda__98,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_uj(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu) ).

% ATP.lambda_98
tff(fact_9893_ATP_Olambda__99,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_iv,Uu),Uua) = produc1086072967326762835nteger(Uu,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),one_one_Code_integer)) ).

% ATP.lambda_99
tff(fact_9894_ATP_Olambda__100,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_iu,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_100
tff(fact_9895_ATP_Olambda__101,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_it,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)) ).

% ATP.lambda_101
tff(fact_9896_ATP_Olambda__102,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_iy,Uu),Uua) = produc1086072967326762835nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_102
tff(fact_9897_ATP_Olambda__103,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ix,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_103
tff(fact_9898_ATP_Olambda__104,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_iw,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),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_104
tff(fact_9899_ATP_Olambda__105,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_uo(Uu),Uua) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) ).

% ATP.lambda_105
tff(fact_9900_ATP_Olambda__106,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uq,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% ATP.lambda_106
tff(fact_9901_ATP_Olambda__107,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_up(Uu),Uua) = aa_rat_rat(plus_plus_rat(Uu),aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% ATP.lambda_107
tff(fact_9902_ATP_Olambda__108,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zp,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_108
tff(fact_9903_ATP_Olambda__109,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zo,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_109
tff(fact_9904_ATP_Olambda__110,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zl,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_110
tff(fact_9905_ATP_Olambda__111,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ya,Uu),Uua)
    <=> ( Uua = aa_nat_nat(suc,Uu) ) ) ).

% ATP.lambda_111
tff(fact_9906_ATP_Olambda__112,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aTP_Lamm_bp(Uu),Uua)
    <=> aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Uua),Uu) ) ).

% ATP.lambda_112
tff(fact_9907_ATP_Olambda__113,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( aa_set_complex_o(aTP_Lamm_bo(Uu),Uua)
    <=> aa_set_complex_o(ord_le211207098394363844omplex(Uua),Uu) ) ).

% ATP.lambda_113
tff(fact_9908_ATP_Olambda__114,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_bn,Uu),Uua)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Uua),Uu) ) ).

% ATP.lambda_114
tff(fact_9909_ATP_Olambda__115,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aTP_Lamm_bq(Uu),Uua)
    <=> aa_set_int_o(ord_less_eq_set_int(Uua),Uu) ) ).

% ATP.lambda_115
tff(fact_9910_ATP_Olambda__116,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bz,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_116
tff(fact_9911_ATP_Olambda__117,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yn,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_117
tff(fact_9912_ATP_Olambda__118,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_qm,Uu),Uua)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),Uu) ) ).

% ATP.lambda_118
tff(fact_9913_ATP_Olambda__119,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_qi,Uu),Uua)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Uua),Uu) ) ).

% ATP.lambda_119
tff(fact_9914_ATP_Olambda__120,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_qj,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uua),Uu) ) ).

% ATP.lambda_120
tff(fact_9915_ATP_Olambda__121,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_qk,Uu),Uua)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Uua),Uu) ) ).

% ATP.lambda_121
tff(fact_9916_ATP_Olambda__122,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_by,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_122
tff(fact_9917_ATP_Olambda__123,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ql,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uu) ) ).

% ATP.lambda_123
tff(fact_9918_ATP_Olambda__124,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zg,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_124
tff(fact_9919_ATP_Olambda__125,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yl,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_125
tff(fact_9920_ATP_Olambda__126,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_yp,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_126
tff(fact_9921_ATP_Olambda__127,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aaf,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_127
tff(fact_9922_ATP_Olambda__128,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_yq,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu) ).

% ATP.lambda_128
tff(fact_9923_ATP_Olambda__129,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_dh,Uu),Uua)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),Uu) ) ).

% ATP.lambda_129
tff(fact_9924_ATP_Olambda__130,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_de,Uu),Uua)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Uua),Uu) ) ).

% ATP.lambda_130
tff(fact_9925_ATP_Olambda__131,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_df,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_131
tff(fact_9926_ATP_Olambda__132,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_dg,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu) ) ).

% ATP.lambda_132
tff(fact_9927_ATP_Olambda__133,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_xq,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu) ).

% ATP.lambda_133
tff(fact_9928_ATP_Olambda__134,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vw,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_134
tff(fact_9929_ATP_Olambda__135,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zj,Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_135
tff(fact_9930_ATP_Olambda__136,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_a(Uu),Uua)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uua),Uu) ) ).

% ATP.lambda_136
tff(fact_9931_ATP_Olambda__137,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_aa(Uu),Uua)
    <=> aa_set_real_o(member_real(Uua),Uu) ) ).

% ATP.lambda_137
tff(fact_9932_ATP_Olambda__138,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ab(Uu),Uua)
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_138
tff(fact_9933_ATP_Olambda__139,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ac(Uu),Uua)
    <=> aa_set_set_nat_o(member_set_nat(Uua),Uu) ) ).

% ATP.lambda_139
tff(fact_9934_ATP_Olambda__140,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ad(Uu),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_140
tff(fact_9935_ATP_Olambda__141,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ae(Uu),Uua)
    <=> aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_141
tff(fact_9936_ATP_Olambda__142,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aTP_Lamm_br(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_142
tff(fact_9937_ATP_Olambda__143,axiom,
    ! [Uu: product_prod_nat_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aTP_Lamm_bu,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_143
tff(fact_9938_ATP_Olambda__144,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_bv,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_144
tff(fact_9939_ATP_Olambda__145,axiom,
    ! [Uu: list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_bs(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_145
tff(fact_9940_ATP_Olambda__146,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_bt,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_146
tff(fact_9941_ATP_Olambda__147,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bw,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_147
tff(fact_9942_ATP_Olambda__148,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bx,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_148
tff(fact_9943_ATP_Olambda__149,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_aao(Uu),Uua) = order_underS_nat(bNF_Ca8665028551170535155natLeq,Uu) ).

% ATP.lambda_149
tff(fact_9944_ATP_Olambda__150,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ro(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_150
tff(fact_9945_ATP_Olambda__151,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_rn(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_151
tff(fact_9946_ATP_Olambda__152,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_st(Uu),Uua) = aa_nat_complex(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_152
tff(fact_9947_ATP_Olambda__153,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nn(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_153
tff(fact_9948_ATP_Olambda__154,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_zr(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_154
tff(fact_9949_ATP_Olambda__155,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_og(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_155
tff(fact_9950_ATP_Olambda__156,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_nm(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_156
tff(fact_9951_ATP_Olambda__157,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_oh(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_157
tff(fact_9952_ATP_Olambda__158,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_yh(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_158
tff(fact_9953_ATP_Olambda__159,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_yf(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu))) ).

% ATP.lambda_159
tff(fact_9954_ATP_Olambda__160,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_xy,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_xx(Uu),Uua)) ).

% ATP.lambda_160
tff(fact_9955_ATP_Olambda__161,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_xw,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_xv(Uu),Uua)) ).

% ATP.lambda_161
tff(fact_9956_ATP_Olambda__162,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_xu,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_xt(Uu),Uua)) ).

% ATP.lambda_162
tff(fact_9957_ATP_Olambda__163,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_xs,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_xr(Uu),Uua)) ).

% ATP.lambda_163
tff(fact_9958_ATP_Olambda__164,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_xp,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_xo(Uu),Uua)) ).

% ATP.lambda_164
tff(fact_9959_ATP_Olambda__165,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_zd(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_zc(Uu),Uua)) ).

% ATP.lambda_165
tff(fact_9960_ATP_Olambda__166,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_aap(Uu),Uua) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu))) ).

% ATP.lambda_166
tff(fact_9961_ATP_Olambda__167,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_ul(Uu),Uua) = times_times_complex(aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_167
tff(fact_9962_ATP_Olambda__168,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_un(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_168
tff(fact_9963_ATP_Olambda__169,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_um(Uu),Uua) = times_times_rat(aa_rat_rat(minus_minus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_169
tff(fact_9964_ATP_Olambda__170,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_ur(Uu),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_170
tff(fact_9965_ATP_Olambda__171,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_uu(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_171
tff(fact_9966_ATP_Olambda__172,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_us(Uu),Uua) = times_times_rat(aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_172
tff(fact_9967_ATP_Olambda__173,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_uv(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua))) ).

% ATP.lambda_173
tff(fact_9968_ATP_Olambda__174,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_ut(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua))) ).

% ATP.lambda_174
tff(fact_9969_ATP_Olambda__175,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_aal(Uu),Uua) = set_ord_atMost_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uua)) ).

% ATP.lambda_175
tff(fact_9970_ATP_Olambda__176,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zn,Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_176
tff(fact_9971_ATP_Olambda__177,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xm,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_177
tff(fact_9972_ATP_Olambda__178,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xl,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_178
tff(fact_9973_ATP_Olambda__179,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xj,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_179
tff(fact_9974_ATP_Olambda__180,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xi,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_180
tff(fact_9975_ATP_Olambda__181,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_ed(Uu),Uua)
    <=> ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uua),Uu) ) ).

% ATP.lambda_181
tff(fact_9976_ATP_Olambda__182,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_ee(Uu),Uua)
    <=> ~ aa_set_real_o(member_real(Uua),Uu) ) ).

% ATP.lambda_182
tff(fact_9977_ATP_Olambda__183,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ef(Uu),Uua)
    <=> ~ member_list_nat(Uua,Uu) ) ).

% ATP.lambda_183
tff(fact_9978_ATP_Olambda__184,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_eg(Uu),Uua)
    <=> ~ aa_set_set_nat_o(member_set_nat(Uua),Uu) ) ).

% ATP.lambda_184
tff(fact_9979_ATP_Olambda__185,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_eh(Uu),Uua)
    <=> ~ aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_185
tff(fact_9980_ATP_Olambda__186,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ei(Uu),Uua)
    <=> ~ aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_186
tff(fact_9981_ATP_Olambda__187,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_tb(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_187
tff(fact_9982_ATP_Olambda__188,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_zi(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_188
tff(fact_9983_ATP_Olambda__189,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] : aa_complex_complex(aTP_Lamm_ph(Uu),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_complex_nat(Uu,Uua)) ).

% ATP.lambda_189
tff(fact_9984_ATP_Olambda__190,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_real(aTP_Lamm_pk(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_190
tff(fact_9985_ATP_Olambda__191,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_pj(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_191
tff(fact_9986_ATP_Olambda__192,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_pi(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_192
tff(fact_9987_ATP_Olambda__193,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_uw(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_193
tff(fact_9988_ATP_Olambda__194,axiom,
    ! [Uu: fun_complex_int,Uua: complex] : aa_complex_complex(aTP_Lamm_ju(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_complex_int(Uu,Uua)) ).

% ATP.lambda_194
tff(fact_9989_ATP_Olambda__195,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_real(aTP_Lamm_jv(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_195
tff(fact_9990_ATP_Olambda__196,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_rat(aTP_Lamm_ux(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_196
tff(fact_9991_ATP_Olambda__197,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_uy(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_197
tff(fact_9992_ATP_Olambda__198,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_real(aTP_Lamm_uz(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_int_int(Uu,Uua)) ).

% ATP.lambda_198
tff(fact_9993_ATP_Olambda__199,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_rat(aTP_Lamm_va(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_int_int(Uu,Uua)) ).

% ATP.lambda_199
tff(fact_9994_ATP_Olambda__200,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_vb(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_200
tff(fact_9995_ATP_Olambda__201,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_kq(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_201
tff(fact_9996_ATP_Olambda__202,axiom,
    ! [Uu: fun_real_o,Uua: real] :
      ( aa_real_o(aTP_Lamm_dy(Uu),Uua)
    <=> ~ aa_real_o(Uu,Uua) ) ).

% ATP.lambda_202
tff(fact_9997_ATP_Olambda__203,axiom,
    ! [Uu: fun_list_nat_o,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_dz(Uu),Uua)
    <=> ~ aa_list_nat_o(Uu,Uua) ) ).

% ATP.lambda_203
tff(fact_9998_ATP_Olambda__204,axiom,
    ! [Uu: fun_set_nat_o,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ea(Uu),Uua)
    <=> ~ aa_set_nat_o(Uu,Uua) ) ).

% ATP.lambda_204
tff(fact_9999_ATP_Olambda__205,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_eb(Uu),Uua)
    <=> ~ aa_nat_o(Uu,Uua) ) ).

% ATP.lambda_205
tff(fact_10000_ATP_Olambda__206,axiom,
    ! [Uu: fun_int_o,Uua: int] :
      ( aa_int_o(aTP_Lamm_ec(Uu),Uua)
    <=> ~ aa_int_o(Uu,Uua) ) ).

% ATP.lambda_206
tff(fact_10001_ATP_Olambda__207,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_aam(Uu),Uua) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_by,Uu)) ).

