%------------------------------------------------------------------------------
% File     : ITP233_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_InsertCorrectness 00603_038671
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0067_VEBT_InsertCorrectness_00603_038671 [Des22]

% Status   : Theorem
% Rating   : 0.00 v8.2.0, 0.67 v8.1.0
% Syntax   : Number of formulae    : 12383 (3531 unt;2616 typ;   0 def)
%            Number of atoms       : 24950 (8774 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 16703 (2366   ~; 447   |;1335   &)
%                                         (2473 <=>;10082  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   15 (   2 avg)
%            Number of FOOLs       : 2151 ( 711 fml;1440 var)
%            Number of X terms     :  778 (   0  []; 705 ite;  73 let)
%            Number of types       :  494 ( 493 usr)
%            Number of type conns  : 2579 (1728   >; 851   *;   0   +;   0  <<)
%            Number of predicates  :  154 ( 151 usr;   3 prp; 0-4 aty)
%            Number of functors    : 1981 (1981 usr; 403 con; 0-4 aty)
%            Number of variables   : 25029 (24339   !; 690   ?;25029   :)
% SPC      : TX0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 21:14:46.382
%------------------------------------------------------------------------------
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tff(ty_n_t__fun_It__Filter__Ofilter_It__Nat__Onat_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__Extended____Nat__Oenat_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__fun_I_Eo_M_Eo_J_Mt__fun_I_Eo_M_Eo_J_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Real__Oreal_J,type,
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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__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
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tff(ty_n_t__Product____Type__Oprod_It__Int__Oint_M_Eo_J,type,
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tff(ty_n_t__fun_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__VEBT____Definitions__OVEBT_M_Eo_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__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Real__Oreal_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__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Real__Oreal_J,type,
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tff(ty_n_t__VEBT____Definitions__OVEBT,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__Product____Type__Ounit,type,
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tff(ty_n_t__Num__Onum,type,
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% Explicit typings (2123)
tff(sy_c_ATP_058Lamm__a____,type,
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tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: set_real > fun_real_o ).

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,
    aTP_Lamm_af: set_complex > fun_set_complex_o ).

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,
    aTP_Lamm_ap: set_int > fun_fu3883539188510166275_int_o ).

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,
    aTP_Lamm_au: fun_o_o ).

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,
    aTP_Lamm_az: fun_real_real ).

tff(sy_c_ATP_058Lamm__ba____,type,
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tff(sy_c_ATP_058Lamm__bb____,type,
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tff(sy_c_ATP_058Lamm__bc____,type,
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tff(sy_c_ATP_058Lamm__bd____,type,
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tff(sy_c_ATP_058Lamm__be____,type,
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tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_Ex5852502975177633731d_enat ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: set_Pr8693737435421807431at_nat > fun_Pr2028404973800837934_nat_o ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: set_Pr7556676689462069481BT_nat > fun_VE6480751511521678896_nat_o ).

tff(sy_c_ATP_058Lamm__bn____,type,
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tff(sy_c_ATP_058Lamm__bo____,type,
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tff(sy_c_ATP_058Lamm__bp____,type,
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tff(sy_c_ATP_058Lamm__bq____,type,
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tff(sy_c_ATP_058Lamm__br____,type,
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tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: set_Extended_enat > fun_na4030263931958639548enat_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__bv____,type,
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tff(sy_c_ATP_058Lamm__bw____,type,
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tff(sy_c_ATP_058Lamm__bx____,type,
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tff(sy_c_ATP_058Lamm__by____,type,
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tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__ca____,type,
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tff(sy_c_ATP_058Lamm__cb____,type,
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tff(sy_c_ATP_058Lamm__cc____,type,
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tff(sy_c_ATP_058Lamm__cd____,type,
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tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: set_o > fun_fu6781074197733401937un_o_o ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: set_o > fun_fu5915781709770235737un_o_o ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: set_real > fun_fu1852998413150313061real_o ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: set_real > fun_fu5575653946431489541real_o ).

tff(sy_c_ATP_058Lamm__ci____,type,
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tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: set_nat > fun_fu6075382464776002713_nat_o ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: set_complex > fun_fu6122667721776870753plex_o ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: set_complex > fun_fu7998770217265696895plex_o ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: set_int > fun_fu1805223966450871909_int_o ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: set_int > fun_fu2760153274138146437_int_o ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: set_Extended_enat > fun_fu7383104390662542801enat_o ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: set_Extended_enat > fun_fu7039103635370308747enat_o ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: set_o > fun_fu2389858412225411757un_o_o ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: set_o > fun_fu8756204934394872281un_o_o ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: set_complex > fun_fu806425790188207109plex_o ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: set_complex > fun_fu7523448966510524487plex_o ).

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

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

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

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

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_Pr4796943543331071514_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_Pr2602117396471556196at_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: fun_Pr2477239776880934807at_nat > fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: fun_Pr2028404973800837934_nat_o > fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: fun_Pr3238374608701176162_int_o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: int > fun_int_fun_int_o ).

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

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

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: fun_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: fun_complex_int > fun_complex_complex ).

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

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: real > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: $o > fun_fu6701067901881939075_o_nat ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: complex > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: extended_enat > fun_fu5112335056569181187at_nat ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: $o > fun_fu8103376058317145219o_real ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: extended_enat > fun_fu7336479014630192387t_real ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: real > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: $o > fun_fu6701067901881939075_o_nat ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: complex > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: extended_enat > fun_fu5112335056569181187at_nat ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: $o > fun_fu8103376058317145219o_real ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: extended_enat > fun_fu7336479014630192387t_real ).

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

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: fun_complex_complex > fun_fu5956788528830221615omplex ).

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

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: $o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: $o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: complex > fun_fu2513798379308225899ex_nat ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: extended_enat > fun_fu8366861321737653343at_nat ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: complex > fun_fu4692242605864086655ex_int ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: extended_enat > fun_fu2469135700067826619at_int ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: complex > fun_fu709861064203714431x_real ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: extended_enat > fun_fu3686806697881729723t_real ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: complex > fun_fu5040975675184911655d_enat ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: extended_enat > fun_fu2295278630349710275d_enat ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: real > fun_fu3768230810189712741al_nat ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: real > fun_fu5645595436833112697al_int ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: fun_o_nat > fun_fu7210065372943547985_o_nat ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: fun_real_nat > fun_fu2707946803925496145al_nat ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: set_complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_complex_nat > fun_fu1553612835185860561ex_nat ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_int_nat > fun_fu1078644964449000785nt_nat ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: set_Extended_enat > fun_fu9148622857633396327enat_o ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_Ex7660298398204468809at_nat > fun_fu4016676428956456173at_nat ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_o_real > fun_fu3042504704906485677o_real ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_real_real > fun_fu4478067837645853357l_real ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: fun_complex_real > fun_fu3043978019632300333x_real ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: fun_int_real > fun_fu7030473000426307245t_real ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: fun_Ex1451620480498224165t_real > fun_fu746971108421118025t_real ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: fun_o_nat > fun_fu6701067901881939075_o_nat ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_nat_nat > fun_fu7002938529518989751at_nat ).

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

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_real_nat > fun_real_o ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_complex_nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_Ex7660298398204468809at_nat > fun_Extended_enat_o ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_real_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_complex_real > fun_complex_o ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_int_real > fun_int_o ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: fun_Ex1451620480498224165t_real > fun_Extended_enat_o ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: fun_real_int > fun_real_o ).

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

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: fun_na8797796813480665199d_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: fun_na8797796813480665199d_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: fun_nat_nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: fun_complex_nat > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: fun_nat_nat > fun_nat_int ).

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

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

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: extended_enat > fun_Ex2441982661509597004d_enat ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_na7542918316964655096d_enat ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: fun_Ex5852502975177633731d_enat ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_real_real ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: fun_int_int ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: fun_Ex239764483819389678enat_o ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: fun_nat_int > fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: fun_nat_real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: fun_nat_real ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: extended_enat > fun_na1154345429118027792d_enat ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: fun_int_nat > fun_int_int ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: fun_int_int > fun_int_real ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: fun_nat_int > fun_na7654559027617327379at_int ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: $o > fun_fu3581538110357139729_o_nat ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: int > fun_fu973771059875708133nt_nat ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: $o > fun_fu9026017786105829997_o_int ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: extended_enat > fun_na7542918316964655096d_enat ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: real > fun_nat_fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_na8797796813480665199d_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: fun_na8797796813480665199d_enat > fun_na1154345429118027792d_enat ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: fun_nat_real ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: fun_nat_int ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_nat_complex ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: fun_nat_real > fun_se2100927751953696370t_real ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_nat_o > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: fun_nat_o > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_nat_o > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: fun_nat_o > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: set_nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: set_nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: set_nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: nat > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: nat > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: nat > fun_co3258106519541737638omplex ).

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

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: filter_nat > fun_filter_nat_o ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: complex > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_nat_complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: fun_nat_real > fun_nat_fun_real_o ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: fun_nat_int > fun_fu1567874110255095356at_int ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_nat_int > fun_fu5519318738670804632at_int ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: fun_nat_complex > fun_fu4528473944208963392omplex ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: fun_nat_complex > fun_fu6652553491520710942omplex ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: fun_nat_real > fun_fu8064956510386623256t_real ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: nat > fun_fu5303113179588540421d_enat ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: nat > fun_fu5303113179588540421d_enat ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: nat > fun_co6147893885205795665omplex ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: nat > fun_in1521351555827773387at_int ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: nat > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: fun_nat_int > fun_in281396392327853336at_int ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: fun_nat_complex > fun_co3194742168270691230omplex ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: fun_nat_real > fun_re7819166435457029784t_real ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: fun_nat_int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: fun_nat_complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: fun_nat_real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: extended_enat > fun_Ex8979681104671372885d_enat ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: real > fun_re4756235295228751087t_real ).

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

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: real > fun_fu2115079063411676732t_real ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: real > fun_fu4540635375566031512t_real ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: real > fun_fu3477889739401271834t_real ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: real > fun_fu2357351296865450298t_real ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: fun_nat_fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: fun_nat_fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: fun_nat_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_nat_complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: fun_o_real > fun_fu8103376058317145219o_real ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: fun_set_nat_real > fun_fu5354524256550068955t_real ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: fun_real_int > fun_fu3737244727311182119al_int ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: fun_o_int > fun_fu6940589326549028739_o_int ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: fun_set_nat_int > fun_fu3492966305048685787at_int ).