% ATP.lambda_207
tff(fact_10002_ATP_Olambda__208,axiom,
    ! [Uu: list_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aTP_Lamm_gm(Uu),Uua)
    <=> ? [I2: nat] :
          ( ( Uua = nth_VEBT_VEBT(Uu,I2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Uu)) ) ) ).

% ATP.lambda_208
tff(fact_10003_ATP_Olambda__209,axiom,
    ! [Uu: list_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_gj(Uu),Uua)
    <=> ? [I2: nat] :
          ( ( Uua = aa_nat_real(nth_real(Uu),I2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Uu)) ) ) ).

% ATP.lambda_209
tff(fact_10004_ATP_Olambda__210,axiom,
    ! [Uu: list_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_gk(Uu),Uua)
    <=> ? [I2: nat] :
          ( ( Uua = nth_list_nat(Uu,I2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3023201423986296836st_nat(Uu)) ) ) ).

% ATP.lambda_210
tff(fact_10005_ATP_Olambda__211,axiom,
    ! [Uu: list_o,Uua: $o] :
      ( aa_o_o(aTP_Lamm_gn(Uu),(Uua))
    <=> ? [I2: nat] :
          ( ( (Uua)
          <=> aa_nat_o(nth_o(Uu),I2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_o_nat(size_size_list_o,Uu)) ) ) ).

% ATP.lambda_211
tff(fact_10006_ATP_Olambda__212,axiom,
    ! [Uu: list_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_gl(Uu),Uua)
    <=> ? [I2: nat] :
          ( ( Uua = nth_set_nat(Uu,I2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3254054031482475050et_nat(Uu)) ) ) ).

% ATP.lambda_212
tff(fact_10007_ATP_Olambda__213,axiom,
    ! [Uu: list_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_go(Uu),Uua)
    <=> ? [I2: nat] :
          ( ( Uua = aa_nat_nat(nth_nat(Uu),I2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Uu)) ) ) ).

% ATP.lambda_213
tff(fact_10008_ATP_Olambda__214,axiom,
    ! [Uu: list_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_gp(Uu),Uua)
    <=> ? [I2: nat] :
          ( ( Uua = aa_nat_int(nth_int(Uu),I2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_int_nat(size_size_list_int,Uu)) ) ) ).

% ATP.lambda_214
tff(fact_10009_ATP_Olambda__215,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_rm(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uu,Uub),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_215
tff(fact_10010_ATP_Olambda__216,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_jq(Uu),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu)),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,numera6620942414471956472nteger(Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_216
tff(fact_10011_ATP_Olambda__217,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_jo(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu)),Uub),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_217
tff(fact_10012_ATP_Olambda__218,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_jp(Uu),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),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,Uu))),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_218
tff(fact_10013_ATP_Olambda__219,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sg(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_219
tff(fact_10014_ATP_Olambda__220,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_sh(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_220
tff(fact_10015_ATP_Olambda__221,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_si(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_221
tff(fact_10016_ATP_Olambda__222,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_sj(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_222
tff(fact_10017_ATP_Olambda__223,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_kl(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_223
tff(fact_10018_ATP_Olambda__224,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_kp(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_224
tff(fact_10019_ATP_Olambda__225,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] :
      aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_kg(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_225
tff(fact_10020_ATP_Olambda__226,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_kj(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_226
tff(fact_10021_ATP_Olambda__227,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_km(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_227
tff(fact_10022_ATP_Olambda__228,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_kh(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_228
tff(fact_10023_ATP_Olambda__229,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_kn(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_229
tff(fact_10024_ATP_Olambda__230,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] :
      aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_ki(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_230
tff(fact_10025_ATP_Olambda__231,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_kk(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_231
tff(fact_10026_ATP_Olambda__232,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ko(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_232
tff(fact_10027_ATP_Olambda__233,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] :
      aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_vp(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_233
tff(fact_10028_ATP_Olambda__234,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_vt(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_real(Uua,Uub),one_one_real) ).

% ATP.lambda_234
tff(fact_10029_ATP_Olambda__235,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] :
      aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_vm(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_235
tff(fact_10030_ATP_Olambda__236,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vq(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_real(Uua,Uub),one_one_real) ).

% ATP.lambda_236
tff(fact_10031_ATP_Olambda__237,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_vu(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_237
tff(fact_10032_ATP_Olambda__238,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vn(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_238
tff(fact_10033_ATP_Olambda__239,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_vr(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_real(Uua,Uub),one_one_real) ).

% ATP.lambda_239
tff(fact_10034_ATP_Olambda__240,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_vv(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_240
tff(fact_10035_ATP_Olambda__241,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] :
      aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_vo(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_241
tff(fact_10036_ATP_Olambda__242,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_vs(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_real(Uua,Uub),one_one_real) ).

% ATP.lambda_242
tff(fact_10037_ATP_Olambda__243,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_kb(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_243
tff(fact_10038_ATP_Olambda__244,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_kf(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_244
tff(fact_10039_ATP_Olambda__245,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] :
      aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_jw(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_245
tff(fact_10040_ATP_Olambda__246,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_jz(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_246
tff(fact_10041_ATP_Olambda__247,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_kc(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_247
tff(fact_10042_ATP_Olambda__248,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_jx(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_248
tff(fact_10043_ATP_Olambda__249,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_rv(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_249
tff(fact_10044_ATP_Olambda__250,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_kd(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_250
tff(fact_10045_ATP_Olambda__251,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rw(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_251
tff(fact_10046_ATP_Olambda__252,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_rx(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_252
tff(fact_10047_ATP_Olambda__253,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] :
      aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_jy(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_253
tff(fact_10048_ATP_Olambda__254,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_ka(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_254
tff(fact_10049_ATP_Olambda__255,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ke(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_255
tff(fact_10050_ATP_Olambda__256,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] :
      aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_vf(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_256
tff(fact_10051_ATP_Olambda__257,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_vj(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_real(Uua,Uub),one_one_real) ).

% ATP.lambda_257
tff(fact_10052_ATP_Olambda__258,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] :
      aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_vc(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_258
tff(fact_10053_ATP_Olambda__259,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vg(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_real(Uua,Uub),one_one_real) ).

% ATP.lambda_259
tff(fact_10054_ATP_Olambda__260,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_vk(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_260
tff(fact_10055_ATP_Olambda__261,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vd(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_261
tff(fact_10056_ATP_Olambda__262,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_vh(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_real(Uua,Uub),one_one_real) ).

% ATP.lambda_262
tff(fact_10057_ATP_Olambda__263,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_vl(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_263
tff(fact_10058_ATP_Olambda__264,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] :
      aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_ve(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_264
tff(fact_10059_ATP_Olambda__265,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_vi(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_real(Uua,Uub),one_one_real) ).

% ATP.lambda_265
tff(fact_10060_ATP_Olambda__266,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_xf(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu),Uub))) ).

% ATP.lambda_266
tff(fact_10061_ATP_Olambda__267,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_xk(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(abs_abs_Code_integer(Uu),Uub))) ).

% ATP.lambda_267
tff(fact_10062_ATP_Olambda__268,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_xe(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uu,Uub))) ).

% ATP.lambda_268
tff(fact_10063_ATP_Olambda__269,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sc(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_269
tff(fact_10064_ATP_Olambda__270,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_sd(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_270
tff(fact_10065_ATP_Olambda__271,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_se(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_271
tff(fact_10066_ATP_Olambda__272,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_sf(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_272
tff(fact_10067_ATP_Olambda__273,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_o,Uub: complex] :
      aa_complex_real(aa_fun8261057609414086622x_real(aTP_Lamm_ms(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_273
tff(fact_10068_ATP_Olambda__274,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_o,Uub: complex] :
      aa_complex_rat(aa_fun4010016270681809034ex_rat(aTP_Lamm_mw(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_274
tff(fact_10069_ATP_Olambda__275,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_o,Uub: real] :
      aa_real_complex(aa_fun1098856492951973088omplex(aTP_Lamm_mm(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_complex(Uu,Uub),zero_zero_complex) ).

% ATP.lambda_275
tff(fact_10070_ATP_Olambda__276,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_o,Uub: real] :
      aa_real_real(aa_fun4448911596306054750l_real(aTP_Lamm_mp(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_276
tff(fact_10071_ATP_Olambda__277,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_o,Uub: real] :
      aa_real_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_mt(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_277
tff(fact_10072_ATP_Olambda__278,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_o,Uub: nat] :
      aa_nat_complex(aa_fun1400689748890728928omplex(aTP_Lamm_mn(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_complex(Uu,Uub),zero_zero_complex) ).

% ATP.lambda_278
tff(fact_10073_ATP_Olambda__279,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] :
      aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_mu(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_279
tff(fact_10074_ATP_Olambda__280,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_o,Uub: int] :
      aa_int_complex(aa_fun6439487526662521568omplex(aTP_Lamm_mo(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_complex(Uu,Uub),zero_zero_complex) ).

% ATP.lambda_280
tff(fact_10075_ATP_Olambda__281,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_o,Uub: int] :
      aa_int_real(aa_fun7506669883404309598t_real(aTP_Lamm_mq(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_281
tff(fact_10076_ATP_Olambda__282,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_o,Uub: int] :
      aa_int_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_mv(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_282
tff(fact_10077_ATP_Olambda__283,axiom,
    ! [Uu: fun_co4998825143550818160real_o,Uua: real,Uub: complex] :
      ( aa_complex_o(aa_rea2382570592068293209plex_o(aTP_Lamm_ba(Uu),Uua),Uub)
    <=> aa_real_o(aa_com1951324360330529881real_o(Uu,Uub),Uua) ) ).

% ATP.lambda_283
tff(fact_10078_ATP_Olambda__284,axiom,
    ! [Uu: fun_co8378349014452122532t_real,Uua: nat,Uub: complex] : aa_complex_real(aa_nat480766339200631653x_real(aTP_Lamm_mf(Uu),Uua),Uub) = aa_nat_real(aa_com20334259897932645t_real(Uu,Uub),Uua) ).

% ATP.lambda_284
tff(fact_10079_ATP_Olambda__285,axiom,
    ! [Uu: fun_co5676208753217392844_nat_o,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_bi(Uu),Uua),Uub)
    <=> aa_nat_o(aa_complex_fun_nat_o(Uu,Uub),Uua) ) ).

% ATP.lambda_285
tff(fact_10080_ATP_Olambda__286,axiom,
    ! [Uu: fun_co636942507509095880at_nat,Uua: nat,Uub: complex] : aa_complex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_lk(Uu),Uua),Uub) = aa_nat_nat(aa_com6383720667567721865at_nat(Uu,Uub),Uua) ).

% ATP.lambda_286
tff(fact_10081_ATP_Olambda__287,axiom,
    ! [Uu: fun_re7208826350674946562omplex,Uua: complex,Uub: real] : aa_real_complex(aa_com4766338864777664963omplex(aTP_Lamm_lo(Uu),Uua),Uub) = aa_complex_complex(aa_rea2618301233319580611omplex(Uu,Uub),Uua) ).

% ATP.lambda_287
tff(fact_10082_ATP_Olambda__288,axiom,
    ! [Uu: fun_real_fun_real_o,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_au(Uu),Uua),Uub)
    <=> aa_real_o(aa_real_fun_real_o(Uu,Uub),Uua) ) ).

% ATP.lambda_288
tff(fact_10083_ATP_Olambda__289,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_lz(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_289
tff(fact_10084_ATP_Olambda__290,axiom,
    ! [Uu: fun_real_fun_nat_o,Uua: nat,Uub: real] :
      ( aa_real_o(aa_nat_fun_real_o(aTP_Lamm_bc(Uu),Uua),Uub)
    <=> aa_nat_o(aa_real_fun_nat_o(Uu,Uub),Uua) ) ).

% ATP.lambda_290
tff(fact_10085_ATP_Olambda__291,axiom,
    ! [Uu: fun_real_fun_nat_nat,Uua: nat,Uub: real] : aa_real_nat(aa_nat_fun_real_nat(aTP_Lamm_lc(Uu),Uua),Uub) = aa_nat_nat(aa_real_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_291
tff(fact_10086_ATP_Olambda__292,axiom,
    ! [Uu: fun_real_fun_int_o,Uua: int,Uub: real] :
      ( aa_real_o(aa_int_fun_real_o(aTP_Lamm_bk(Uu),Uua),Uub)
    <=> aa_int_o(aa_real_fun_int_o(Uu,Uub),Uua) ) ).

% ATP.lambda_292
tff(fact_10087_ATP_Olambda__293,axiom,
    ! [Uu: fun_na1591161823130732966omplex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ls(Uu),Uua),Uub) = aa_complex_complex(aa_nat3829984526574172007omplex(Uu,Uub),Uua) ).

% ATP.lambda_293
tff(fact_10088_ATP_Olambda__294,axiom,
    ! [Uu: fun_nat_fun_real_o,Uua: real,Uub: nat] :
      ( aa_nat_o(aa_real_fun_nat_o(aTP_Lamm_aw(Uu),Uua),Uub)
    <=> aa_real_o(aa_nat_fun_real_o(Uu,Uub),Uua) ) ).

% ATP.lambda_294
tff(fact_10089_ATP_Olambda__295,axiom,
    ! [Uu: fun_nat_fun_real_nat,Uua: real,Uub: nat] : aa_nat_nat(aa_real_fun_nat_nat(aTP_Lamm_mj(Uu),Uua),Uub) = aa_real_nat(aa_nat_fun_real_nat(Uu,Uub),Uua) ).

% ATP.lambda_295
tff(fact_10090_ATP_Olambda__296,axiom,
    ! [Uu: fun_nat_fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_be(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(Uu,Uub),Uua) ) ).

% ATP.lambda_296
tff(fact_10091_ATP_Olambda__297,axiom,
    ! [Uu: fun_nat_fun_int_o,Uua: int,Uub: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aTP_Lamm_bm(Uu),Uua),Uub)
    <=> aa_int_o(aa_nat_fun_int_o(Uu,Uub),Uua) ) ).

% ATP.lambda_297
tff(fact_10092_ATP_Olambda__298,axiom,
    ! [Uu: fun_in1307110748760623106omplex,Uua: complex,Uub: int] : aa_int_complex(aa_com9156536935073639875omplex(aTP_Lamm_lw(Uu),Uua),Uub) = aa_complex_complex(aa_int3545933452204062147omplex(Uu,Uub),Uua) ).

% ATP.lambda_298
tff(fact_10093_ATP_Olambda__299,axiom,
    ! [Uu: fun_int_fun_real_o,Uua: real,Uub: int] :
      ( aa_int_o(aa_real_fun_int_o(aTP_Lamm_ay(Uu),Uua),Uub)
    <=> aa_real_o(aa_int_fun_real_o(Uu,Uub),Uua) ) ).

% ATP.lambda_299
tff(fact_10094_ATP_Olambda__300,axiom,
    ! [Uu: fun_int_fun_nat_real,Uua: nat,Uub: int] : aa_int_real(aa_nat_fun_int_real(aTP_Lamm_mc(Uu),Uua),Uub) = aa_nat_real(aa_int_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_300
tff(fact_10095_ATP_Olambda__301,axiom,
    ! [Uu: fun_int_fun_nat_o,Uua: nat,Uub: int] :
      ( aa_int_o(aa_nat_fun_int_o(aTP_Lamm_bg(Uu),Uua),Uub)
    <=> aa_nat_o(aa_int_fun_nat_o(Uu,Uub),Uua) ) ).

% ATP.lambda_301
tff(fact_10096_ATP_Olambda__302,axiom,
    ! [Uu: fun_int_fun_nat_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_lg(Uu),Uua),Uub) = aa_nat_nat(aa_int_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_302
tff(fact_10097_ATP_Olambda__303,axiom,
    ! [Uu: nat,Uua: complex,Uub: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_tc(Uu),Uua),Uub) = aa_complex_complex(times_times_complex($ite(Uub = Uu,one_one_complex,zero_zero_complex)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_303
tff(fact_10098_ATP_Olambda__304,axiom,
    ! [Uu: nat,Uua: real,Uub: nat] :
      aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_td(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
          $ite(Uub = Uu,one_one_real,zero_zero_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_304
tff(fact_10099_ATP_Olambda__305,axiom,
    ! [Uu: nat,Uua: int,Uub: nat] :
      aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_te(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,
          $ite(Uub = Uu,one_one_int,zero_zero_int)),aa_nat_int(power_power_int(Uua),Uub)) ).

% ATP.lambda_305
tff(fact_10100_ATP_Olambda__306,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_xd(Uu),Uua),Uub) = produc6677183202524767010eger_o($ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub)),Uub = one_one_Code_integer) ).

% ATP.lambda_306
tff(fact_10101_ATP_Olambda__307,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ww(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_wv(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_307
tff(fact_10102_ATP_Olambda__308,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_wt(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ws(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_308
tff(fact_10103_ATP_Olambda__309,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_wr(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_wq(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_309
tff(fact_10104_ATP_Olambda__310,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_wp(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_wo(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_310
tff(fact_10105_ATP_Olambda__311,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_wn(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_wm(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_311
tff(fact_10106_ATP_Olambda__312,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yt(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_312
tff(fact_10107_ATP_Olambda__313,axiom,
    ! [Uu: real,Uua: fun_na5723442673449079972x_real,Uub: nat] : aa_nat_real(aa_fun5746040925223427547t_real(aTP_Lamm_ts(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(aa_nat480766339200631653x_real(Uua,Uub),zero_zero_complex)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_313
tff(fact_10108_ATP_Olambda__314,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_tt(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_314
tff(fact_10109_ATP_Olambda__315,axiom,
    ! [Uu: real,Uua: fun_nat_fun_rat_real,Uub: nat] : aa_nat_real(aa_fun507281639146497029t_real(aTP_Lamm_tu(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_315
tff(fact_10110_ATP_Olambda__316,axiom,
    ! [Uu: real,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_tv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_316
tff(fact_10111_ATP_Olambda__317,axiom,
    ! [Uu: real,Uua: fun_nat_fun_int_real,Uub: nat] : aa_nat_real(aa_fun241848180832411225t_real(aTP_Lamm_tw(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_317
tff(fact_10112_ATP_Olambda__318,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ku(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(zero_zero_complex),Uub))),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_318
tff(fact_10113_ATP_Olambda__319,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_kw(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uub))),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_319
tff(fact_10114_ATP_Olambda__320,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_kv(Uu),Uua),Uub) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(zero_zero_rat),Uub))),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_320
tff(fact_10115_ATP_Olambda__321,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_321
tff(fact_10116_ATP_Olambda__322,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_tx(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_322
tff(fact_10117_ATP_Olambda__323,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_aai(Uu),Uua),Uub)
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),one_one_nat) )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_323
tff(fact_10118_ATP_Olambda__324,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_tl(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),aa_nat_complex(Uu,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_324
tff(fact_10119_ATP_Olambda__325,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tn(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),aa_nat_real(Uu,Uub))),aa_nat_real(aa_real_fun_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_325
tff(fact_10120_ATP_Olambda__326,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_aaj(Uu),Uua),Uub)
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_326
tff(fact_10121_ATP_Olambda__327,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_gy(Uu),Uua),Uub)
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Uub)),Uua) ) ) ).

% ATP.lambda_327
tff(fact_10122_ATP_Olambda__328,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: list_P6011104703257516679at_nat] :
      ( aa_lis2793599718760257805_nat_o(aa_nat1970911183850433186_nat_o(aTP_Lamm_gx(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s5460976970255530739at_nat(Uub)),Uua) ) ) ).

% ATP.lambda_328
tff(fact_10123_ATP_Olambda__329,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_gw(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua) ) ) ).

% ATP.lambda_329
tff(fact_10124_ATP_Olambda__330,axiom,
    ! [Uu: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_gz(Uu),Uua),Uub)
    <=> ( ord_less_eq_set_o(set_o2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_o_nat(size_size_list_o,Uub)),Uua) ) ) ).

% ATP.lambda_330
tff(fact_10125_ATP_Olambda__331,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_ha(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Uub)),Uua) ) ) ).

% ATP.lambda_331
tff(fact_10126_ATP_Olambda__332,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_hb(Uu),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Uub)),Uua) ) ) ).

% ATP.lambda_332
tff(fact_10127_ATP_Olambda__333,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_gs(Uu),Uua),Uub)
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & ( aa_lis7044684121217598542BT_nat(size_s6755466524823107622T_VEBT,Uub) = Uua ) ) ) ).

% ATP.lambda_333
tff(fact_10128_ATP_Olambda__334,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: list_P6011104703257516679at_nat] :
      ( aa_lis2793599718760257805_nat_o(aa_nat1970911183850433186_nat_o(aTP_Lamm_gr(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Uub)),Uu)
        & ( size_s5460976970255530739at_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_334
tff(fact_10129_ATP_Olambda__335,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_gq(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu)
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_335
tff(fact_10130_ATP_Olambda__336,axiom,
    ! [Uu: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_gt(Uu),Uua),Uub)
    <=> ( ord_less_eq_set_o(set_o2(Uub),Uu)
        & ( aa_list_o_nat(size_size_list_o,Uub) = Uua ) ) ) ).

% ATP.lambda_336
tff(fact_10131_ATP_Olambda__337,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_gu(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Uub)),Uu)
        & ( aa_list_nat_nat(size_size_list_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_337
tff(fact_10132_ATP_Olambda__338,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_gv(Uu),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu)
        & ( aa_list_int_nat(size_size_list_int,Uub) = Uua ) ) ) ).

% ATP.lambda_338
tff(fact_10133_ATP_Olambda__339,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_aah(Uu),Uua),Uub)
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_339
tff(fact_10134_ATP_Olambda__340,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_wz(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_340
tff(fact_10135_ATP_Olambda__341,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_xh(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),pow_nat(set_or4665077453230672383an_nat(zero_zero_nat,Uu)))
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_341
tff(fact_10136_ATP_Olambda__342,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aan(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uub) ) ) ).

% ATP.lambda_342
tff(fact_10137_ATP_Olambda__343,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_wy(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),aa_nat_nat(suc,Uua)) ) ) ).

% ATP.lambda_343
tff(fact_10138_ATP_Olambda__344,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_tk(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uu),Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_344
tff(fact_10139_ATP_Olambda__345,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tm(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(diffs_real(Uu),Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_345
tff(fact_10140_ATP_Olambda__346,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_tp(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uua),Uub)),aa_nat_complex(power_power_complex(Uu),Uub)) ).

% ATP.lambda_346
tff(fact_10141_ATP_Olambda__347,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_to(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(diffs_real(Uua),Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_347
tff(fact_10142_ATP_Olambda__348,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vz(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uub)) ).

% ATP.lambda_348
tff(fact_10143_ATP_Olambda__349,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: set_Pr4329608150637261639at_nat,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_set6353399069419443519_nat_o(aTP_Lamm_ep(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Uub),Uua) ) ) ).

% ATP.lambda_349
tff(fact_10144_ATP_Olambda__350,axiom,
    ! [Uu: product_prod_nat_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_set1857767853180004543_nat_o(aTP_Lamm_eq(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uua) ) ) ).

% ATP.lambda_350
tff(fact_10145_ATP_Olambda__351,axiom,
    ! [Uu: real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_er(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_351
tff(fact_10146_ATP_Olambda__352,axiom,
    ! [Uu: list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_es(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_352
tff(fact_10147_ATP_Olambda__353,axiom,
    ! [Uu: set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_et(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_set_nat_o(member_set_nat(Uub),Uua) ) ) ).

% ATP.lambda_353
tff(fact_10148_ATP_Olambda__354,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_eu(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_354
tff(fact_10149_ATP_Olambda__355,axiom,
    ! [Uu: int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_ev(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_355
tff(fact_10150_ATP_Olambda__356,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_wg(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uua)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_356
tff(fact_10151_ATP_Olambda__357,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ca(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_357
tff(fact_10152_ATP_Olambda__358,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_wf(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_358
tff(fact_10153_ATP_Olambda__359,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_cc(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_359
tff(fact_10154_ATP_Olambda__360,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_cd(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_360
tff(fact_10155_ATP_Olambda__361,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_cb(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_361
tff(fact_10156_ATP_Olambda__362,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_zx(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uua)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uu) ) ) ).

% ATP.lambda_362
tff(fact_10157_ATP_Olambda__363,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_zy(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uub),Uua)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uub),Uu) ) ) ).