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

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: fun_int_int > fun_fu1143798476471108391nt_int ).

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

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: complex > fun_na1142814060743658952omplex ).

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

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

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: set_complex > fun_fu2445682883699234307plex_o ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: set_Extended_enat > fun_fu521531014127253423enat_o ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: set_real > fun_fu7444430165986413063real_o ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: set_o > fun_fu3569133559346579503un_o_o ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: set_int > fun_fu7994259084669964039_int_o ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: set_complex > fun_fu6229866931165912581plex_o ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: set_Extended_enat > fun_fu8278438252182731309enat_o ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: set_real > fun_fu3729927046167673097real_o ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: set_o > fun_fu7756609662939579309un_o_o ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: set_int > fun_fu4534828536049688329_int_o ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: fun_o_nat > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: fun_real_nat > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: fun_nat_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: fun_int_nat > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: fun_o_real > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: fun_real_real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: fun_nat_real > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: fun_int_real > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: fun_o_int > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_real_int > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: fun_complex_nat > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: fun_Ex7660298398204468809at_nat > fun_fu5112335056569181187at_nat ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: fun_complex_int > fun_fu4213752561377164075ex_int ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: fun_Ex3482447378695272101at_int > fun_fu6685514224203069699at_int ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: fun_complex_real > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: fun_o_nat > fun_fu6701067901881939075_o_nat ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: fun_o_real > fun_fu8103376058317145219o_real ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: fun_complex_nat > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: fun_Ex7660298398204468809at_nat > fun_se6072594010446775118enat_o ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: fun_complex_real > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: fun_Ex1451620480498224165t_real > fun_se6072594010446775118enat_o ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_nat_fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: fun_nat_int > fun_na7654559027617327379at_int ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: nat > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: fun_VEBT_VEBT_int > fun_in3204970175496335219at_int ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: fun_int_int > fun_in5653523039088217947at_int ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: fun_nat_int > fun_in2640216123419753727at_int ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: fun_VE5569070535936029429omplex > fun_co6469028316994822131omplex ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: fun_int_complex > fun_co5214255584322676319omplex ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: fun_nat_complex > fun_co7890082732823579651omplex ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: fun_VE3569950477627547253d_enat > fun_Ex1869439690531410417d_enat ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: fun_in5914218760682845459d_enat > fun_Ex4306801469811899753d_enat ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: fun_na8797796813480665199d_enat > fun_Ex6561714628001473733d_enat ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: fun_VEBT_VEBT_nat > fun_na1689058941480031347at_nat ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: nat > fun_li8515462164684499182VEBT_o ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: nat > fun_li1334793589135804498_int_o ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: nat > fun_li1025265955262268170_nat_o ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: fun_o_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: fun_o_nat > fun_fu318872218122388049at_nat ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: fun_nat_o > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: fun_nat_o ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: set_Extended_enat > fun_na4030263931958639548enat_o ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: fun_VEBT_VEBT_int > fun_in3204970175496335219at_int ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: fun_int_int > fun_in5653523039088217947at_int ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_nat_int > fun_in2640216123419753727at_int ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: fun_VE5569070535936029429omplex > fun_co6469028316994822131omplex ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: fun_int_complex > fun_co5214255584322676319omplex ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: fun_nat_complex > fun_co7890082732823579651omplex ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: fun_VE3569950477627547253d_enat > fun_Ex1869439690531410417d_enat ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: fun_in5914218760682845459d_enat > fun_Ex4306801469811899753d_enat ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: fun_na8797796813480665199d_enat > fun_Ex6561714628001473733d_enat ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: fun_VEBT_VEBT_nat > fun_na1689058941480031347at_nat ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: set_set_nat > fun_na3711450577076729278_nat_o ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: set_list_nat > fun_na1108963418173300452_nat_o ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: set_complex > fun_na6069412773753413270plex_o ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: set_int > fun_na8635581178960477080_int_o ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: set_Extended_enat > fun_na9114950421169325794enat_o ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: set_nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: set_set_nat > fun_na6948449159361463396_nat_o ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: set_list_nat > fun_na4138138649119996554_nat_o ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: fun_real_real ).

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

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: fun_nat_real > fun_real_real ).

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

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

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: fun_nat_real > fun_fu6899955116716589935t_real ).

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

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: fun_nat_real ).

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

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

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

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__yf____,type,
    aTP_Lamm_yf: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__yh____,type,
    aTP_Lamm_yh: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__yi____,type,
    aTP_Lamm_yi: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yj____,type,
    aTP_Lamm_yj: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__yk____,type,
    aTP_Lamm_yk: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ym____,type,
    aTP_Lamm_ym: nat > fun_li6914387844523535256_nat_o ).

tff(sy_c_ATP_058Lamm__yn____,type,
    aTP_Lamm_yn: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__yo____,type,
    aTP_Lamm_yo: fun_real_o ).

tff(sy_c_ATP_058Lamm__yp____,type,
    aTP_Lamm_yp: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yq____,type,
    aTP_Lamm_yq: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yr____,type,
    aTP_Lamm_yr: nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__ys____,type,
    aTP_Lamm_ys: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yt____,type,
    aTP_Lamm_yt: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yu____,type,
    aTP_Lamm_yu: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__yv____,type,
    aTP_Lamm_yv: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__yw____,type,
    aTP_Lamm_yw: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__yx____,type,
    aTP_Lamm_yx: fun_na6128946301132549280et_nat ).

tff(sy_c_ATP_058Lamm__yy____,type,
    aTP_Lamm_yy: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__yz____,type,
    aTP_Lamm_yz: fun_na6128946301132549280et_nat ).

tff(sy_c_ATP_058Lamm__za____,type,
    aTP_Lamm_za: fun_na6128946301132549280et_nat ).

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

tff(sy_c_ATP_058Lamm__zc____,type,
    aTP_Lamm_zc: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__zd____,type,
    aTP_Lamm_zd: extended_enat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ze____,type,
    aTP_Lamm_ze: extended_enat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__zf____,type,
    aTP_Lamm_zf: fun_na819858167508000762st_nat ).

tff(sy_c_ATP_058Lamm__zg____,type,
    aTP_Lamm_zg: fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__zh____,type,
    aTP_Lamm_zh: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__zi____,type,
    aTP_Lamm_zi: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__zj____,type,
    aTP_Lamm_zj: nat > fun_VE3165209603783583043T_VEBT ).

tff(sy_c_ATP_058Lamm__zk____,type,
    aTP_Lamm_zk: set_Extended_enat > fun_Extended_enat_o ).

tff(sy_c_ATP_058Lamm__zl____,type,
    aTP_Lamm_zl: nat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__zm____,type,
    aTP_Lamm_zm: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__zn____,type,
    aTP_Lamm_zn: nat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__zo____,type,
    aTP_Lamm_zo: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__zp____,type,
    aTP_Lamm_zp: fun_na8797796813480665199d_enat ).

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

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__Real__Oreal,type,
    archim6058952711729229775r_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
    archim2898591450579166408c_real: fun_real_real ).

tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: fun_real_int ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    bNF_re6830278522597306478at_int: ( fun_nat_fun_nat_o * fun_Pr4738963024209204295_int_o * fun_na1476630412504255114at_nat * fun_nat_int ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_Eo_001_Eo,type,
    bNF_re6644619430987730960nt_o_o: ( fun_Pr4738963024209204295_int_o * fun_o_fun_o_o ) > fun_fu9090758901314268781nt_o_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re4555766996558763186at_nat: ( fun_Pr4738963024209204295_int_o * fun_nat_fun_nat_o * fun_Pr2602117396471556196at_nat * fun_int_nat ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    bNF_re7400052026677387805at_int: ( fun_Pr4738963024209204295_int_o * fun_Pr4738963024209204295_int_o ) > fun_fu4638078822479986034_int_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001t__fun_It__Int__Oint_M_Eo_J,type,
    bNF_re4195927319312321177_int_o: ( fun_Pr4738963024209204295_int_o * fun_fu9090758901314268781nt_o_o * fun_Pr2028404973800837934_nat_o * fun_int_fun_int_o ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re8869769828555196650nt_int: ( fun_Pr4738963024209204295_int_o * fun_fu4638078822479986034_int_o * fun_Pr2477239776880934807at_nat * fun_int_fun_int_int ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
    bNF_re3666534408544137501at_o_o: ( fun_Pr2028404973800837934_nat_o * fun_o_fun_o_o ) > fun_fu9045186299510176150at_o_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re8246922863344978751at_nat: ( fun_Pr2028404973800837934_nat_o * fun_nat_fun_nat_o * fun_Pr2602117396471556196at_nat * fun_Pr2602117396471556196at_nat ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re2241393799969408733at_nat: ( fun_Pr2028404973800837934_nat_o * fun_Pr2028404973800837934_nat_o ) > fun_fu3373238862511017284_nat_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    bNF_re4103222659376994899_nat_o: ( fun_Pr2028404973800837934_nat_o * fun_fu9045186299510176150at_o_o * fun_Pr2028404973800837934_nat_o * fun_Pr2028404973800837934_nat_o ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_re6214786843070381989at_nat: ( fun_Pr2028404973800837934_nat_o * fun_fu3373238862511017284_nat_o * fun_Pr2477239776880934807at_nat * fun_Pr2477239776880934807at_nat ) > $o ).

tff(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
    bNF_We3818239936649020644el_nat: set_Pr1261947904930325089at_nat > $o ).

tff(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod_001t__Int__Oint_001t__Int__Oint,type,
    basic_1872990034501187214nt_int: ( fun_int_nat * fun_int_nat * product_prod_int_int ) > nat ).

tff(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod_001t__Nat__Onat_001t__Nat__Onat,type,
    basic_876126793109182934at_nat: ( fun_nat_nat * fun_nat_nat ) > fun_Pr2602117396471556196at_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__Real__Oreal,type,
    gbinomial_real: real > fun_nat_real ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun_Pr9018117161326028078_int_o ).

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

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

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
    bit_se727722235901077358nd_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
    bit_se8570568707652914677it_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
    bit_se2161824704523386999it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
    bit_se2000444600071755411sk_int: fun_nat_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
    bit_se2002935070580805687sk_nat: fun_nat_nat ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
    bit_se1412395901928357646or_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
    bit_se545348938243370406it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
    bit_se547839408752420682it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
    bit_se7879613467334960850it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
    bit_se7882103937844011126it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
    bit_se2923211474154528505it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
    bit_se2925701944663578781it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
    bit_se4203085406695923979it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
    bit_se4205575877204974255it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
    bit_se6526347334894502574or_int: int > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
    bit_se6528837805403552850or_nat: fun_nat_fun_nat_nat ).