% ATP.lambda_363
tff(fact_10158_ATP_Olambda__364,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_di(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uu)
        & aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_364
tff(fact_10159_ATP_Olambda__365,axiom,
    ! [Uu: set_complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_mr(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_365
tff(fact_10160_ATP_Olambda__366,axiom,
    ! [Uu: set_real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_dj(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_366
tff(fact_10161_ATP_Olambda__367,axiom,
    ! [Uu: set_list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_dk(Uu),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu)
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_367
tff(fact_10162_ATP_Olambda__368,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_dl(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_368
tff(fact_10163_ATP_Olambda__369,axiom,
    ! [Uu: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_dm(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_369
tff(fact_10164_ATP_Olambda__370,axiom,
    ! [Uu: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_dn(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_370
tff(fact_10165_ATP_Olambda__371,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_ck(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_371
tff(fact_10166_ATP_Olambda__372,axiom,
    ! [Uu: product_prod_nat_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_cn(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_372
tff(fact_10167_ATP_Olambda__373,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_co(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_373
tff(fact_10168_ATP_Olambda__374,axiom,
    ! [Uu: list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_cl(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_374
tff(fact_10169_ATP_Olambda__375,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_cm(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_375
tff(fact_10170_ATP_Olambda__376,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_cp(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_376
tff(fact_10171_ATP_Olambda__377,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_cq(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_377
tff(fact_10172_ATP_Olambda__378,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_cr(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_378
tff(fact_10173_ATP_Olambda__379,axiom,
    ! [Uu: product_prod_nat_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_cu(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_379
tff(fact_10174_ATP_Olambda__380,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_cv(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_380
tff(fact_10175_ATP_Olambda__381,axiom,
    ! [Uu: list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_cs(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_381
tff(fact_10176_ATP_Olambda__382,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ct(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_382
tff(fact_10177_ATP_Olambda__383,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_cw(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_383
tff(fact_10178_ATP_Olambda__384,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_cx(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_384
tff(fact_10179_ATP_Olambda__385,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aTP_Lamm_if(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_385
tff(fact_10180_ATP_Olambda__386,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aTP_Lamm_in(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_386
tff(fact_10181_ATP_Olambda__387,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aTP_Lamm_hz(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_387
tff(fact_10182_ATP_Olambda__388,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_ih(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_388
tff(fact_10183_ATP_Olambda__389,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_ip(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_389
tff(fact_10184_ATP_Olambda__390,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aTP_Lamm_ib(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_390
tff(fact_10185_ATP_Olambda__391,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_ij(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_391
tff(fact_10186_ATP_Olambda__392,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_ir(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_392
tff(fact_10187_ATP_Olambda__393,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aTP_Lamm_id(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_393
tff(fact_10188_ATP_Olambda__394,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_il(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_394
tff(fact_10189_ATP_Olambda__395,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aTP_Lamm_hl(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_395
tff(fact_10190_ATP_Olambda__396,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aTP_Lamm_ht(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_396
tff(fact_10191_ATP_Olambda__397,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aTP_Lamm_hf(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_397
tff(fact_10192_ATP_Olambda__398,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_hn(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_398
tff(fact_10193_ATP_Olambda__399,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_hv(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_399
tff(fact_10194_ATP_Olambda__400,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aTP_Lamm_hh(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_400
tff(fact_10195_ATP_Olambda__401,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_hp(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_401
tff(fact_10196_ATP_Olambda__402,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_hx(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_402
tff(fact_10197_ATP_Olambda__403,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aTP_Lamm_hj(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_403
tff(fact_10198_ATP_Olambda__404,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_hr(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_404
tff(fact_10199_ATP_Olambda__405,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_set1857767853180004543_nat_o(aTP_Lamm_ej(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uu)
        & ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uua) ) ) ).

% ATP.lambda_405
tff(fact_10200_ATP_Olambda__406,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_ek(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ~ aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_406
tff(fact_10201_ATP_Olambda__407,axiom,
    ! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_el(Uu),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu)
        & ~ member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_407
tff(fact_10202_ATP_Olambda__408,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_em(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu)
        & ~ aa_set_set_nat_o(member_set_nat(Uub),Uua) ) ) ).

% ATP.lambda_408
tff(fact_10203_ATP_Olambda__409,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_eo(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ~ aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_409
tff(fact_10204_ATP_Olambda__410,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_en(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ~ aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_410
tff(fact_10205_ATP_Olambda__411,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_zw(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uu) ) ).

% ATP.lambda_411
tff(fact_10206_ATP_Olambda__412,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aak(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu) ) ).

% ATP.lambda_412
tff(fact_10207_ATP_Olambda__413,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_zz(Uu),Uua),Uub)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V3694042436643373181omplex(Uua),Uub)),Uu) ) ).

% ATP.lambda_413
tff(fact_10208_ATP_Olambda__414,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_aab(Uu),Uua),Uub)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,Uua),Uub)),Uu) ) ).

% ATP.lambda_414
tff(fact_10209_ATP_Olambda__415,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_cy(Uu),Uua),Uub)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)),Uu) ) ).

% ATP.lambda_415
tff(fact_10210_ATP_Olambda__416,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_cz(Uu),Uua),Uub)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub),Uu) ) ).

% ATP.lambda_416
tff(fact_10211_ATP_Olambda__417,axiom,
    ! [Uu: set_Pr1200682188532975830nteger,Uua: fun_Co3460408610320954864e_term,Uub: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun5125889547937825198eger_o(aTP_Lamm_da(Uu),Uua),Uub)
    <=> member6901806533512420791nteger(aa_Pro4709850991492583807nteger(aa_fun6437779379678770976nteger(produc7264323995540650962nteger,Uua),Uub),Uu) ) ).

% ATP.lambda_417
tff(fact_10212_ATP_Olambda__418,axiom,
    ! [Uu: set_Pr3486312109254157132nteger,Uua: fun_Pr5098669352780829690e_term,Uub: produc8923325533196201883nteger] :
      ( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(aTP_Lamm_db(Uu),Uua),Uub)
    <=> member8978266922497061677nteger(aa_Pro4291442876242880245nteger(aa_fun8303988899834042124nteger(produc565448451787809096nteger,Uua),Uub),Uu) ) ).

% ATP.lambda_418
tff(fact_10213_ATP_Olambda__419,axiom,
    ! [Uu: set_Pr6700895561065273913nt_int,Uua: fun_Pr7763680713840462443e_term,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(aTP_Lamm_dc(Uu),Uua),Uub)
    <=> member753276645541486746nt_int(aa_Pro5748361610964203648nt_int(aa_fun3265472288063244902nt_int(produc7320969046608631221nt_int,Uua),Uub),Uu) ) ).

% ATP.lambda_419
tff(fact_10214_ATP_Olambda__420,axiom,
    ! [Uu: set_Pr4166680992137330533nt_int,Uua: fun_in5730379987311621695e_term,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(aTP_Lamm_dd(Uu),Uua),Uub)
    <=> member5890090881338784710nt_int(aa_Pro8800328608558162860nt_int(aa_fun260840925927807678nt_int(produc4729913460834754785nt_int,Uua),Uub),Uu) ) ).

% ATP.lambda_420
tff(fact_10215_ATP_Olambda__421,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_hc(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_421
tff(fact_10216_ATP_Olambda__422,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_xa(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_422
tff(fact_10217_ATP_Olambda__423,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_sx(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(suc,Uub))),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_423
tff(fact_10218_ATP_Olambda__424,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sy(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(suc,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_424
tff(fact_10219_ATP_Olambda__425,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_425
tff(fact_10220_ATP_Olambda__426,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_sp(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_426
tff(fact_10221_ATP_Olambda__427,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sn(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_427
tff(fact_10222_ATP_Olambda__428,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wc(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uub)) ).

% ATP.lambda_428
tff(fact_10223_ATP_Olambda__429,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_gh(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_429
tff(fact_10224_ATP_Olambda__430,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_wk(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_430
tff(fact_10225_ATP_Olambda__431,axiom,
    ! [Uu: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_nc(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_real_Code_integer(Uua,Uub)),aa_real_Code_integer(Uu,Uub)) ).

% ATP.lambda_431
tff(fact_10226_ATP_Olambda__432,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_432
tff(fact_10227_ATP_Olambda__433,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_nh(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uub)),aa_real_rat(Uu,Uub)) ).

% ATP.lambda_433
tff(fact_10228_ATP_Olambda__434,axiom,
    ! [Uu: fun_real_int,Uua: fun_real_int,Uub: real] : aa_real_int(aa_fun8270430325945277160al_int(aTP_Lamm_nk(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(Uua,Uub)),aa_real_int(Uu,Uub)) ).

% ATP.lambda_434
tff(fact_10229_ATP_Olambda__435,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_nd(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu,Uub)) ).

% ATP.lambda_435
tff(fact_10230_ATP_Olambda__436,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ni(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu,Uub)) ).

% ATP.lambda_436
tff(fact_10231_ATP_Olambda__437,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_nl(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uua,Uub)),aa_nat_int(Uu,Uub)) ).

% ATP.lambda_437
tff(fact_10232_ATP_Olambda__438,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_ne(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu,Uub)) ).

% ATP.lambda_438
tff(fact_10233_ATP_Olambda__439,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_ng(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uub)),aa_int_real(Uu,Uub)) ).

% ATP.lambda_439
tff(fact_10234_ATP_Olambda__440,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_nj(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_int_rat(Uua,Uub)),aa_int_rat(Uu,Uub)) ).

% ATP.lambda_440
tff(fact_10235_ATP_Olambda__441,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_zh(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_441
tff(fact_10236_ATP_Olambda__442,axiom,
    ! [Uu: fun_Pr2602117396471556196at_nat,Uua: fun_Pr2602117396471556196at_nat,Uub: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(aa_fun2351638177499875132at_nat(aTP_Lamm_mx(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Pro4243173770044397973at_nat(Uua,Uub)),aa_Pro4243173770044397973at_nat(Uu,Uub)) ).

% ATP.lambda_442
tff(fact_10237_ATP_Olambda__443,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_my(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_real_nat(Uua,Uub)),aa_real_nat(Uu,Uub)) ).

% ATP.lambda_443
tff(fact_10238_ATP_Olambda__444,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_mz(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(Uua,Uub)),aa_set_nat_nat(Uu,Uub)) ).

% ATP.lambda_444
tff(fact_10239_ATP_Olambda__445,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_tf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uua,Uub)),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_445
tff(fact_10240_ATP_Olambda__446,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nb(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uua,Uub)),aa_nat_nat(Uu,Uub)) ).

% ATP.lambda_446
tff(fact_10241_ATP_Olambda__447,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_na(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(Uua,Uub)),aa_int_nat(Uu,Uub)) ).

% ATP.lambda_447
tff(fact_10242_ATP_Olambda__448,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_ky(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_448
tff(fact_10243_ATP_Olambda__449,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_kz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_449
tff(fact_10244_ATP_Olambda__450,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kx(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_450
tff(fact_10245_ATP_Olambda__451,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_sk(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_451
tff(fact_10246_ATP_Olambda__452,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ys(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_452
tff(fact_10247_ATP_Olambda__453,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_as(Uu),Uua),Uub)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu,Uub)
        | aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_453
tff(fact_10248_ATP_Olambda__454,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_ar(Uu),Uua),Uub)
    <=> ( aa_complex_o(Uu,Uub)
        | aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_454
tff(fact_10249_ATP_Olambda__455,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_am(Uu),Uua),Uub)
    <=> ( aa_real_o(Uu,Uub)
        | aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_455
tff(fact_10250_ATP_Olambda__456,axiom,
    ! [Uu: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_an(Uu),Uua),Uub)
    <=> ( aa_list_nat_o(Uu,Uub)
        | aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_456
tff(fact_10251_ATP_Olambda__457,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ao(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        | aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_457
tff(fact_10252_ATP_Olambda__458,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ap(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        | aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_458
tff(fact_10253_ATP_Olambda__459,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_aq(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        | aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_459
tff(fact_10254_ATP_Olambda__460,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_al(Uu),Uua),Uub)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu,Uub)
        & aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_460
tff(fact_10255_ATP_Olambda__461,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_ak(Uu),Uua),Uub)
    <=> ( aa_complex_o(Uu,Uub)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_461
tff(fact_10256_ATP_Olambda__462,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_af(Uu),Uua),Uub)
    <=> ( aa_real_o(Uu,Uub)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_462
tff(fact_10257_ATP_Olambda__463,axiom,
    ! [Uu: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_ag(Uu),Uua),Uub)
    <=> ( aa_list_nat_o(Uu,Uub)
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_463
tff(fact_10258_ATP_Olambda__464,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ah(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_464
tff(fact_10259_ATP_Olambda__465,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ai(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_465
tff(fact_10260_ATP_Olambda__466,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_aj(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_466
tff(fact_10261_ATP_Olambda__467,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_gi(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_467
tff(fact_10262_ATP_Olambda__468,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_sa(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_468
tff(fact_10263_ATP_Olambda__469,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_469
tff(fact_10264_ATP_Olambda__470,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sr(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uub)),Uu) ).

% ATP.lambda_470
tff(fact_10265_ATP_Olambda__471,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ss(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uub)),Uu) ).