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    bit_se1146084159140164899it_int: int > fun_nat_o ).

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tff(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
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tff(sy_c_Fun__Def_Opair__less,type,
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tff(sy_c_GCD_Obezw,type,
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tff(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
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tff(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_M_Eo_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_M_Eo_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
    set_real2: list_real > set_real ).

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

tff(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist_Osize__list_001_Eo,type,
    size_list_o: ( fun_o_nat * list_o ) > nat ).

tff(sy_c_List_Olist_Osize__list_001t__Int__Oint,type,
    size_list_int: ( fun_int_nat * list_int ) > nat ).

tff(sy_c_List_Olist_Osize__list_001t__Nat__Onat,type,
    size_list_nat: ( fun_nat_nat * list_nat ) > nat ).

tff(sy_c_List_Olist_Osize__list_001t__Real__Oreal,type,
    size_list_real: ( fun_real_nat * list_real ) > nat ).

tff(sy_c_List_Olist_Osize__list_001t__Set__Oset_It__Nat__Onat_J,type,
    size_list_set_nat: ( fun_set_nat_nat * list_set_nat ) > nat ).

tff(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
    tl_nat: list_nat > list_nat ).

tff(sy_c_List_Olist__ex1_001_Eo,type,
    list_ex1_o: ( fun_o_o * list_o ) > $o ).

tff(sy_c_List_Olist__ex1_001t__Int__Oint,type,
    list_ex1_int: fun_int_o > fun_list_int_o ).

tff(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
    list_ex1_nat: fun_nat_o > fun_list_nat_o ).

tff(sy_c_List_Olist__ex1_001t__Real__Oreal,type,
    list_ex1_real: ( fun_real_o * list_real ) > $o ).

tff(sy_c_List_Olist__ex1_001t__Set__Oset_It__Nat__Onat_J,type,
    list_ex1_set_nat: ( fun_set_nat_o * list_set_nat ) > $o ).

tff(sy_c_List_Olist__ex1_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist__update_001_Eo,type,
    list_update_o: ( list_o * nat * $o ) > list_o ).

tff(sy_c_List_Olist__update_001t__Extended____Nat__Oenat,type,
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tff(sy_c_List_Olist__update_001t__Int__Oint,type,
    list_update_int: ( list_int * nat * int ) > list_int ).

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

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
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tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Olist__update_001t__Real__Oreal,type,
    list_update_real: ( list_real * nat * real ) > list_real ).

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

tff(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olistrel1_001t__Int__Oint,type,
    listrel1_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).

tff(sy_c_List_Olistrel1_001t__Nat__Onat,type,
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tff(sy_c_List_Olistrel1_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Olistrel1_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olistrel1p_001t__Int__Oint,type,
    listrel1p_int: ( fun_int_fun_int_o * list_int ) > fun_list_int_o ).

tff(sy_c_List_Olistrel1p_001t__Nat__Onat,type,
    listrel1p_nat: ( fun_nat_fun_nat_o * list_nat ) > fun_list_nat_o ).

tff(sy_c_List_Omember_001_Eo,type,
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tff(sy_c_List_Omember_001t__Int__Oint,type,
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tff(sy_c_List_Omember_001t__Nat__Onat,type,
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tff(sy_c_List_Omember_001t__Real__Oreal,type,
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tff(sy_c_List_Omember_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Omember_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_On__lists_001t__Int__Oint,type,
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tff(sy_c_List_On__lists_001t__Nat__Onat,type,
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tff(sy_c_List_On__lists_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Onth_001_Eo,type,
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tff(sy_c_List_Onth_001t__Extended____Nat__Oenat,type,
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tff(sy_c_List_Onth_001t__Int__Oint,type,
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tff(sy_c_List_Onth_001t__Nat__Onat,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Real__Oreal,type,
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tff(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Extended____Nat__Oenat,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oremdups_001t__Nat__Onat,type,
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tff(sy_c_List_Oremove1_001_Eo,type,
    remove1_o: ( $o * list_o ) > list_o ).

tff(sy_c_List_Oremove1_001t__Int__Oint,type,
    remove1_int: ( int * list_int ) > list_int ).

tff(sy_c_List_Oremove1_001t__Nat__Onat,type,
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tff(sy_c_List_Oremove1_001t__Real__Oreal,type,
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tff(sy_c_List_Oremove1_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Oremove1_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oreplicate_001_Eo,type,
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tff(sy_c_List_Oreplicate_001t__Int__Oint,type,
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tff(sy_c_List_Oreplicate_001t__Nat__Onat,type,
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tff(sy_c_List_Oreplicate_001t__Real__Oreal,type,
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tff(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
    replicate_VEBT_VEBT: ( nat * vEBT_VEBT ) > list_VEBT_VEBT ).

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

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

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

tff(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
    sorted_wrt_int: fun_int_fun_int_o > 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_Osubseqs_001t__Int__Oint,type,
    subseqs_int: list_int > list_list_int ).

tff(sy_c_List_Osubseqs_001t__Nat__Onat,type,
    subseqs_nat: list_nat > list_list_nat ).

tff(sy_c_List_Osubseqs_001t__VEBT____Definitions__OVEBT,type,
    subseqs_VEBT_VEBT: list_VEBT_VEBT > list_list_VEBT_VEBT ).

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

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

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

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

tff(sy_c_List_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_001_Eo,type,
    zip_int_o: ( list_int * list_o ) > list_P5087981734274514673_int_o ).

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__Nat__Onat,type,
    zip_int_nat: ( list_int * list_nat ) > list_P8198026277950538467nt_nat ).

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_001t__Int__Oint,type,
    zip_nat_int: ( list_nat * list_int ) > list_P3521021558325789923at_int ).

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__Extended____Nat__Oenat,type,
    zip_VE7205001627739651817d_enat: ( list_VEBT_VEBT * list_Extended_enat ) > list_P6356568628958627295d_enat ).

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_Nat_OSuc,type,
    suc: fun_nat_nat ).

tff(sy_c_Nat_Ocompow_001t__fun_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
    compow7134260122181328870omplex: ( nat * fun_complex_complex ) > fun_complex_complex ).

tff(sy_c_Nat_Ocompow_001t__fun_It__Extended____Nat__Oenat_Mt__Extended____Nat__Oenat_J,type,
    compow2216990816977393616d_enat: ( nat * fun_Ex5852502975177633731d_enat ) > fun_Ex5852502975177633731d_enat ).

tff(sy_c_Nat_Ocompow_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    compow_fun_int_int: ( nat * fun_int_int ) > fun_int_int ).

tff(sy_c_Nat_Ocompow_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    compow_fun_nat_nat: fun_na6032788992908162308at_nat ).

tff(sy_c_Nat_Ocompow_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J,type,
    compow_fun_real_real: ( nat * fun_real_real ) > fun_real_real ).

tff(sy_c_Nat_Ofunpow_001t__Nat__Onat,type,
    funpow_nat: fun_na6032788992908162308at_nat ).

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

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

tff(sy_c_Nat_Onat_Opred,type,
    pred: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
    semiring_1_Nats_int: set_int ).

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__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 * nat ) > fun_complex_complex ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Extended____Nat__Oenat,type,
    semiri8563196900006977889d_enat: ( fun_Ex5852502975177633731d_enat * nat ) > fun_Ex5852502975177633731d_enat ).

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__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: 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__Extended____Nat__Oenat_J,type,
    size_s3941691890525107288d_enat: list_Extended_enat > nat ).

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
    size_s533118279054570080st_int: list_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__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
    size_s8217280938318005548T_VEBT: list_list_VEBT_VEBT > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
    size_size_list_nat: 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_It__Int__Oint_Mt__Int__Oint_J_J,type,
    size_s5157815400016825771nt_int: list_P5707943133018811711nt_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J_J,type,
    size_s7647898544948552527nt_nat: list_P8198026277950538467nt_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s6639371672096860321T_VEBT: list_P7524865323317820941T_VEBT > 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_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: list_VEBT_VEBT > 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__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__decode,type,
    nat_list_decode: nat > list_nat ).

tff(sy_c_Nat__Bijection_Olist__decode__rel,type,
    nat_list_decode_rel: fun_nat_fun_nat_o ).

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,type,
    nat_prod_decode: fun_na1476630412504255114at_nat ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: nat > fun_na1476630412504255114at_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: fun_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_Num_Oinc,type,
    inc: num > num ).

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__Real__Oreal,type,
    neg_nu6075765906172075777c_real: fun_real_real ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: 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_Osize__num,type,
    size_num: fun_num_nat ).

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

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: 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__Real__Oreal,type,
    numeral_numeral_real: num > real ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: fun_num_nat ).

tff(sy_c_Option_Ooption_ONone_001_Eo,type,
    none_o: option_o ).

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,
    none_real: option_real ).

tff(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Nat__Onat_J,type,
    none_set_nat: option_set_nat ).

tff(sy_c_Option_Ooption_ONone_001t__VEBT____Definitions__OVEBT,type,
    none_VEBT_VEBT: option_VEBT_VEBT ).

tff(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
    some_int: int > option_int ).

tff(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
    some_nat: nat > option_nat ).

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

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

tff(sy_c_Option_Ooption_OSome_001t__VEBT____Definitions__OVEBT,type,
    some_VEBT_VEBT: vEBT_VEBT > option_VEBT_VEBT ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o184042715313410164at_nat: ( $o * fun_Pr4796943543331071514_nat_o ) > fun_op8028897214545316308_nat_o ).

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

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

tff(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
    size_option_num: ( fun_num_nat * option_num ) > nat ).

tff(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    size_o8335143837870341156at_nat: ( fun_Pr2602117396471556196at_nat * option4927543243414619207at_nat ) > nat ).

tff(sy_c_Order__Relation_OunderS_001t__Nat__Onat,type,
    order_underS_nat: set_Pr1261947904930325089at_nat > fun_nat_set_nat ).

tff(sy_c_Order__Relation_Owell__order__on_001t__Nat__Onat,type,
    order_2888998067076097458on_nat: ( set_nat * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
    bot_bot_o: $o ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    bot_bot_filter_nat: filter_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__fun_I_Eo_M_Eo_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__Int__Oint_M_Eo_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__Nat__Onat_M_Eo_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__Real__Oreal_M_Eo_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_OLeast_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_Oless_001_Eo,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Filter__Ofilter_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,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__Num__Onum,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_I_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Extended____Nat__Oenat_J,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_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_I_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Real__Oreal_J,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_Oordering__top_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,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_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,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__Int__Oint,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__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__Int__Oint_001_Eo,type,
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tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Nat__Onat,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__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,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,
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tff(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
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tff(sy_c_Set_OCollect_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Set_OCollect_001t__Filter__Ofilter_It__Nat__Onat_J,type,
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tff(sy_c_Set_OCollect_001t__Int__Oint,type,
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tff(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
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tff(sy_c_Set_OCollect_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
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tff(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
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tff(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Set_OCollect_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_Set_OCollect_001t__Nat__Onat,type,
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tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Set_OCollect_001t__Real__Oreal,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Set_OPow_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001t__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
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tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
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tff(sy_c_Set_Oinsert_001_Eo,type,
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tff(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
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tff(sy_c_Set_Oinsert_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Set_Oinsert_001t__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_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_Oinsert_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Set_Ois__singleton_001_Eo,type,
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tff(sy_c_Set_Ois__singleton_001t__Int__Oint,type,
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tff(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
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tff(sy_c_Set_Ois__singleton_001t__Real__Oreal,type,
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tff(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_Othe__elem_001_Eo,type,
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tff(sy_c_Set_Othe__elem_001t__Int__Oint,type,
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tff(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
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tff(sy_c_Set_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__Extended____Nat__Oenat,type,
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tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,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__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001_Eo,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Filter__Ofilter_It__Nat__Onat_J,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__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001_Eo,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Filter__Ofilter_It__Nat__Onat_J,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Nat__Onat_J,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_001_Eo,type,
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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Filter__Ofilter_It__Nat__Onat_J,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__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or4236626031148496127et_nat: set_nat > set_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__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_OlessThan_001_Eo,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Extended____Nat__Oenat,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,
    set_ord_lessThan_nat: fun_nat_set_nat ).

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_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: fun_char_nat ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: fun_nat_char ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    topolo6427056007704750605er_nat: fun_nat_filter_nat > $o ).

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

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

tff(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Nat__Onat_J,type,
    topolo7278393974255667507et_nat: fun_nat_set_nat > $o ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex,type,
    topolo6517432010174082258omplex: fun_nat_complex > $o ).

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

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
    cos_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Complex__Ocomplex,type,
    cosh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
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tff(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
    diffs_int: fun_nat_int > fun_nat_int ).

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

tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

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

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

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

tff(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Complex__Ocomplex,type,
    sinh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
    tan_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
    tanh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

tff(sy_c_Transfer_Obi__total_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    bi_tot896582865486249351at_int: fun_Pr4738963024209204295_int_o > $o ).

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

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

tff(sy_c_Typedef_Otype__definition_001t__Product____Type__Ounit_001_Eo,type,
    type_d6188575255521822967unit_o: ( fun_Product_unit_o * fun_o_Product_unit ) > fun_set_o_o ).

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

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

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: fun_VE6480751511521678896_nat_o ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
    vEBT_V312737461966249ad_rel: fun_Pr1263128411319240110enat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: fun_nat_fun_nat_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: fun_VE6480751511521678896_nat_o ).

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

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

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_o ).

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

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun_Pr8877776288780692068_nat_o ).

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

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_o ).

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

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_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,
    accp_nat: fun_nat_fun_nat_o > fun_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    accp_P1096762738010456898nt_int: fun_Pr9018117161326028078_int_o > fun_Pr3238374608701176162_int_o ).

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

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
    accp_P6183159247885693666d_enat: fun_Pr1263128411319240110enat_o > fun_Pr5843119925985263810enat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    accp_P2887432264394892906BT_nat: fun_Pr8877776288780692068_nat_o > fun_Pr7546709192626087934_nat_o ).

tff(sy_c_Wellfounded_Oless__than,type,
    less_than: set_Pr1261947904930325089at_nat ).

tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

tff(sy_c_Wellfounded_Owf_001t__Int__Oint,type,
    wf_int: set_Pr958786334691620121nt_int > $o ).

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

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

tff(sy_c_aa_001_Eo_001t__Complex__Ocomplex,type,
    aa_o_complex: ( fun_o_complex * $o ) > complex ).

tff(sy_c_aa_001_Eo_001t__Extended____Nat__Oenat,type,
    aa_o_Extended_enat: ( fun_o_Extended_enat * $o ) > extended_enat ).

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

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

tff(sy_c_aa_001_Eo_001t__Product____Type__Ounit,type,
    aa_o_Product_unit: ( fun_o_Product_unit * $o ) > product_unit ).

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

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

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

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

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

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Extended____Nat__Oenat,type,
    aa_com3683005661447327866d_enat: ( fun_co6496365337113880337d_enat * complex ) > extended_enat ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Int__Oint,type,
    aa_complex_int: ( fun_complex_int * complex ) > int ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Nat__Onat,type,
    aa_complex_nat: ( fun_complex_nat * complex ) > nat ).

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

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_I_Eo_M_Eo_J,type,
    aa_complex_fun_o_o: ( fun_complex_fun_o_o * complex ) > fun_o_o ).

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

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Complex__Ocomplex_Mt__fun_It__Complex__Ocomplex_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J_J,type,
    aa_com177695874307951165omplex: ( fun_co3732688428756473980omplex * complex ) > fun_co6147893885205795665omplex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Complex__Ocomplex_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J,type,
    aa_com8177640530696437650omplex: ( fun_co6147893885205795665omplex * complex ) > fun_co3258106519541737638omplex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Complex__Ocomplex_Mt__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J_J,type,
    aa_com7713930848417683039omplex: ( fun_co3194742168270691230omplex * complex ) > fun_co6883864892131739507omplex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__List__Olist_It__Int__Oint_J_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J,type,
    aa_com4579324164465666976omplex: ( fun_co5214255584322676319omplex * complex ) > fun_li3213232893379486388omplex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__List__Olist_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J,type,
    aa_com7804175510677644100omplex: ( fun_co7890082732823579651omplex * complex ) > fun_li6438084239591463512omplex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J,type,
    aa_com9025189225015940188omplex: ( fun_co6469028316994822131omplex * complex ) > fun_li3032794628081365576omplex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J,type,
    aa_com8181001469867462503omplex: ( fun_co3258106519541737638omplex * complex ) > fun_nat_complex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J,type,
    aa_com4347742694490471604omplex: ( fun_co6883864892131739507omplex * complex ) > fun_na1142814060743658952omplex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Real__Oreal_M_Eo_J,type,
    aa_com1951324360330529881real_o: ( fun_co4998825143550818160real_o * complex ) > fun_real_o ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001_Eo,type,
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tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__Complex__Ocomplex,type,
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tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
    aa_Ext7969882492699152700d_enat: ( fun_Ex5852502975177633731d_enat * extended_enat ) > extended_enat ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__Int__Oint,type,
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tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__Nat__Onat,type,
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tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
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tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__Real__Oreal,type,
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tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__Extended____Nat__Oenat_M_Eo_J,type,
    aa_Ext6368937152127170207enat_o: ( fun_Ex239764483819389678enat_o * extended_enat ) > fun_Extended_enat_o ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__List__Olist_It__Int__Oint_J_Mt__fun_It__Nat__Onat_Mt__Extended____Nat__Oenat_J_J,type,
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tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__List__Olist_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Extended____Nat__Oenat_J_J,type,
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tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__fun_It__Nat__Onat_Mt__Extended____Nat__Oenat_J_J,type,
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tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__Nat__Onat_Mt__Extended____Nat__Oenat_J,type,
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tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Extended____Nat__Oenat_J_J,type,
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tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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tff(sy_c_aa_001t__Filter__Ofilter_It__Nat__Onat_J_001_Eo,type,
    aa_filter_nat_o: ( fun_filter_nat_o * filter_nat ) > $o ).

tff(sy_c_aa_001t__Filter__Ofilter_It__Nat__Onat_J_001t__Real__Oreal,type,
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tff(sy_c_aa_001t__Int__Oint_001_Eo,type,
    aa_int_o: ( fun_int_o * int ) > $o ).

tff(sy_c_aa_001t__Int__Oint_001t__Complex__Ocomplex,type,
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tff(sy_c_aa_001t__Int__Oint_001t__Extended____Nat__Oenat,type,
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tff(sy_c_aa_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_aa_001t__Int__Oint_001t__Nat__Onat,type,
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tff(sy_c_aa_001t__Int__Oint_001t__Real__Oreal,type,
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tff(sy_c_aa_001t__Int__Oint_001t__Set__Oset_I_Eo_J,type,
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tff(sy_c_aa_001t__Int__Oint_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

tff(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (9754)
tff(fact_0_True,axiom,
    i = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na) ).

% True
tff(fact_1__C166_C,axiom,
    aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na)) ).

% "166"
tff(fact_2_bit__split__inv,axiom,
    ! [Xb: 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,Xb),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),D)),D) = Xb ).

% bit_split_inv
tff(fact_3__C164_C,axiom,
    nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na),vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na))),i) = vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na)) ).

% "164"
tff(fact_4__C162_C,axiom,
    ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na))),X_1) ).

% "162"
tff(fact_5_insprop,axiom,
    nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na),vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)) = vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na)) ).

% insprop
tff(fact_6_tc,axiom,
    aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na))) ).

% tc
tff(fact_7_in__children__def,axiom,
    ! [Nb: nat,TreeLista: list_VEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V5917875025757280293ildren(Nb,TreeLista),Xb)
    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb)) ) ).

% in_children_def
tff(fact_8__C161_C,axiom,
    ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na))),X_1) ).

% "161"
tff(fact_9__C165_C,axiom,
    aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na))),na) ).

% "165"
tff(fact_10_list__update__id,axiom,
    ! [Xs: list_nat,Ia: nat] : list_update_nat(Xs,Ia,aa_nat_nat(nth_nat(Xs),Ia)) = Xs ).

% list_update_id
tff(fact_11_list__update__id,axiom,
    ! [Xs: list_int,Ia: nat] : list_update_int(Xs,Ia,aa_nat_int(nth_int(Xs),Ia)) = Xs ).

% list_update_id
tff(fact_12_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,Ia: nat] : list_u1324408373059187874T_VEBT(Xs,Ia,nth_VEBT_VEBT(Xs,Ia)) = Xs ).