% ATP.lambda_471
tff(fact_10266_ATP_Olambda__472,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zu(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_472
tff(fact_10267_ATP_Olambda__473,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_yr(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_473
tff(fact_10268_ATP_Olambda__474,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_ex(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_474
tff(fact_10269_ATP_Olambda__475,axiom,
    ! [Uu: product_prod_nat_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_ew(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_475
tff(fact_10270_ATP_Olambda__476,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_ey(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_476
tff(fact_10271_ATP_Olambda__477,axiom,
    ! [Uu: list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_ez(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_477
tff(fact_10272_ATP_Olambda__478,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_fa(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_478
tff(fact_10273_ATP_Olambda__479,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_fb(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_479
tff(fact_10274_ATP_Olambda__480,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_fc(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_480
tff(fact_10275_ATP_Olambda__481,axiom,
    ! [Uu: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_ti(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uua,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_481
tff(fact_10276_ATP_Olambda__482,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_ps(Uu),Uua),Uub) = plus_p5714425477246183910nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Uub)),Uua)) ).

% ATP.lambda_482
tff(fact_10277_ATP_Olambda__483,axiom,
    ! [Uu: extended_enat,Uua: extended_enat,Uub: nat] : aa_nat_Extended_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_pn(Uu),Uua),Uub) = plus_p3455044024723400733d_enat(Uu,times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uub),Uua)) ).

% ATP.lambda_483
tff(fact_10278_ATP_Olambda__484,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_pm(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_484
tff(fact_10279_ATP_Olambda__485,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pr(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),Uua)) ).

% ATP.lambda_485
tff(fact_10280_ATP_Olambda__486,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_po(Uu),Uua),Uub) = aa_rat_rat(plus_plus_rat(Uu),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ).

% ATP.lambda_486
tff(fact_10281_ATP_Olambda__487,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pq(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Uub)),Uua)) ).

% ATP.lambda_487
tff(fact_10282_ATP_Olambda__488,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_pp(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Uub)),Uua)) ).

% ATP.lambda_488
tff(fact_10283_ATP_Olambda__489,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ri(Uu),Uua),Uub) = aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_489
tff(fact_10284_ATP_Olambda__490,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_rl(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_490
tff(fact_10285_ATP_Olambda__491,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_rj(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_491
tff(fact_10286_ATP_Olambda__492,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_rk(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_492
tff(fact_10287_ATP_Olambda__493,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zm(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_493
tff(fact_10288_ATP_Olambda__494,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_oa(Uu),Uua),Uub) = aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_494
tff(fact_10289_ATP_Olambda__495,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_od(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_495
tff(fact_10290_ATP_Olambda__496,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_ob(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_496
tff(fact_10291_ATP_Olambda__497,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_oc(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_497
tff(fact_10292_ATP_Olambda__498,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_js(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ).

% ATP.lambda_498
tff(fact_10293_ATP_Olambda__499,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_zt(Uu),Uua),Uub)
    <=> aa_set_real_o(member_real(Uub),set_or1633881224788618240n_real(Uu,Uua)) ) ).

% ATP.lambda_499
tff(fact_10294_ATP_Olambda__500,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ry(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_500
tff(fact_10295_ATP_Olambda__501,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_rz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_501
tff(fact_10296_ATP_Olambda__502,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_qo(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_502
tff(fact_10297_ATP_Olambda__503,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qn(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_503
tff(fact_10298_ATP_Olambda__504,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_th(Uu),Uua),Uub) = aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_504
tff(fact_10299_ATP_Olambda__505,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_np(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_505
tff(fact_10300_ATP_Olambda__506,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_zs(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_506
tff(fact_10301_ATP_Olambda__507,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_no(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_507
tff(fact_10302_ATP_Olambda__508,axiom,
    ! [Uu: fun_Pr2602117396471556196at_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jj(Uu),Uua),Uub) = aa_Pro4243173770044397973at_nat(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_508
tff(fact_10303_ATP_Olambda__509,axiom,
    ! [Uu: fun_Pr2477239776880934807at_nat,Uua: nat,Uub: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_jk(Uu),Uua),Uub) = aa_Pro8023792847356852296at_nat(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_509
tff(fact_10304_ATP_Olambda__510,axiom,
    ! [Uu: fun_Pr2028404973800837934_nat_o,Uua: nat,Uub: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_jl(Uu),Uua),Uub) = aa_Pro989595756709598047_nat_o(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_510
tff(fact_10305_ATP_Olambda__511,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_jm(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_511
tff(fact_10306_ATP_Olambda__512,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_jn(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ) ).

% ATP.lambda_512
tff(fact_10307_ATP_Olambda__513,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_su(Uu),Uua),Uub) = aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_513
tff(fact_10308_ATP_Olambda__514,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_sv(Uu),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_514
tff(fact_10309_ATP_Olambda__515,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_sq(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub))) ).

% ATP.lambda_515
tff(fact_10310_ATP_Olambda__516,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_so(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub))) ).

% ATP.lambda_516
tff(fact_10311_ATP_Olambda__517,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_complex,Uub: complex] :
      ( aa_complex_o(aa_fun1011824800444484462plex_o(aTP_Lamm_ga(Uu),Uua),Uub)
    <=> ? [X4: real] :
          ( ( Uub = aa_real_complex(Uua,X4) )
          & aa_real_o(Uu,X4) ) ) ).

% ATP.lambda_517
tff(fact_10312_ATP_Olambda__518,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_fx(Uu),Uua),Uub)
    <=> ? [X4: real] :
          ( ( Uub = aa_real_real(Uua,X4) )
          & aa_real_o(Uu,X4) ) ) ).

% ATP.lambda_518
tff(fact_10313_ATP_Olambda__519,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_nat,Uub: nat] :
      ( aa_nat_o(aa_fun2333486890442116906_nat_o(aTP_Lamm_fy(Uu),Uua),Uub)
    <=> ? [X4: real] :
          ( ( Uub = aa_real_nat(Uua,X4) )
          & aa_real_o(Uu,X4) ) ) ).

% ATP.lambda_519
tff(fact_10314_ATP_Olambda__520,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_int,Uub: int] :
      ( aa_int_o(aa_fun3242049114608245362_int_o(aTP_Lamm_fz(Uu),Uua),Uub)
    <=> ? [X4: real] :
          ( ( Uub = aa_real_int(Uua,X4) )
          & aa_real_o(Uu,X4) ) ) ).

% ATP.lambda_520
tff(fact_10315_ATP_Olambda__521,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_complex,Uub: complex] :
      ( aa_complex_o(aa_fun2681562978987535434plex_o(aTP_Lamm_ge(Uu),Uua),Uub)
    <=> ? [X4: nat] :
          ( ( Uub = aa_nat_complex(Uua,X4) )
          & aa_nat_o(Uu,X4) ) ) ).

% ATP.lambda_521
tff(fact_10316_ATP_Olambda__522,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_real,Uub: real] :
      ( aa_real_o(aa_fun6953832176698369998real_o(aTP_Lamm_gb(Uu),Uua),Uub)
    <=> ? [X4: nat] :
          ( ( Uub = aa_nat_real(Uua,X4) )
          & aa_nat_o(Uu,X4) ) ) ).

% ATP.lambda_522
tff(fact_10317_ATP_Olambda__523,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_gc(Uu),Uua),Uub)
    <=> ? [X4: nat] :
          ( ( Uub = aa_nat_nat(Uua,X4) )
          & aa_nat_o(Uu,X4) ) ) ).

% ATP.lambda_523
tff(fact_10318_ATP_Olambda__524,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_int,Uub: int] :
      ( aa_int_o(aa_fun359159518738836302_int_o(aTP_Lamm_gd(Uu),Uua),Uub)
    <=> ? [X4: nat] :
          ( ( Uub = aa_nat_int(Uua,X4) )
          & aa_nat_o(Uu,X4) ) ) ).

% ATP.lambda_524
tff(fact_10319_ATP_Olambda__525,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_real,Uub: real] :
      ( aa_real_o(aa_fun7191409318348945010real_o(aTP_Lamm_gf(Uu),Uua),Uub)
    <=> ? [X4: int] :
          ( ( Uub = aa_int_real(Uua,X4) )
          & aa_int_o(Uu,X4) ) ) ).

% ATP.lambda_525
tff(fact_10320_ATP_Olambda__526,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_nat,Uub: nat] :
      ( aa_nat_o(aa_fun7246820503061074602_nat_o(aTP_Lamm_gg(Uu),Uua),Uub)
    <=> ? [X4: int] :
          ( ( Uub = aa_int_nat(Uua,X4) )
          & aa_int_o(Uu,X4) ) ) ).

% ATP.lambda_526
tff(fact_10321_ATP_Olambda__527,axiom,
    ! [Uu: fun_real_o,Uua: fun_co4998825143550818160real_o,Uub: complex] :
      ( aa_complex_o(aa_fun4501470310227681423plex_o(aTP_Lamm_az(Uu),Uua),Uub)
    <=> ? [Y5: real] :
          ( aa_real_o(Uu,Y5)
          & aa_real_o(aa_com1951324360330529881real_o(Uua,Uub),Y5) ) ) ).

% ATP.lambda_527
tff(fact_10322_ATP_Olambda__528,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun8588574755773516687real_o(aTP_Lamm_at(Uu),Uua),Uub)
    <=> ? [Y5: real] :
          ( aa_real_o(Uu,Y5)
          & aa_real_o(aa_real_fun_real_o(Uua,Uub),Y5) ) ) ).

% ATP.lambda_528
tff(fact_10323_ATP_Olambda__529,axiom,
    ! [Uu: fun_real_o,Uua: fun_nat_fun_real_o,Uub: nat] :
      ( aa_nat_o(aa_fun1522457866930155023_nat_o(aTP_Lamm_av(Uu),Uua),Uub)
    <=> ? [Y5: real] :
          ( aa_real_o(Uu,Y5)
          & aa_real_o(aa_nat_fun_real_o(Uua,Uub),Y5) ) ) ).

% ATP.lambda_529
tff(fact_10324_ATP_Olambda__530,axiom,
    ! [Uu: fun_real_o,Uua: fun_int_fun_real_o,Uub: int] :
      ( aa_int_o(aa_fun8063507495001755407_int_o(aTP_Lamm_ax(Uu),Uua),Uub)
    <=> ? [Y5: real] :
          ( aa_real_o(Uu,Y5)
          & aa_real_o(aa_int_fun_real_o(Uua,Uub),Y5) ) ) ).

% ATP.lambda_530
tff(fact_10325_ATP_Olambda__531,axiom,
    ! [Uu: fun_nat_o,Uua: fun_co5676208753217392844_nat_o,Uub: complex] :
      ( aa_complex_o(aa_fun3010802591474140851plex_o(aTP_Lamm_bh(Uu),Uua),Uub)
    <=> ? [Y5: nat] :
          ( aa_nat_o(Uu,Y5)
          & aa_nat_o(aa_complex_fun_nat_o(Uua,Uub),Y5) ) ) ).

% ATP.lambda_531
tff(fact_10326_ATP_Olambda__532,axiom,
    ! [Uu: fun_nat_o,Uua: fun_real_fun_nat_o,Uub: real] :
      ( aa_real_o(aa_fun8982401453414893235real_o(aTP_Lamm_bb(Uu),Uua),Uub)
    <=> ? [Y5: nat] :
          ( aa_nat_o(Uu,Y5)
          & aa_nat_o(aa_real_fun_nat_o(Uua,Uub),Y5) ) ) ).

% ATP.lambda_532
tff(fact_10327_ATP_Olambda__533,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun6854469565811772723_nat_o(aTP_Lamm_bd(Uu),Uua),Uub)
    <=> ? [Y5: nat] :
          ( aa_nat_o(Uu,Y5)
          & aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Y5) ) ) ).

% ATP.lambda_533
tff(fact_10328_ATP_Olambda__534,axiom,
    ! [Uu: fun_nat_o,Uua: fun_int_fun_nat_o,Uub: int] :
      ( aa_int_o(aa_fun7458983532533439539_int_o(aTP_Lamm_bf(Uu),Uua),Uub)
    <=> ? [Y5: nat] :
          ( aa_nat_o(Uu,Y5)
          & aa_nat_o(aa_int_fun_nat_o(Uua,Uub),Y5) ) ) ).

% ATP.lambda_534
tff(fact_10329_ATP_Olambda__535,axiom,
    ! [Uu: fun_int_o,Uua: fun_real_fun_int_o,Uub: real] :
      ( aa_real_o(aa_fun3710474044975826703real_o(aTP_Lamm_bj(Uu),Uua),Uub)
    <=> ? [Y5: int] :
          ( aa_int_o(Uu,Y5)
          & aa_int_o(aa_real_fun_int_o(Uua,Uub),Y5) ) ) ).

% ATP.lambda_535
tff(fact_10330_ATP_Olambda__536,axiom,
    ! [Uu: fun_int_o,Uua: fun_nat_fun_int_o,Uub: nat] :
      ( aa_nat_o(aa_fun3042179710660459407_nat_o(aTP_Lamm_bl(Uu),Uua),Uub)
    <=> ? [Y5: int] :
          ( aa_int_o(Uu,Y5)
          & aa_int_o(aa_nat_fun_int_o(Uua,Uub),Y5) ) ) ).