% list_update_id
tff(fact_13_nth__list__update__neq,axiom,
    ! [Ia: nat,J: nat,Xs: list_nat,Xb: nat] :
      ( ( Ia != J )
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,Ia,Xb)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_14_nth__list__update__neq,axiom,
    ! [Ia: nat,J: nat,Xs: list_int,Xb: int] :
      ( ( Ia != J )
     => ( aa_nat_int(nth_int(list_update_int(Xs,Ia,Xb)),J) = aa_nat_int(nth_int(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_15_nth__list__update__neq,axiom,
    ! [Ia: nat,J: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( ( Ia != J )
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,Ia,Xb),J) = nth_VEBT_VEBT(Xs,J) ) ) ).

% nth_list_update_neq
tff(fact_16__C163_C,axiom,
    ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,summary),i) ).

% "163"
tff(fact_17_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBT,Ia: nat,Xb: vEBT_VEBT,Y: vEBT_VEBT] : list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,Ia,Xb),Ia,Y) = list_u1324408373059187874T_VEBT(Xs,Ia,Y) ).

% list_update_overwrite
tff(fact_18_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),X_12) ) ).

% not_min_Null_member
tff(fact_19_nsprop,axiom,
    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na))),vEBT_vebt_insert(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),summary) = vEBT_vebt_insert(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)) ).

% nsprop
tff(fact_20__C11_C,axiom,
    ! [X: vEBT_VEBT] :
      ( member_VEBT_VEBT2(X,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na),vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na)))))
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),na) ) ).

% "11"
tff(fact_21_list__update__swap,axiom,
    ! [Ia: nat,I: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT,X2: vEBT_VEBT] :
      ( ( Ia != I )
     => ( list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,Ia,Xb),I,X2) = list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X2),Ia,Xb) ) ) ).

% list_update_swap
tff(fact_22_False,axiom,
    ~ ( ( xa = mi )
      | ( xa = ma ) ) ).

% False
tff(fact_23_min__Null__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Xb) ) ).

% min_Null_member
tff(fact_24_valid__member__both__member__options,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),Xb)
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Xb) ) ) ).

% valid_member_both_member_options
tff(fact_25_both__member__options__equiv__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),Xb)
      <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Xb) ) ) ).

% both_member_options_equiv_member
tff(fact_26__C5_C,axiom,
    ( ( mi = ma )
   => ! [X: vEBT_VEBT] :
        ( member_VEBT_VEBT2(X,set_VEBT_VEBT2(treeList))
       => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_1) ) ) ).

% "5"
tff(fact_27_member__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Xb)
      <=> aa_set_nat_o(member_nat2(Xb),vEBT_set_vebt(Ta)) ) ) ).

% member_correct
tff(fact_28__C0_C,axiom,
    ! [X: vEBT_VEBT] :
      ( member_VEBT_VEBT2(X,set_VEBT_VEBT2(treeList))
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),na) ) ).

% "0"
tff(fact_29_mimaxrel,axiom,
    ( ( xa != mi )
    & ( xa != ma ) ) ).

% mimaxrel
tff(fact_30__C1_C,axiom,
    aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,summary),m) ).

% "1"
tff(fact_31_abcdef,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,mi),xa) ).

% abcdef
tff(fact_32__C5_Ohyps_C_I7_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,mi),ma) ).

% "5.hyps"(7)
tff(fact_33__C12_C,axiom,
    aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,
        $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na))),vEBT_vebt_insert(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),summary)),m) ).

% "12"
tff(fact_34_set__vebt__set__vebt_H__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( vEBT_set_vebt(Ta) = vEBT_VEBT_set_vebt(Ta) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_35_valid__eq,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Ta),D)
    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),D) ) ).

% valid_eq
tff(fact_36_valid__eq1,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),D)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Ta),D) ) ).

% valid_eq1
tff(fact_37_valid__eq2,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Ta),D)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),D) ) ).

% valid_eq2
tff(fact_38_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),zero_zero_nat) ).

% valid_0_not
tff(fact_39_valid__tree__deg__neq__0,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),zero_zero_nat) ).

% valid_tree_deg_neq_0
tff(fact_40__C10_C,axiom,
    vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),ma)),deg,treeList,summary),xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,mi),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xa),ma))),deg,list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na),vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na))),
      $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na))),vEBT_vebt_insert(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),summary)) ).

% "10"
tff(fact_41_in__set__member,axiom,
    ! [Xb: real,Xs: list_real] :
      ( aa_set_real_o(member_real2(Xb),set_real2(Xs))
    <=> aa_real_o(member_real(Xs),Xb) ) ).

% in_set_member
tff(fact_42_in__set__member,axiom,
    ! [Xb: $o,Xs: list_o] :
      ( aa_set_o_o(member_o2((Xb)),set_o2(Xs))
    <=> aa_o_o(member_o(Xs),(Xb)) ) ).

% in_set_member
tff(fact_43_in__set__member,axiom,
    ! [Xb: set_nat,Xs: list_set_nat] :
      ( aa_set_set_nat_o(member_set_nat2(Xb),set_set_nat2(Xs))
    <=> aa_set_nat_o(member_set_nat(Xs),Xb) ) ).

% in_set_member
tff(fact_44_in__set__member,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(Xs))
    <=> aa_VEBT_VEBT_o(member_VEBT_VEBT(Xs),Xb) ) ).

% in_set_member
tff(fact_45_in__set__member,axiom,
    ! [Xb: int,Xs: list_int] :
      ( aa_set_int_o(member_int2(Xb),set_int2(Xs))
    <=> aa_int_o(member_int(Xs),Xb) ) ).

% in_set_member
tff(fact_46_in__set__member,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat2(Xb),set_nat2(Xs))
    <=> aa_nat_o(member_nat(Xs),Xb) ) ).

% in_set_member
tff(fact_47_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_Node(Info,Dega,TreeLista,Summarya)),Nb)
     => ( Dega = Nb ) ) ).

% deg_deg_n
tff(fact_48_in__set__insert,axiom,
    ! [Xb: real,Xs: list_real] :
      ( aa_set_real_o(member_real2(Xb),set_real2(Xs))
     => ( insert_real(Xb,Xs) = Xs ) ) ).

% in_set_insert
tff(fact_49_in__set__insert,axiom,
    ! [Xb: $o,Xs: list_o] :
      ( aa_set_o_o(member_o2((Xb)),set_o2(Xs))
     => ( insert_o((Xb),Xs) = Xs ) ) ).

% in_set_insert
tff(fact_50_in__set__insert,axiom,
    ! [Xb: set_nat,Xs: list_set_nat] :
      ( aa_set_set_nat_o(member_set_nat2(Xb),set_set_nat2(Xs))
     => ( insert_set_nat(Xb,Xs) = Xs ) ) ).

% in_set_insert
tff(fact_51_in__set__insert,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(Xs))
     => ( insert_VEBT_VEBT(Xb,Xs) = Xs ) ) ).

% in_set_insert
tff(fact_52_in__set__insert,axiom,
    ! [Xb: int,Xs: list_int] :
      ( aa_set_int_o(member_int2(Xb),set_int2(Xs))
     => ( insert_int(Xb,Xs) = Xs ) ) ).

% in_set_insert
tff(fact_53_in__set__insert,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat2(Xb),set_nat2(Xs))
     => ( insert_nat(Xb,Xs) = Xs ) ) ).

% in_set_insert
tff(fact_54_list__ex1__iff,axiom,
    ! [P: fun_real_o,Xs: list_real] :
      ( list_ex1_real(P,Xs)
    <=> ? [X3: real] :
          ( aa_set_real_o(member_real2(X3),set_real2(Xs))
          & aa_real_o(P,X3)
          & ! [Y2: real] :
              ( ( aa_set_real_o(member_real2(Y2),set_real2(Xs))
                & aa_real_o(P,Y2) )
             => ( Y2 = X3 ) ) ) ) ).

% list_ex1_iff
tff(fact_55_list__ex1__iff,axiom,
    ! [P: fun_o_o,Xs: list_o] :
      ( list_ex1_o(P,Xs)
    <=> ? [X3: $o] :
          ( aa_set_o_o(member_o2((X3)),set_o2(Xs))
          & aa_o_o(P,(X3))
          & ! [Y2: $o] :
              ( ( aa_set_o_o(member_o2((Y2)),set_o2(Xs))
                & aa_o_o(P,(Y2)) )
             => ( (Y2)
              <=> (X3) ) ) ) ) ).

% list_ex1_iff
tff(fact_56_list__ex1__iff,axiom,
    ! [P: fun_set_nat_o,Xs: list_set_nat] :
      ( list_ex1_set_nat(P,Xs)
    <=> ? [X3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat2(X3),set_set_nat2(Xs))
          & aa_set_nat_o(P,X3)
          & ! [Y2: set_nat] :
              ( ( aa_set_set_nat_o(member_set_nat2(Y2),set_set_nat2(Xs))
                & aa_set_nat_o(P,Y2) )
             => ( Y2 = X3 ) ) ) ) ).

% list_ex1_iff
tff(fact_57_list__ex1__iff,axiom,
    ! [P: fun_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(list_ex1_VEBT_VEBT(P),Xs)
    <=> ? [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(Xs))
          & aa_VEBT_VEBT_o(P,X3)
          & ! [Y2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT2(Y2,set_VEBT_VEBT2(Xs))
                & aa_VEBT_VEBT_o(P,Y2) )
             => ( Y2 = X3 ) ) ) ) ).

% list_ex1_iff
tff(fact_58_list__ex1__iff,axiom,
    ! [P: fun_int_o,Xs: list_int] :
      ( aa_list_int_o(list_ex1_int(P),Xs)
    <=> ? [X3: int] :
          ( aa_set_int_o(member_int2(X3),set_int2(Xs))
          & aa_int_o(P,X3)
          & ! [Y2: int] :
              ( ( aa_set_int_o(member_int2(Y2),set_int2(Xs))
                & aa_int_o(P,Y2) )
             => ( Y2 = X3 ) ) ) ) ).

% list_ex1_iff
tff(fact_59_list__ex1__iff,axiom,
    ! [P: fun_nat_o,Xs: list_nat] :
      ( aa_list_nat_o(list_ex1_nat(P),Xs)
    <=> ? [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),set_nat2(Xs))
          & aa_nat_o(P,X3)
          & ! [Y2: nat] :
              ( ( aa_set_nat_o(member_nat2(Y2),set_nat2(Xs))
                & aa_nat_o(P,Y2) )
             => ( Y2 = X3 ) ) ) ) ).

% list_ex1_iff
tff(fact_60_deg__not__0,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% deg_not_0
tff(fact_61_mi__eq__ma__no__ch,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya)),Dega)
     => ( ( Mia = Maa )
       => ( ! [X: vEBT_VEBT] :
              ( member_VEBT_VEBT2(X,set_VEBT_VEBT2(TreeLista))
             => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_1) )
          & ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summarya),X_1) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_62__C3_C,axiom,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

% "3"
tff(fact_63_buildup__gives__valid,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_vebt_buildup(Nb)),Nb) ) ).

% buildup_gives_valid
tff(fact_64_mem__Collect__eq,axiom,
    ! [A: $o,P: fun_o_o] :
      ( aa_set_o_o(member_o2((A)),collect_o(P))
    <=> aa_o_o(P,(A)) ) ).

% mem_Collect_eq
tff(fact_65_mem__Collect__eq,axiom,
    ! [A: real,P: fun_real_o] :
      ( aa_set_real_o(member_real2(A),collect_real(P))
    <=> aa_real_o(P,A) ) ).

% mem_Collect_eq
tff(fact_66_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_67_mem__Collect__eq,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( aa_set_set_nat_o(member_set_nat2(A),collect_set_nat(P))
    <=> aa_set_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_68_mem__Collect__eq,axiom,
    ! [A: nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat2(A),collect_nat(P))
    <=> aa_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_69_mem__Collect__eq,axiom,
    ! [A: int,P: fun_int_o] :
      ( aa_set_int_o(member_int2(A),collect_int(P))
    <=> aa_int_o(P,A) ) ).

% mem_Collect_eq
tff(fact_70_Collect__mem__eq,axiom,
    ! [A2: set_o] : collect_o(aTP_Lamm_a(A2)) = A2 ).

% Collect_mem_eq
tff(fact_71_Collect__mem__eq,axiom,
    ! [A2: set_real] : collect_real(aTP_Lamm_aa(A2)) = A2 ).

% Collect_mem_eq
tff(fact_72_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] : collect_list_nat(aTP_Lamm_ab(A2)) = A2 ).

% Collect_mem_eq
tff(fact_73_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] : collect_set_nat(aTP_Lamm_ac(A2)) = A2 ).

% Collect_mem_eq
tff(fact_74_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aTP_Lamm_ad(A2)) = A2 ).

% Collect_mem_eq
tff(fact_75_Collect__mem__eq,axiom,
    ! [A2: set_int] : collect_int(aTP_Lamm_ae(A2)) = A2 ).

% Collect_mem_eq
tff(fact_76_Collect__cong,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( ! [X4: real] :
          ( aa_real_o(P,X4)
        <=> aa_real_o(Q,X4) )
     => ( collect_real(P) = collect_real(Q) ) ) ).

% Collect_cong
tff(fact_77_Collect__cong,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ! [X4: list_nat] :
          ( aa_list_nat_o(P,X4)
        <=> aa_list_nat_o(Q,X4) )
     => ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_78_Collect__cong,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X4: set_nat] :
          ( aa_set_nat_o(P,X4)
        <=> aa_set_nat_o(Q,X4) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_79_Collect__cong,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_nat_o(P,X4)
        <=> aa_nat_o(Q,X4) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_80_Collect__cong,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X4: int] :
          ( aa_int_o(P,X4)
        <=> aa_int_o(Q,X4) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_81_max__0R,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),zero_zero_nat) = Nb ).

% max_0R
tff(fact_82_max__0L,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Nb) = Nb ).

% max_0L
tff(fact_83_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_84_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_85_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_86_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_87_max__less__iff__conj,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)),Z)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_88_max__less__iff__conj,axiom,
    ! [Xb: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(Xb),Y)),Z)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Z)
        & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_89_max__less__iff__conj,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ord_max_real(Xb),Y)),Z)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_90_max__less__iff__conj,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y)),Z)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_91_max_Oabsorb4,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(ord_max_nat,A),B) = B ) ) ).

% max.absorb4
tff(fact_92_max_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = B ) ) ).

% max.absorb4
tff(fact_93_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(ord_max_real(A),B) = B ) ) ).

% max.absorb4
tff(fact_94_max_Oabsorb4,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(ord_max_int,A),B) = B ) ) ).

% max.absorb4
tff(fact_95_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb3
tff(fact_96_max_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
     => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = A ) ) ).

% max.absorb3
tff(fact_97_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_real(ord_max_real(A),B) = A ) ) ).

% max.absorb3
tff(fact_98_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb3
tff(fact_99_max_Obounded__iff,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(B),C)),A)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A)
        & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),A) ) ) ).

% max.bounded_iff
tff(fact_100_max_Obounded__iff,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ord_max_real(B),C)),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,C),A) ) ) ).

% max.bounded_iff
tff(fact_101_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% max.bounded_iff
tff(fact_102_max_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% max.bounded_iff
tff(fact_103_max_Oabsorb2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
     => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = B ) ) ).

% max.absorb2
tff(fact_104_max_Oabsorb2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_real(ord_max_real(A),B) = B ) ) ).

% max.absorb2
tff(fact_105_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb2
tff(fact_106_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb2
tff(fact_107_max_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),A) = A ).

% max.idem
tff(fact_108_max_Oidem,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,A),A) = A ).

% max.idem
tff(fact_109_max_Oidem,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),A) = A ).

% max.idem
tff(fact_110_max_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ).

% max.left_idem
tff(fact_111_max_Oleft__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ).

% max.left_idem
tff(fact_112_max_Oleft__idem,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B)) = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) ).

% max.left_idem
tff(fact_113_max_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ).

% max.right_idem
tff(fact_114_max_Oright__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ).

% max.right_idem
tff(fact_115_max_Oright__idem,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B)),B) = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) ).

% max.right_idem
tff(fact_116_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_117_neq0__conv,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% neq0_conv
tff(fact_118_less__nat__zero__code,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_119_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_120_le0,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% le0
tff(fact_121_add__is__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = zero_zero_nat )
    <=> ( ( Mb = zero_zero_nat )
        & ( Nb = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_122_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_123_nat__add__left__cancel__less,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_124_max_Oabsorb1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A)
     => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = A ) ) ).

% max.absorb1
tff(fact_125_max_Oabsorb1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(ord_max_real(A),B) = A ) ) ).

% max.absorb1
tff(fact_126_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb1
tff(fact_127_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb1
tff(fact_128_nat__add__left__cancel__le,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_129_add__gr__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% add_gr_0
tff(fact_130_plus__nat_Oadd__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Nb) = Nb ).

% plus_nat.add_0
tff(fact_131_add__eq__self__zero,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = Mb )
     => ( Nb = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_132_add__lessD1,axiom,
    ! [Ia: 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,Ia),J)),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),K) ) ).

% add_lessD1
tff(fact_133_add__less__mono,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),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,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_less_mono
tff(fact_134_not__add__less1,axiom,
    ! [Ia: 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,Ia),J)),Ia) ).

% not_add_less1
tff(fact_135_not__add__less2,axiom,
    ! [J: nat,Ia: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Ia)),Ia) ).

% not_add_less2
tff(fact_136_add__less__mono1,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_less_mono1
tff(fact_137_trans__less__add1,axiom,
    ! [Ia: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Mb)) ) ).

% trans_less_add1
tff(fact_138_trans__less__add2,axiom,
    ! [Ia: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),J)) ) ).

% trans_less_add2
tff(fact_139_less__add__eq__less,axiom,
    ! [K: nat,L: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% less_add_eq_less
tff(fact_140_add__leE,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),Nb)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ) ).

% add_leE
tff(fact_141_le__add1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) ).

% le_add1
tff(fact_142_le__add2,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% le_add2
tff(fact_143_add__leD1,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% add_leD1
tff(fact_144_add__leD2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ).

% add_leD2
tff(fact_145_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_146_add__le__mono,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),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,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_le_mono
tff(fact_147_add__le__mono1,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_le_mono1
tff(fact_148_trans__le__add1,axiom,
    ! [Ia: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Mb)) ) ).

% trans_le_add1
tff(fact_149_trans__le__add2,axiom,
    ! [Ia: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),J)) ) ).

% trans_le_add2
tff(fact_150_nat__le__iff__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> ? [K2: nat] : Nb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2) ) ).

% nat_le_iff_add
tff(fact_151_nat__add__max__left,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Q2)) ).

% nat_add_max_left
tff(fact_152_nat__add__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q2)) ).

% nat_add_max_right
tff(fact_153_subset__code_I1_J,axiom,
    ! [Xs: list_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_real2(Xs)),B2)
    <=> ! [X3: real] :
          ( aa_set_real_o(member_real2(X3),set_real2(Xs))
         => aa_set_real_o(member_real2(X3),B2) ) ) ).

% subset_code(1)
tff(fact_154_subset__code_I1_J,axiom,
    ! [Xs: list_o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(set_o2(Xs)),B2)
    <=> ! [X3: $o] :
          ( aa_set_o_o(member_o2((X3)),set_o2(Xs))
         => aa_set_o_o(member_o2((X3)),B2) ) ) ).

% subset_code(1)
tff(fact_155_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)
    <=> ! [X3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat2(X3),set_set_nat2(Xs))
         => aa_set_set_nat_o(member_set_nat2(X3),B2) ) ) ).

% subset_code(1)
tff(fact_156_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),B2)
    <=> ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(Xs))
         => member_VEBT_VEBT2(X3,B2) ) ) ).

% subset_code(1)
tff(fact_157_subset__code_I1_J,axiom,
    ! [Xs: list_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),B2)
    <=> ! [X3: int] :
          ( aa_set_int_o(member_int2(X3),set_int2(Xs))
         => aa_set_int_o(member_int2(X3),B2) ) ) ).

% subset_code(1)
tff(fact_158_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)
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),set_nat2(Xs))
         => aa_set_nat_o(member_nat2(X3),B2) ) ) ).

% subset_code(1)
tff(fact_159_less__imp__add__positive,axiom,
    ! [Ia: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K3)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K3) = J ) ) ) ).

% less_imp_add_positive
tff(fact_160_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_161_set__update__subsetI,axiom,
    ! [Xs: list_real,A2: set_real,Xb: real,Ia: nat] :
      ( aa_set_real_o(ord_less_eq_set_real(set_real2(Xs)),A2)
     => ( aa_set_real_o(member_real2(Xb),A2)
       => aa_set_real_o(ord_less_eq_set_real(set_real2(list_update_real(Xs,Ia,Xb))),A2) ) ) ).