% ATP.lambda_536
tff(fact_10331_ATP_Olambda__537,axiom,
    ! [Uu: fun_nat_real,Uua: set_nat,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_tg(Uu),Uua),Uub),Uuc) = $ite(aa_set_nat_o(member_nat(Uuc),Uua),aa_nat_real(Uub,Uuc),aa_nat_real(Uu,Uuc)) ).

% ATP.lambda_537
tff(fact_10332_ATP_Olambda__538,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_ol(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_real(Uua,Uuc),aa_complex_real(Uub,Uuc)) ).

% ATP.lambda_538
tff(fact_10333_ATP_Olambda__539,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_om(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_rat(Uua,Uuc),aa_complex_rat(Uub,Uuc)) ).

% ATP.lambda_539
tff(fact_10334_ATP_Olambda__540,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] :
      aa_complex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_on(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_nat(Uua,Uuc),aa_complex_nat(Uub,Uuc)) ).

% ATP.lambda_540
tff(fact_10335_ATP_Olambda__541,axiom,
    ! [Uu: complex,Uua: fun_complex_int,Uub: fun_complex_int,Uuc: complex] :
      aa_complex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_oo(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_int(Uua,Uuc),aa_complex_int(Uub,Uuc)) ).

% ATP.lambda_541
tff(fact_10336_ATP_Olambda__542,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_op(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_real_real(Uua,Uuc),aa_real_real(Uub,Uuc)) ).

% ATP.lambda_542
tff(fact_10337_ATP_Olambda__543,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_oq(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_real_rat(Uua,Uuc),aa_real_rat(Uub,Uuc)) ).

% ATP.lambda_543
tff(fact_10338_ATP_Olambda__544,axiom,
    ! [Uu: real,Uua: fun_real_nat,Uub: fun_real_nat,Uuc: real] :
      aa_real_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_or(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_real_nat(Uua,Uuc),aa_real_nat(Uub,Uuc)) ).

% ATP.lambda_544
tff(fact_10339_ATP_Olambda__545,axiom,
    ! [Uu: real,Uua: fun_real_int,Uub: fun_real_int,Uuc: real] :
      aa_real_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_os(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_real_int(Uua,Uuc),aa_real_int(Uub,Uuc)) ).

% ATP.lambda_545
tff(fact_10340_ATP_Olambda__546,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_ot(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_int_real(Uua,Uuc),aa_int_real(Uub,Uuc)) ).

% ATP.lambda_546
tff(fact_10341_ATP_Olambda__547,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_ou(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_int_rat(Uua,Uuc),aa_int_rat(Uub,Uuc)) ).

% ATP.lambda_547
tff(fact_10342_ATP_Olambda__548,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qg(Uu),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qf(Uu),Uua),Uub),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_548
tff(fact_10343_ATP_Olambda__549,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_qe(Uu),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_qd(Uu),Uua),Uub),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_549
tff(fact_10344_ATP_Olambda__550,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_qc(Uu),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_qb(Uu),Uua),Uub),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_550
tff(fact_10345_ATP_Olambda__551,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_real_nat,Uub: fun_nat_fun_real_o,Uuc: real] : aa_real_nat(aa_fun1455215566758156043al_nat(aa_fun1180509458258223190al_nat(aTP_Lamm_ml(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_real_fun_nat_nat(aTP_Lamm_mj(Uua),Uuc)),collect_nat(aa_real_fun_nat_o(aa_fun7270066119382671164_nat_o(aTP_Lamm_mk(Uu),Uub),Uuc))) ).

% ATP.lambda_551
tff(fact_10346_ATP_Olambda__552,axiom,
    ! [Uu: set_complex,Uua: fun_co8378349014452122532t_real,Uub: fun_co5676208753217392844_nat_o,Uuc: nat] : aa_nat_real(aa_fun8607523022212034825t_real(aa_fun982078519811717750t_real(aTP_Lamm_mg(Uu),Uua),Uub),Uuc) = groups5808333547571424918x_real(aa_nat480766339200631653x_real(aTP_Lamm_mf(Uua),Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_fm(Uu),Uub),Uuc))) ).

% ATP.lambda_552
tff(fact_10347_ATP_Olambda__553,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_real,Uub: fun_int_fun_nat_o,Uuc: nat] : aa_nat_real(aa_fun5195266948229132043t_real(aa_fun3400660822709534074t_real(aTP_Lamm_md(Uu),Uua),Uub),Uuc) = groups8778361861064173332t_real(aa_nat_fun_int_real(aTP_Lamm_mc(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_fj(Uu),Uub),Uuc))) ).

% ATP.lambda_553
tff(fact_10348_ATP_Olambda__554,axiom,
    ! [Uu: set_real,Uua: fun_re4238438383274954274t_real,Uub: fun_real_fun_nat_o,Uuc: nat] : aa_nat_real(aa_fun6247686994378451851t_real(aa_fun7274880274332377722t_real(aTP_Lamm_ma(Uu),Uua),Uub),Uuc) = groups8097168146408367636l_real(aa_nat_fun_real_real(aTP_Lamm_lz(Uua),Uuc),collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_fd(Uu),Uub),Uuc))) ).

% ATP.lambda_554
tff(fact_10349_ATP_Olambda__555,axiom,
    ! [Uu: set_int,Uua: fun_in1307110748760623106omplex,Uub: fun_in5065666251602127472plex_o,Uuc: complex] : aa_complex_complex(aa_fun7450476157521870733omplex(aa_fun6156949752960786844omplex(aTP_Lamm_lx(Uu),Uua),Uub),Uuc) = groups3049146728041665814omplex(aa_com9156536935073639875omplex(aTP_Lamm_lw(Uua),Uuc),collect_int(aa_complex_fun_int_o(aa_fun7455058932602313788_int_o(aTP_Lamm_fl(Uu),Uub),Uuc))) ).

% ATP.lambda_555
tff(fact_10350_ATP_Olambda__556,axiom,
    ! [Uu: set_nat,Uua: fun_na1591161823130732966omplex,Uub: fun_na1554462459066888140plex_o,Uuc: complex] : aa_complex_complex(aa_fun1423577888295648489omplex(aa_fun601428455815558228omplex(aTP_Lamm_lt(Uu),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ls(Uua),Uuc),collect_nat(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_fi(Uu),Uub),Uuc))) ).

% ATP.lambda_556
tff(fact_10351_ATP_Olambda__557,axiom,
    ! [Uu: set_real,Uua: fun_re7208826350674946562omplex,Uub: fun_re7319081385389557872plex_o,Uuc: complex] : aa_complex_complex(aa_fun9084337531551611917omplex(aa_fun6739043078600750748omplex(aTP_Lamm_lp(Uu),Uua),Uub),Uuc) = groups5754745047067104278omplex(aa_com4766338864777664963omplex(aTP_Lamm_lo(Uua),Uuc),collect_real(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_ff(Uu),Uub),Uuc))) ).

% ATP.lambda_557
tff(fact_10352_ATP_Olambda__558,axiom,
    ! [Uu: set_complex,Uua: fun_co636942507509095880at_nat,Uub: fun_co5676208753217392844_nat_o,Uuc: nat] : aa_nat_nat(aa_fun5527826185635525933at_nat(aa_fun4894236392271416310at_nat(aTP_Lamm_ll(Uu),Uua),Uub),Uuc) = groups5693394587270226106ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_lk(Uua),Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_fm(Uu),Uub),Uuc))) ).

% ATP.lambda_558
tff(fact_10353_ATP_Olambda__559,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_lh(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_lg(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_fj(Uu),Uub),Uuc))) ).

% ATP.lambda_559
tff(fact_10354_ATP_Olambda__560,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_nat,Uub: fun_real_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_ld(Uu),Uua),Uub),Uuc) = groups1935376822645274424al_nat(aa_nat_fun_real_nat(aTP_Lamm_lc(Uua),Uuc),collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_fd(Uu),Uub),Uuc))) ).

% ATP.lambda_560
tff(fact_10355_ATP_Olambda__561,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_we(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_wd(Uu),Uua),Uuc)),set_ord_atMost_nat(Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uub),Uuc)) ).

% ATP.lambda_561
tff(fact_10356_ATP_Olambda__562,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_yy(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_562
tff(fact_10357_ATP_Olambda__563,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_yw(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_563
tff(fact_10358_ATP_Olambda__564,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_yu(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uub),Uua)),Uuc)) ).

% ATP.lambda_564
tff(fact_10359_ATP_Olambda__565,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_yv(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uua),Uub)),Uuc)) ).

% ATP.lambda_565
tff(fact_10360_ATP_Olambda__566,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_qs(Uu),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(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uu),Uub)) ).

% ATP.lambda_566
tff(fact_10361_ATP_Olambda__567,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qy(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_567
tff(fact_10362_ATP_Olambda__568,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_qu(Uu),Uua),Uub),Uuc) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),Uuc))),aa_nat_rat(power_power_rat(Uu),Uub)) ).

% ATP.lambda_568
tff(fact_10363_ATP_Olambda__569,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_qw(Uu),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,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu),Uuc))),aa_nat_int(power_power_int(Uu),Uub)) ).

% ATP.lambda_569
tff(fact_10364_ATP_Olambda__570,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_xo(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uub))) ).

% ATP.lambda_570
tff(fact_10365_ATP_Olambda__571,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_qb(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uua),Uu)),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_571
tff(fact_10366_ATP_Olambda__572,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qf(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uua),Uu)),Uuc)),aa_nat_real(aa_real_fun_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_572
tff(fact_10367_ATP_Olambda__573,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_qd(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uua),Uu)),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_573
tff(fact_10368_ATP_Olambda__574,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_wb(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_574
tff(fact_10369_ATP_Olambda__575,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_re(Uu),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(Uu),Uuc)) ).

% ATP.lambda_575
tff(fact_10370_ATP_Olambda__576,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_rh(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_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(aa_real_fun_nat_real(power_power_real,Uu),Uuc)) ).

% ATP.lambda_576
tff(fact_10371_ATP_Olambda__577,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_rf(Uu),Uua),Uub),Uuc) = aa_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(Uu),Uuc)) ).

% ATP.lambda_577
tff(fact_10372_ATP_Olambda__578,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_rg(Uu),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(Uu),Uuc)) ).