% set_update_subsetI
tff(fact_162_set__update__subsetI,axiom,
    ! [Xs: list_o,A2: set_o,Xb: $o,Ia: nat] :
      ( aa_set_o_o(ord_less_eq_set_o(set_o2(Xs)),A2)
     => ( aa_set_o_o(member_o2((Xb)),A2)
       => aa_set_o_o(ord_less_eq_set_o(set_o2(list_update_o(Xs,Ia,(Xb)))),A2) ) ) ).

% set_update_subsetI
tff(fact_163_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A2: set_set_nat,Xb: set_nat,Ia: nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),A2)
     => ( aa_set_set_nat_o(member_set_nat2(Xb),A2)
       => aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_set_nat2(list_update_set_nat(Xs,Ia,Xb))),A2) ) ) ).

% set_update_subsetI
tff(fact_164_set__update__subsetI,axiom,
    ! [Xs: list_int,A2: set_int,Xb: int,Ia: nat] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),A2)
     => ( aa_set_int_o(member_int2(Xb),A2)
       => aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,Ia,Xb))),A2) ) ) ).

% set_update_subsetI
tff(fact_165_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,Xb: vEBT_VEBT,Ia: nat] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),A2)
     => ( member_VEBT_VEBT2(Xb,A2)
       => ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,Ia,Xb)),A2) ) ) ).

% set_update_subsetI
tff(fact_166_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,Xb: nat,Ia: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Xs)),A2)
     => ( aa_set_nat_o(member_nat2(Xb),A2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(list_update_nat(Xs,Ia,Xb))),A2) ) ) ).

% set_update_subsetI
tff(fact_167_nat__neq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb != Nb )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% nat_neq_iff
tff(fact_168_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_not_refl
tff(fact_169_less__not__refl2,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( Mb != Nb ) ) ).

% less_not_refl2
tff(fact_170_less__not__refl3,axiom,
    ! [S: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S),Ta)
     => ( S != Ta ) ) ).

% less_not_refl3
tff(fact_171_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_irrefl_nat
tff(fact_172_nat__less__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
             => aa_nat_o(P,M2) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Nb) ) ).

% nat_less_induct
tff(fact_173_infinite__descent,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(P,N)
         => ? [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
              & ~ aa_nat_o(P,M2) ) )
     => aa_nat_o(P,Nb) ) ).

% infinite_descent
tff(fact_174_linorder__neqE__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).

% linorder_neqE_nat
tff(fact_175_le__refl,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Nb) ).

% le_refl
tff(fact_176_le__trans,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),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,Ia),K) ) ) ).

% le_trans
tff(fact_177_eq__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = Nb )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% eq_imp_le
tff(fact_178_le__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( Mb = Nb ) ) ) ).

% le_antisym
tff(fact_179_nat__le__linear,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ).

% nat_le_linear
tff(fact_180_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) )
       => ? [X4: nat] :
            ( aa_nat_o(P,X4)
            & ! [Y4: nat] :
                ( aa_nat_o(P,Y4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),X4) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_181_max_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)) ).

% max.assoc
tff(fact_182_max_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(ord_max_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)) ).

% max.assoc
tff(fact_183_max_Oassoc,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B)),C) = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(B),C)) ).

% max.assoc
tff(fact_184_max_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),A) ).

% max.commute
tff(fact_185_max_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = aa_int_int(aa_int_fun_int_int(ord_max_int,B),A) ).

% max.commute
tff(fact_186_max_Ocommute,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(B),A) ).

% max.commute
tff(fact_187_max_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)) ).

% max.left_commute
tff(fact_188_max_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,B),aa_int_int(aa_int_fun_int_int(ord_max_int,A),C)) = aa_int_int(aa_int_fun_int_int(ord_max_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)) ).

% max.left_commute
tff(fact_189_max_Oleft__commute,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(B),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),C)) = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(B),C)) ).

% max.left_commute
tff(fact_190_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_191_gr0I,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% gr0I
tff(fact_192_not__gr0,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% not_gr0
tff(fact_193_not__less0,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less0
tff(fact_194_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_zeroE
tff(fact_195_gr__implies__not0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_196_infinite__descent0,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( ~ aa_nat_o(P,N)
             => ? [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
                  & ~ aa_nat_o(P,M2) ) ) )
       => aa_nat_o(P,Nb) ) ) ).

% infinite_descent0
tff(fact_197_less__eq__nat_Osimps_I1_J,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% less_eq_nat.simps(1)
tff(fact_198_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_199_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_200_le__0__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_201_nat__less__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & ( Mb != Nb ) ) ) ).

% nat_less_le
tff(fact_202_less__imp__le__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_imp_le_nat
tff(fact_203_le__eq__less__or__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) ) ) ).

% le_eq_less_or_eq
tff(fact_204_less__or__eq__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_or_eq_imp_le
tff(fact_205_le__neq__implies__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( ( Mb != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% le_neq_implies_less
tff(fact_206_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,Ia: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Ia)),aa_nat_nat(F,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_207_max_Omono,axiom,
    ! [C: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),A)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,D),B)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(C),D)),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B)) ) ) ).

% max.mono
tff(fact_208_max_Omono,axiom,
    ! [C: real,A: real,D: real,B: real] :
      ( 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,D),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ord_max_real(C),D)),aa_real_real(ord_max_real(A),B)) ) ) ).

% max.mono
tff(fact_209_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C),D)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ) ).

% max.mono
tff(fact_210_max_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,C),D)),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ) ).

% max.mono
tff(fact_211_max_OorderE,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A)
     => ( A = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) ) ) ).

% max.orderE
tff(fact_212_max_OorderE,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( A = aa_real_real(ord_max_real(A),B) ) ) ).

% max.orderE
tff(fact_213_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.orderE
tff(fact_214_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.orderE
tff(fact_215_max_OorderI,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) )
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A) ) ).

% max.orderI
tff(fact_216_max_OorderI,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(ord_max_real(A),B) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% max.orderI
tff(fact_217_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% max.orderI
tff(fact_218_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% max.orderI
tff(fact_219_max_OboundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(B),C)),A)
     => ~ ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A)
         => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),A) ) ) ).

% max.boundedE
tff(fact_220_max_OboundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ord_max_real(B),C)),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,C),A) ) ) ).

% max.boundedE
tff(fact_221_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% max.boundedE
tff(fact_222_max_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% max.boundedE
tff(fact_223_max_OboundedI,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),A)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(B),C)),A) ) ) ).

% max.boundedI
tff(fact_224_max_OboundedI,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),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ord_max_real(B),C)),A) ) ) ).

% max.boundedI
tff(fact_225_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A) ) ) ).

% max.boundedI
tff(fact_226_max_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A) ) ) ).

% max.boundedI
tff(fact_227_max_Oorder__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A)
    <=> ( A = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) ) ) ).

% max.order_iff
tff(fact_228_max_Oorder__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
    <=> ( A = aa_real_real(ord_max_real(A),B) ) ) ).

% max.order_iff
tff(fact_229_max_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.order_iff
tff(fact_230_max_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.order_iff
tff(fact_231_max_Ocobounded1,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B)) ).

% max.cobounded1
tff(fact_232_max_Ocobounded1,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(ord_max_real(A),B)) ).

% max.cobounded1
tff(fact_233_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded1
tff(fact_234_max_Ocobounded1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ).

% max.cobounded1
tff(fact_235_max_Ocobounded2,axiom,
    ! [B: extended_enat,A: extended_enat] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B)) ).

% max.cobounded2
tff(fact_236_max_Ocobounded2,axiom,
    ! [B: real,A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(ord_max_real(A),B)) ).

% max.cobounded2
tff(fact_237_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded2
tff(fact_238_max_Ocobounded2,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ).

% max.cobounded2
tff(fact_239_le__max__iff__disj,axiom,
    ! [Z: extended_enat,Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Z),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(Xb),Y))
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Z),Xb)
        | aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_240_le__max__iff__disj,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),aa_real_real(ord_max_real(Xb),Y))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),Xb)
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_241_le__max__iff__disj,axiom,
    ! [Z: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_242_le__max__iff__disj,axiom,
    ! [Z: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xb)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_243_max_Oabsorb__iff1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A)
    <=> ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_244_max_Oabsorb__iff1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
    <=> ( aa_real_real(ord_max_real(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_245_max_Oabsorb__iff1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_246_max_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_247_max_Oabsorb__iff2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
    <=> ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_248_max_Oabsorb__iff2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
    <=> ( aa_real_real(ord_max_real(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_249_max_Oabsorb__iff2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_250_max_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_251_max_OcoboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),A)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B)) ) ).

% max.coboundedI1
tff(fact_252_max_OcoboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( 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,C),aa_real_real(ord_max_real(A),B)) ) ).

% max.coboundedI1
tff(fact_253_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI1
tff(fact_254_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.coboundedI1
tff(fact_255_max_OcoboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),B)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B)) ) ).

% max.coboundedI2
tff(fact_256_max_OcoboundedI2,axiom,
    ! [C: real,B: real,A: 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,C),aa_real_real(ord_max_real(A),B)) ) ).

% max.coboundedI2
tff(fact_257_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI2
tff(fact_258_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.coboundedI2
tff(fact_259_less__max__iff__disj,axiom,
    ! [Z: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_260_less__max__iff__disj,axiom,
    ! [Z: extended_enat,Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(Xb),Y))
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z),Xb)
        | aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_261_less__max__iff__disj,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(ord_max_real(Xb),Y))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xb)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_262_less__max__iff__disj,axiom,
    ! [Z: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Xb)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_263_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
     => ~ ( 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),A) ) ) ).

% max.strict_boundedE
tff(fact_264_max_Ostrict__boundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(B),C)),A)
     => ~ ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
         => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),A) ) ) ).

% max.strict_boundedE
tff(fact_265_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ord_max_real(B),C)),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,C),A) ) ) ).

% max.strict_boundedE
tff(fact_266_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)
     => ~ ( 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),A) ) ) ).

% max.strict_boundedE
tff(fact_267_max_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_268_max_Ostrict__order__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
    <=> ( ( A = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_269_max_Ostrict__order__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( ( A = aa_real_real(ord_max_real(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_270_max_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_271_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_272_max_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),A)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B)) ) ).

% max.strict_coboundedI1
tff(fact_273_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(ord_max_real(A),B)) ) ).