% ATP.lambda_578
tff(fact_10373_ATP_Olambda__579,axiom,
    ! [Uu: set_complex,Uua: fun_re7319081385389557872plex_o,Uub: real,Uuc: complex] :
      ( aa_complex_o(aa_rea2382570592068293209plex_o(aa_fun789424914597169340plex_o(aTP_Lamm_lm(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_complex_o(aa_rea2382570592068293209plex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_579
tff(fact_10374_ATP_Olambda__580,axiom,
    ! [Uu: set_complex,Uua: fun_na1554462459066888140plex_o,Uub: nat,Uuc: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aa_fun6810266893190567484plex_o(aTP_Lamm_lq(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_complex_o(aa_nat_fun_complex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_580
tff(fact_10375_ATP_Olambda__581,axiom,
    ! [Uu: set_complex,Uua: fun_in5065666251602127472plex_o,Uub: int,Uuc: complex] :
      ( aa_complex_o(aa_int_fun_complex_o(aa_fun9089661398657494332plex_o(aTP_Lamm_lu(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_complex_o(aa_int_fun_complex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_581
tff(fact_10376_ATP_Olambda__582,axiom,
    ! [Uu: set_real,Uua: fun_nat_fun_real_o,Uub: nat,Uuc: real] :
      ( aa_real_o(aa_nat_fun_real_o(aa_fun1719600354486425148real_o(aTP_Lamm_mh(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & aa_real_o(aa_nat_fun_real_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_582
tff(fact_10377_ATP_Olambda__583,axiom,
    ! [Uu: set_nat,Uua: fun_co5676208753217392844_nat_o,Uub: complex,Uuc: nat] :
      ( aa_nat_o(aa_complex_fun_nat_o(aa_fun8732346023647761980_nat_o(aTP_Lamm_li(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_complex_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_583
tff(fact_10378_ATP_Olambda__584,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_o,Uub: real,Uuc: nat] :
      ( aa_nat_o(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_la(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_real_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_584
tff(fact_10379_ATP_Olambda__585,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_o,Uub: int,Uuc: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_le(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_int_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_585
tff(fact_10380_ATP_Olambda__586,axiom,
    ! [Uu: set_complex,Uua: fun_co5676208753217392844_nat_o,Uub: nat,Uuc: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_fm(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_nat_o(aa_complex_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_586
tff(fact_10381_ATP_Olambda__587,axiom,
    ! [Uu: set_real,Uua: fun_re7319081385389557872plex_o,Uub: complex,Uuc: real] :
      ( aa_real_o(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_ff(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & aa_complex_o(aa_rea2382570592068293209plex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_587
tff(fact_10382_ATP_Olambda__588,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_o,Uub: nat,Uuc: real] :
      ( aa_real_o(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_fd(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & aa_nat_o(aa_real_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_588
tff(fact_10383_ATP_Olambda__589,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_int_o,Uub: int,Uuc: real] :
      ( aa_real_o(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_fe(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & aa_int_o(aa_real_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_589
tff(fact_10384_ATP_Olambda__590,axiom,
    ! [Uu: set_nat,Uua: fun_na1554462459066888140plex_o,Uub: complex,Uuc: nat] :
      ( aa_nat_o(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_fi(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_complex_o(aa_nat_fun_complex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_590
tff(fact_10385_ATP_Olambda__591,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_real_o,Uub: real,Uuc: nat] :
      ( aa_nat_o(aa_real_fun_nat_o(aa_fun7270066119382671164_nat_o(aTP_Lamm_mk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_real_o(aa_nat_fun_real_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_591
tff(fact_10386_ATP_Olambda__592,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_o,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_fg(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_592
tff(fact_10387_ATP_Olambda__593,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_o,Uub: int,Uuc: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_fh(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_int_o(aa_nat_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_593
tff(fact_10388_ATP_Olambda__594,axiom,
    ! [Uu: set_int,Uua: fun_in5065666251602127472plex_o,Uub: complex,Uuc: int] :
      ( aa_int_o(aa_complex_fun_int_o(aa_fun7455058932602313788_int_o(aTP_Lamm_fl(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_complex_o(aa_int_fun_complex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_594
tff(fact_10389_ATP_Olambda__595,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_o,Uub: nat,Uuc: int] :
      ( aa_int_o(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_fj(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_nat_o(aa_int_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_595
tff(fact_10390_ATP_Olambda__596,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_fk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_596
tff(fact_10391_ATP_Olambda__597,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_qt(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu),Uuc)),aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_597
tff(fact_10392_ATP_Olambda__598,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qz(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_598
tff(fact_10393_ATP_Olambda__599,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_qv(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu),Uuc)),aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_599
tff(fact_10394_ATP_Olambda__600,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_qx(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu),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,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_600
tff(fact_10395_ATP_Olambda__601,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ra(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu),Uuc)),aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_601
tff(fact_10396_ATP_Olambda__602,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_rd(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_602
tff(fact_10397_ATP_Olambda__603,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_rb(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu),Uuc)),aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_603
tff(fact_10398_ATP_Olambda__604,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_rc(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu),Uuc)),aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_604
tff(fact_10399_ATP_Olambda__605,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_wa(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_605
tff(fact_10400_ATP_Olambda__606,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_xt(Uu),Uua),Uub),Uuc)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_606
tff(fact_10401_ATP_Olambda__607,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_wv(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).

% ATP.lambda_607
tff(fact_10402_ATP_Olambda__608,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_xr(Uu),Uua),Uub),Uuc)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_608
tff(fact_10403_ATP_Olambda__609,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_xv(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)) ).

% ATP.lambda_609
tff(fact_10404_ATP_Olambda__610,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_xx(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_610
tff(fact_10405_ATP_Olambda__611,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_hm(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_complex_complex(times_times_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_611
tff(fact_10406_ATP_Olambda__612,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_hu(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_612
tff(fact_10407_ATP_Olambda__613,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_hg(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_complex_complex(times_times_complex(aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_613
tff(fact_10408_ATP_Olambda__614,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_ho(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_614
tff(fact_10409_ATP_Olambda__615,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_hw(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_615
tff(fact_10410_ATP_Olambda__616,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_hi(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_616
tff(fact_10411_ATP_Olambda__617,axiom,
    ! [Uu: 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_hq(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_617
tff(fact_10412_ATP_Olambda__618,axiom,
    ! [Uu: 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_hy(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_618
tff(fact_10413_ATP_Olambda__619,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_hk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_complex_complex(times_times_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_619
tff(fact_10414_ATP_Olambda__620,axiom,
    ! [Uu: 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_hs(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_620
tff(fact_10415_ATP_Olambda__621,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_ig(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_complex_complex(plus_plus_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_621
tff(fact_10416_ATP_Olambda__622,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_io(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_622
tff(fact_10417_ATP_Olambda__623,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_ia(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_complex_complex(plus_plus_complex(aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_623
tff(fact_10418_ATP_Olambda__624,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_ii(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_624
tff(fact_10419_ATP_Olambda__625,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_iq(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_rat_rat(plus_plus_rat(aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_625
tff(fact_10420_ATP_Olambda__626,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_ic(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_complex_complex(plus_plus_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_626
tff(fact_10421_ATP_Olambda__627,axiom,
    ! [Uu: 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_ik(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_627
tff(fact_10422_ATP_Olambda__628,axiom,
    ! [Uu: 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_is(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_rat_rat(plus_plus_rat(aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_628
tff(fact_10423_ATP_Olambda__629,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_ie(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_complex_complex(plus_plus_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_629
tff(fact_10424_ATP_Olambda__630,axiom,
    ! [Uu: 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_im(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_630
tff(fact_10425_ATP_Olambda__631,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_sz(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_631
tff(fact_10426_ATP_Olambda__632,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ta(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_632
tff(fact_10427_ATP_Olambda__633,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_wd(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uuc)),aa_nat_nat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_633
tff(fact_10428_ATP_Olambda__634,axiom,
    ! [Uu: set_real,Uua: fun_nat_fun_real_nat,Uub: fun_nat_fun_real_o,Uuc: nat] : aa_nat_nat(aa_fun6477086565944264623at_nat(aa_fun888394026836741882at_nat(aTP_Lamm_mi(Uu),Uua),Uub),Uuc) = groups1935376822645274424al_nat(aa_nat_fun_real_nat(Uua,Uuc),collect_real(aa_nat_fun_real_o(aa_fun1719600354486425148real_o(aTP_Lamm_mh(Uu),Uub),Uuc))) ).

% ATP.lambda_634
tff(fact_10429_ATP_Olambda__635,axiom,
    ! [Uu: set_complex,Uua: fun_in1307110748760623106omplex,Uub: fun_in5065666251602127472plex_o,Uuc: int] : aa_int_complex(aa_fun3141780957025892107omplex(aa_fun323909820831547930omplex(aTP_Lamm_lv(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_int3545933452204062147omplex(Uua,Uuc),collect_complex(aa_int_fun_complex_o(aa_fun9089661398657494332plex_o(aTP_Lamm_lu(Uu),Uub),Uuc))) ).

% ATP.lambda_635
tff(fact_10430_ATP_Olambda__636,axiom,
    ! [Uu: set_complex,Uua: fun_na1591161823130732966omplex,Uub: fun_na1554462459066888140plex_o,Uuc: nat] : aa_nat_complex(aa_fun4713049822795992587omplex(aa_fun3479084698764006518omplex(aTP_Lamm_lr(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(Uua,Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun6810266893190567484plex_o(aTP_Lamm_lq(Uu),Uub),Uuc))) ).

% ATP.lambda_636
tff(fact_10431_ATP_Olambda__637,axiom,
    ! [Uu: set_complex,Uua: fun_re7208826350674946562omplex,Uub: fun_re7319081385389557872plex_o,Uuc: real] : aa_real_complex(aa_fun401690246322682763omplex(aa_fun7501666206714588442omplex(aTP_Lamm_ln(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_rea2618301233319580611omplex(Uua,Uuc),collect_complex(aa_rea2382570592068293209plex_o(aa_fun789424914597169340plex_o(aTP_Lamm_lm(Uu),Uub),Uuc))) ).

% ATP.lambda_637
tff(fact_10432_ATP_Olambda__638,axiom,
    ! [Uu: set_nat,Uua: fun_co8378349014452122532t_real,Uub: fun_co5676208753217392844_nat_o,Uuc: complex] : aa_complex_real(aa_fun2831877435191659751x_real(aa_fun3061107605873050452x_real(aTP_Lamm_me(Uu),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_com20334259897932645t_real(Uua,Uuc),collect_nat(aa_complex_fun_nat_o(aa_fun8732346023647761980_nat_o(aTP_Lamm_li(Uu),Uub),Uuc))) ).

% ATP.lambda_638
tff(fact_10433_ATP_Olambda__639,axiom,
    ! [Uu: set_nat,Uua: fun_co636942507509095880at_nat,Uub: fun_co5676208753217392844_nat_o,Uuc: complex] : aa_complex_nat(aa_fun2265263913897893899ex_nat(aa_fun1127489917733235668ex_nat(aTP_Lamm_lj(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_com6383720667567721865at_nat(Uua,Uuc)),collect_nat(aa_complex_fun_nat_o(aa_fun8732346023647761980_nat_o(aTP_Lamm_li(Uu),Uub),Uuc))) ).

% ATP.lambda_639
tff(fact_10434_ATP_Olambda__640,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_real,Uub: fun_int_fun_nat_o,Uuc: int] : aa_int_real(aa_fun7382188522922153831t_real(aa_fun6619364416873904086t_real(aTP_Lamm_mb(Uu),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_int_fun_nat_real(Uua,Uuc),collect_nat(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_le(Uu),Uub),Uuc))) ).

% ATP.lambda_640
tff(fact_10435_ATP_Olambda__641,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_o,Uuc: int] : aa_int_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_lf(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_int_fun_nat_nat(Uua,Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_le(Uu),Uub),Uuc))) ).

% ATP.lambda_641
tff(fact_10436_ATP_Olambda__642,axiom,
    ! [Uu: set_nat,Uua: fun_re4238438383274954274t_real,Uub: fun_real_fun_nat_o,Uuc: real] : aa_real_real(aa_fun6320755270648597735l_real(aa_fun6240130701872165846l_real(aTP_Lamm_ly(Uu),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_real_fun_nat_real(Uua,Uuc),collect_nat(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_la(Uu),Uub),Uuc))) ).

% ATP.lambda_642
tff(fact_10437_ATP_Olambda__643,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_nat,Uub: fun_real_fun_nat_o,Uuc: real] : aa_real_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_lb(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_real_fun_nat_nat(Uua,Uuc)),collect_nat(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_la(Uu),Uub),Uuc))) ).

% ATP.lambda_643
tff(fact_10438_ATP_Olambda__644,axiom,
    ! [Uu: $o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_pg((Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_644
tff(fact_10439_ATP_Olambda__645,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4152872853588681340nt_int(aTP_Lamm_jf(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_645
tff(fact_10440_ATP_Olambda__646,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun4545072666309863515_int_o(aTP_Lamm_jb(Uu),Uua),Uub),Uuc)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_646
tff(fact_10441_ATP_Olambda__647,axiom,
    ! [Uu: fun_o_8988054265482332260nt_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4023131440832735197nt_int(aTP_Lamm_jc(Uu),Uua),Uub),Uuc) = aa_o_P3134438092330336605nt_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_647
tff(fact_10442_ATP_Olambda__648,axiom,
    ! [Uu: fun_o_o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_ja(Uu),Uua),Uub),Uuc)
    <=> aa_o_o(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_648
tff(fact_10443_ATP_Olambda__649,axiom,
    ! [Uu: 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_iz(Uu),Uua),Uub),Uuc) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(Uua,Uub),Uuc)) ).

% ATP.lambda_649
tff(fact_10444_ATP_Olambda__650,axiom,
    ! [Uu: fun_na8713714802964096592at_nat,Uua: fun_nat_fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aa_fun2484538639213223237at_nat(aTP_Lamm_jg(Uu),Uua),Uub),Uuc) = aa_nat7997280740862404369at_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(Uua,Uub),Uuc)) ).

% ATP.lambda_650
tff(fact_10445_ATP_Olambda__651,axiom,
    ! [Uu: 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_jd(Uu),Uua),Uub),Uuc) = aa_nat7695548938202568488_nat_o(Uu,aa_nat_nat(aa_nat_fun_nat_nat(Uua,Uub),Uuc)) ).

% ATP.lambda_651
tff(fact_10446_ATP_Olambda__652,axiom,
    ! [Uu: fun_fu1413608398870722896at_nat,Uua: fun_na1668100129766719005at_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun8817627472026143685at_nat(aTP_Lamm_jh(Uu),Uua),Uub),Uuc) = aa_fun7582795104022253201at_nat(Uu,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(Uua,Uub),Uuc)) ).

% ATP.lambda_652
tff(fact_10447_ATP_Olambda__653,axiom,
    ! [Uu: 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_je(Uu),Uua),Uub),Uuc) = aa_fun1432809138467554728_o_nat(Uu,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uua,Uub),Uuc)) ).

% ATP.lambda_653
tff(fact_10448_ATP_Olambda__654,axiom,
    ! [Uu: fun_fu3275390434952296369_nat_o,Uua: fun_na5914473681949881652_nat_o,Uub: nat,Uuc: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aa_fun824319262034445606_nat_o(aTP_Lamm_ji(Uu),Uua),Uub),Uuc) = aa_fun359341089595923442_nat_o(Uu,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uua,Uub),Uuc)) ).

% ATP.lambda_654
tff(fact_10449_ATP_Olambda__655,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_of(Uu),Uua),Uub),Uuc) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_655
tff(fact_10450_ATP_Olambda__656,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_oe(Uu),Uua),Uub),Uuc) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_656
tff(fact_10451_ATP_Olambda__657,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_wq(Uu),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,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_657
tff(fact_10452_ATP_Olambda__658,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ws(Uu),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,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_658
tff(fact_10453_ATP_Olambda__659,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_wo(Uu),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,Uu),Uub)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_659
tff(fact_10454_ATP_Olambda__660,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_wm(Uu),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,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ).

% ATP.lambda_660
tff(fact_10455_ATP_Olambda__661,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_o,Uub: fun_re7812779276277946752omplex,Uuc: complex] :
      ( aa_complex_o(aa_fun890640625120065285plex_o(aa_fun6537733294925973843plex_o(aTP_Lamm_fq(Uu),Uua),Uub),Uuc)
    <=> ? [X4: real,Y5: real] :
          ( ( Uuc = aa_real_complex(aa_rea2649881322057434177omplex(Uub,X4),Y5) )
          & aa_real_o(Uu,X4)
          & aa_real_o(Uua,Y5) ) ) ).

% ATP.lambda_661
tff(fact_10456_ATP_Olambda__662,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_o,Uub: fun_re7681423839302404990l_real,Uuc: real] :
      ( aa_real_o(aa_fun6833002213068288777real_o(aa_fun8577581014811940695real_o(aTP_Lamm_fn(Uu),Uua),Uub),Uuc)
    <=> ? [X4: real,Y5: real] :
          ( ( Uuc = aa_real_real(aa_rea7340605660300375871l_real(Uub,X4),Y5) )
          & aa_real_o(Uu,X4)
          & aa_real_o(Uua,Y5) ) ) ).

% ATP.lambda_662
tff(fact_10457_ATP_Olambda__663,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_o,Uub: fun_re7428432522468557474al_nat,Uuc: nat] :
      ( aa_nat_o(aa_fun6822315102276328897_nat_o(aa_fun4096426214187263247_nat_o(aTP_Lamm_fo(Uu),Uua),Uub),Uuc)
    <=> ? [X4: real,Y5: real] :
          ( ( Uuc = aa_real_nat(aa_real_fun_real_nat(Uub,X4),Y5) )
          & aa_real_o(Uu,X4)
          & aa_real_o(Uua,Y5) ) ) ).

% ATP.lambda_663
tff(fact_10458_ATP_Olambda__664,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_o,Uub: fun_re4938349377536830718al_int,Uuc: int] :
      ( aa_int_o(aa_fun6878201389861000969_int_o(aa_fun3728209879534177367_int_o(aTP_Lamm_fp(Uu),Uua),Uub),Uuc)
    <=> ? [X4: real,Y5: real] :
          ( ( Uuc = aa_real_int(aa_real_fun_real_int(Uub,X4),Y5) )
          & aa_real_o(Uu,X4)
          & aa_real_o(Uua,Y5) ) ) ).

% ATP.lambda_664
tff(fact_10459_ATP_Olambda__665,axiom,
    ! [Uu: fun_real_o,Uua: fun_nat_o,Uub: fun_re2084383175013312036omplex,Uuc: complex] :
      ( aa_complex_o(aa_fun2810168320254036705plex_o(aa_fun467192419720471251plex_o(aTP_Lamm_fu(Uu),Uua),Uub),Uuc)
    <=> ? [X4: real,Y5: nat] :
          ( ( Uuc = aa_nat_complex(aa_rea1531280491672885477omplex(Uub,X4),Y5) )
          & aa_real_o(Uu,X4)
          & aa_nat_o(Uua,Y5) ) ) ).

% ATP.lambda_665
tff(fact_10460_ATP_Olambda__666,axiom,
    ! [Uu: fun_real_o,Uua: fun_nat_o,Uub: fun_re4238438383274954274t_real,Uuc: real] :
      ( aa_real_o(aa_fun3910182461874428645real_o(aa_fun8527292551407841495real_o(aTP_Lamm_fr(Uu),Uua),Uub),Uuc)
    <=> ? [X4: real,Y5: nat] :
          ( ( Uuc = aa_nat_real(aa_real_fun_nat_real(Uub,X4),Y5) )
          & aa_real_o(Uu,X4)
          & aa_nat_o(Uua,Y5) ) ) ).

% ATP.lambda_666
tff(fact_10461_ATP_Olambda__667,axiom,
    ! [Uu: fun_real_o,Uua: fun_nat_o,Uub: fun_real_fun_nat_nat,Uuc: nat] :
      ( aa_nat_o(aa_fun1675651493036827037_nat_o(aa_fun957765429596782223_nat_o(aTP_Lamm_fs(Uu),Uua),Uub),Uuc)
    <=> ? [X4: real,Y5: nat] :
          ( ( Uuc = aa_nat_nat(aa_real_fun_nat_nat(Uub,X4),Y5) )
          & aa_real_o(Uu,X4)
          & aa_nat_o(Uua,Y5) ) ) ).

% ATP.lambda_667
tff(fact_10462_ATP_Olambda__668,axiom,
    ! [Uu: fun_real_o,Uua: fun_nat_o,Uub: fun_real_fun_nat_int,Uuc: int] :
      ( aa_int_o(aa_fun1731537780621499109_int_o(aa_fun589549094943696343_int_o(aTP_Lamm_ft(Uu),Uua),Uub),Uuc)
    <=> ? [X4: real,Y5: nat] :
          ( ( Uuc = aa_nat_int(aa_real_fun_nat_int(Uub,X4),Y5) )
          & aa_real_o(Uu,X4)
          & aa_nat_o(Uua,Y5) ) ) ).

% ATP.lambda_668
tff(fact_10463_ATP_Olambda__669,axiom,
    ! [Uu: fun_real_o,Uua: fun_int_o,Uub: fun_re7457141977439324286t_real,Uuc: real] :
      ( aa_real_o(aa_fun2318301334355154825real_o(aa_fun8106147379196233687real_o(aTP_Lamm_fv(Uu),Uua),Uub),Uuc)
    <=> ? [X4: real,Y5: int] :
          ( ( Uuc = aa_int_real(aa_real_fun_int_real(Uub,X4),Y5) )
          & aa_real_o(Uu,X4)
          & aa_int_o(Uua,Y5) ) ) ).

% ATP.lambda_669
tff(fact_10464_ATP_Olambda__670,axiom,
    ! [Uu: fun_real_o,Uua: fun_int_o,Uub: fun_real_fun_int_nat,Uuc: nat] :
      ( aa_nat_o(aa_fun1484945577804563009_nat_o(aa_fun4067268449253971343_nat_o(aTP_Lamm_fw(Uu),Uua),Uub),Uuc)
    <=> ? [X4: real,Y5: int] :
          ( ( Uuc = aa_int_nat(aa_real_fun_int_nat(Uub,X4),Y5) )
          & aa_real_o(Uu,X4)
          & aa_int_o(Uua,Y5) ) ) ).

% ATP.lambda_670
tff(fact_10465_ATP_Olambda__671,axiom,
    ! [Uu: nat,Uua: fun_na6724076748692969250l_real,Uub: real,Uuc: nat,Uud: real] : aa_real_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_yx(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_yw(Uua),Uuc),Uud),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uub),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))))) ).

% ATP.lambda_671
tff(fact_10466_ATP_Olambda__672,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_yb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uue),Uuf)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuf),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu))
        & ! [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))),Uud))
           => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(Uua,I2)),X_13)
            <=> aa_nat_o(vEBT_V8194947554948674370ptions(Uub),I2) ) )
        & $ite(
            Uue = Uuf,
            ! [X4: vEBT_VEBT] :
              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Uua))
             => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X6) ),
            ( 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))),Uu))
               => ( 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_672
tff(fact_10467_ATP_Olambda__673,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_sl(Uu),Uua) = Uu ).

% ATP.lambda_673
tff(fact_10468_ATP_Olambda__674,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sm,Uu),Uua) = Uu ).

% ATP.lambda_674
tff(fact_10469_ATP_Olambda__675,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_pl,Uu),Uua)
    <=> $true ) ).

% ATP.lambda_675
tff(fact_10470_ATP_Olambda__676,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_do,Uu) = Uu ).

% ATP.lambda_676
tff(fact_10471_ATP_Olambda__677,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_dp,Uu) = Uu ).

% ATP.lambda_677
tff(fact_10472_ATP_Olambda__678,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_dq,Uu) = Uu ).

% ATP.lambda_678
tff(fact_10473_ATP_Olambda__679,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_dr,Uu) = Uu ).

% ATP.lambda_679
tff(fact_10474_ATP_Olambda__680,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ds,Uu) = Uu ).

% ATP.lambda_680
tff(fact_10475_ATP_Olambda__681,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_dt,Uu) = zero_zero_complex ).

% ATP.lambda_681
tff(fact_10476_ATP_Olambda__682,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_du,Uu) = zero_zero_real ).

% ATP.lambda_682
tff(fact_10477_ATP_Olambda__683,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_dv,Uu) = zero_zero_rat ).

% ATP.lambda_683
tff(fact_10478_ATP_Olambda__684,axiom,
    ! [Uu: nat] : aa_nat_complex(aTP_Lamm_rp,Uu) = zero_zero_complex ).

% ATP.lambda_684
tff(fact_10479_ATP_Olambda__685,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_jt,Uu) = zero_zero_real ).

% ATP.lambda_685
tff(fact_10480_ATP_Olambda__686,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_dw,Uu) = zero_zero_nat ).

% ATP.lambda_686
tff(fact_10481_ATP_Olambda__687,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_rq,Uu) = zero_zero_int ).

% ATP.lambda_687
tff(fact_10482_ATP_Olambda__688,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_dx,Uu) = zero_zero_int ).

% ATP.lambda_688
tff(fact_10483_ATP_Olambda__689,axiom,
    ! [Uu: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_cg,Uu)
    <=> $false ) ).

% ATP.lambda_689
tff(fact_10484_ATP_Olambda__690,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_ch,Uu)
    <=> $false ) ).

% ATP.lambda_690
tff(fact_10485_ATP_Olambda__691,axiom,
    ! [Uu: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ce,Uu)
    <=> $false ) ).

% ATP.lambda_691
tff(fact_10486_ATP_Olambda__692,axiom,
    ! [Uu: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_cf,Uu)
    <=> $false ) ).

% ATP.lambda_692
tff(fact_10487_ATP_Olambda__693,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_ci,Uu)
    <=> $false ) ).

% ATP.lambda_693
tff(fact_10488_ATP_Olambda__694,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_cj,Uu)
    <=> $false ) ).

% ATP.lambda_694
tff(fact_10489_ATP_Olambda__695,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_xg,Uu)
    <=> $true ) ).

% ATP.lambda_695

% Helper facts (15)
tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      | aa_int_o(aa_int_fun_int_o(fequal_int,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      | aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
      | aa_real_o(aa_real_fun_real_o(fequal_real,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(fequal_real,X),Y)
      | ( X = Y ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: fun_real_o] :
      ( aa_real_o(P,fChoice_real(P))
      = ( ? [X6: real] : aa_real_o(P,X6) ) ) ).

tff(help_fequal_2_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X != Y )
      | aa_set_nat_o(aa_set7861427066675100071_nat_o(fequal_set_nat,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(fequal_set_nat,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( X != Y )
      | aa_list_nat_o(fequal_list_nat(X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ~ aa_list_nat_o(fequal_list_nat(X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( X != Y )
      | aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(fequal9188845342289841878at_nat,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ~ aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(fequal9188845342289841878at_nat,X),Y)
      | ( X = 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,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( X != Y )
      | aa_Pro5947980084102686253_nat_o(fequal7949547063978738364at_nat(X),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,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ~ aa_Pro5947980084102686253_nat_o(fequal7949547063978738364at_nat(X),Y)
      | ( X = Y ) ) ).

% Conjectures (2)
tff(conj_0,hypothesis,
    ! [X2: nat] :
      ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,maxs)),X2)
     => thesis ) ).

tff(conj_1,conjecture,
    thesis ).

%------------------------------------------------------------------------------