% max.strict_coboundedI1
tff(fact_274_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_275_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( 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),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_276_max_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),B)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B)) ) ).

% max.strict_coboundedI2
tff(fact_277_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( 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),aa_real_real(ord_max_real(A),B)) ) ).

% max.strict_coboundedI2
tff(fact_278_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( 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),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_279_ex__least__nat__le,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),Nb)
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K3)
               => ~ aa_nat_o(P,I3) )
            & aa_nat_o(P,K3) ) ) ) ).

% ex_least_nat_le
tff(fact_280_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_281_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel1
tff(fact_282_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_283_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_284_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel2
tff(fact_285_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_286_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_287_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(plus_plus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel1
tff(fact_288_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_289_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_290_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(plus_plus_real(B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel2
tff(fact_291_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_292_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_293_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_294_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_295_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_296_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel1
tff(fact_297_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_298_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_299_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel2
tff(fact_300_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_301_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_302_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(plus_plus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel1
tff(fact_303_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_304_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_305_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(plus_plus_real(B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel2
tff(fact_306_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_307_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_308_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_309_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_310_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_311_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_312_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_313_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_314_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(plus_plus_real(B),A) = aa_real_real(plus_plus_real(C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_315_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_316_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_317_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_318_le__zero__eq,axiom,
    ! [Nb: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Nb),zero_z5237406670263579293d_enat)
    <=> ( Nb = zero_z5237406670263579293d_enat ) ) ).

% le_zero_eq
tff(fact_319_le__zero__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_320_not__gr__zero,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_321_not__gr__zero,axiom,
    ! [Nb: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),Nb)
    <=> ( Nb = zero_z5237406670263579293d_enat ) ) ).

% not_gr_zero
tff(fact_322_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_right
tff(fact_323_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_324_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_325_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_left
tff(fact_326_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_327_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_328_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_329_add__0,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(zero_zero_real),A) = A ).

% add_0
tff(fact_330_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_331_add__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add_0
tff(fact_332_add__0,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(zero_z5237406670263579293d_enat),A) = A ).

% add_0
tff(fact_333_zero__eq__add__iff__both__eq__0,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) )
    <=> ( ( Xb = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_334_zero__eq__add__iff__both__eq__0,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( ( zero_z5237406670263579293d_enat = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Xb),Y) )
    <=> ( ( Xb = zero_z5237406670263579293d_enat )
        & ( Y = zero_z5237406670263579293d_enat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_335_add__eq__0__iff__both__eq__0,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = zero_zero_nat )
    <=> ( ( Xb = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_336_add__eq__0__iff__both__eq__0,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Xb),Y) = zero_z5237406670263579293d_enat )
    <=> ( ( Xb = zero_z5237406670263579293d_enat )
        & ( Y = zero_z5237406670263579293d_enat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_337_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_338_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(plus_plus_real(A),B) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_339_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_340_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_341_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_342_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(plus_plus_real(B),A) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_343_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_344_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_345_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_346_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_347_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_348_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_349_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_350_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(plus_plus_real(B),A) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_351_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_352_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_353_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(plus_plus_real(A),A) )
    <=> ( A = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_354_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_355_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_356_add_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),zero_zero_real) = A ).

% add.right_neutral
tff(fact_357_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_358_add_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.right_neutral
tff(fact_359_add_Oright__neutral,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),zero_z5237406670263579293d_enat) = A ).

% add.right_neutral
tff(fact_360_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_361_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_right
tff(fact_362_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_363_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_364_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_left
tff(fact_365_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_366_zero__reorient,axiom,
    ! [Xb: nat] :
      ( ( zero_zero_nat = Xb )
    <=> ( Xb = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_367_zero__reorient,axiom,
    ! [Xb: real] :
      ( ( zero_zero_real = Xb )
    <=> ( Xb = zero_zero_real ) ) ).

% zero_reorient
tff(fact_368_zero__reorient,axiom,
    ! [Xb: int] :
      ( ( zero_zero_int = Xb )
    <=> ( Xb = zero_zero_int ) ) ).

% zero_reorient
tff(fact_369_zero__reorient,axiom,
    ! [Xb: complex] :
      ( ( zero_zero_complex = Xb )
    <=> ( Xb = zero_zero_complex ) ) ).

% zero_reorient
tff(fact_370_zero__reorient,axiom,
    ! [Xb: extended_enat] :
      ( ( zero_z5237406670263579293d_enat = Xb )
    <=> ( Xb = zero_z5237406670263579293d_enat ) ) ).

% zero_reorient
tff(fact_371_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_372_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_373_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(plus_plus_real(B),A) = aa_real_real(plus_plus_real(C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_374_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_375_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_376_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_377_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_378_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_379_add_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(plus_plus_real(B),aa_real_real(plus_plus_real(A),C)) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% add.left_commute
tff(fact_380_add_Oleft__commute,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B),C)) ).

% add.left_commute
tff(fact_381_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_382_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_383_add_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(B),A) ).

% add.commute
tff(fact_384_add_Ocommute,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B),A) ).

% add.commute
tff(fact_385_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_386_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(plus_plus_real(B),A) = aa_real_real(plus_plus_real(C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_387_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_388_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_389_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_390_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_391_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% add.assoc
tff(fact_392_add_Oassoc,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)),C) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B),C)) ).

% add.assoc
tff(fact_393_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_394_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_395_group__cancel_Oadd2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2 = aa_real_real(plus_plus_real(K),B) )
     => ( aa_real_real(plus_plus_real(A),B2) = aa_real_real(plus_plus_real(K),aa_real_real(plus_plus_real(A),B)) ) ) ).

% group_cancel.add2
tff(fact_396_group__cancel_Oadd2,axiom,
    ! [B2: extended_enat,K: extended_enat,B: extended_enat,A: extended_enat] :
      ( ( B2 = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(K),B) )
     => ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B2) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(K),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)) ) ) ).

% group_cancel.add2
tff(fact_397_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_398_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_399_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(plus_plus_real(K),A) )
     => ( aa_real_real(plus_plus_real(A2),B) = aa_real_real(plus_plus_real(K),aa_real_real(plus_plus_real(A),B)) ) ) ).

% group_cancel.add1
tff(fact_400_group__cancel_Oadd1,axiom,
    ! [A2: extended_enat,K: extended_enat,A: extended_enat,B: extended_enat] :
      ( ( A2 = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(K),A) )
     => ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A2),B) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(K),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)) ) ) ).

% group_cancel.add1
tff(fact_401_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( ( ( Ia = J )
        & ( K = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_402_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [Ia: int,J: int,K: int,L: int] :
      ( ( ( Ia = J )
        & ( K = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Ia),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_403_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [Ia: real,J: real,K: real,L: real] :
      ( ( ( Ia = J )
        & ( K = L ) )
     => ( aa_real_real(plus_plus_real(Ia),K) = aa_real_real(plus_plus_real(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_404_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [Ia: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
      ( ( ( Ia = J )
        & ( K = L ) )
     => ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Ia),K) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_405_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_406_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_407_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_408_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)),C) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_409_zero__le,axiom,
    ! [Xb: extended_enat] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),Xb) ).

% zero_le
tff(fact_410_zero__le,axiom,
    ! [Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb) ).

% zero_le
tff(fact_411_zero__less__iff__neq__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_412_zero__less__iff__neq__zero,axiom,
    ! [Nb: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),Nb)
    <=> ( Nb != zero_z5237406670263579293d_enat ) ) ).

% zero_less_iff_neq_zero
tff(fact_413_gr__implies__not__zero,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_414_gr__implies__not__zero,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Mb),Nb)
     => ( Nb != zero_z5237406670263579293d_enat ) ) ).

% gr_implies_not_zero
tff(fact_415_not__less__zero,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less_zero
tff(fact_416_not__less__zero,axiom,
    ! [Nb: extended_enat] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Nb),zero_z5237406670263579293d_enat) ).

% not_less_zero
tff(fact_417_gr__zeroI,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% gr_zeroI
tff(fact_418_gr__zeroI,axiom,
    ! [Nb: extended_enat] :
      ( ( Nb != zero_z5237406670263579293d_enat )
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),Nb) ) ).

% gr_zeroI
tff(fact_419_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_right
tff(fact_420_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_421_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_422_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_left
tff(fact_423_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_424_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_425_le__iff__add,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
    <=> ? [C2: extended_enat] : B = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C2) ) ).

% le_iff_add
tff(fact_426_le__iff__add,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) ) ).

% le_iff_add
tff(fact_427_add__right__mono,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B),C)) ) ).

% add_right_mono
tff(fact_428_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C)) ) ).

% add_right_mono
tff(fact_429_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_430_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_431_less__eqE,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
     => ~ ! [C3: extended_enat] : B != aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C3) ) ).

% less_eqE
tff(fact_432_less__eqE,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) ) ).

% less_eqE
tff(fact_433_add__left__mono,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(C),A)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(C),B)) ) ).

% add_left_mono
tff(fact_434_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B)) ) ).

% add_left_mono
tff(fact_435_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_436_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_437_add__mono,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),D)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B),D)) ) ) ).

% add_mono
tff(fact_438_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) ) ) ).

% add_mono
tff(fact_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__mono__thms__linordered__semiring_I1_J,axiom,
    ! [Ia: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
      ( ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Ia),J)
        & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,K),L) )
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Ia),K)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_442_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [Ia: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ia),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(plus_plus_real(Ia),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_443_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),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,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_444_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [Ia: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),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,Ia),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_445_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [Ia: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
      ( ( ( Ia = J )
        & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,K),L) )
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Ia),K)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_446_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [Ia: real,J: real,K: real,L: real] :
      ( ( ( Ia = 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(plus_plus_real(Ia),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_447_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( ( ( Ia = 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,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_448_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [Ia: int,J: int,K: int,L: int] :
      ( ( ( Ia = 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,Ia),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_449_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [Ia: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
      ( ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Ia),J)
        & ( K = L ) )
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Ia),K)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_450_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [Ia: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ia),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(Ia),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_451_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),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,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_452_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [Ia: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),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,Ia),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_453_add_Ogroup__left__neutral,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(zero_zero_real),A) = A ).

% add.group_left_neutral
tff(fact_454_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_455_add_Ogroup__left__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add.group_left_neutral
tff(fact_456_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_457_add_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),zero_zero_real) = A ).

% add.comm_neutral
tff(fact_458_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_459_add_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.comm_neutral
tff(fact_460_add_Ocomm__neutral,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),zero_z5237406670263579293d_enat) = A ).

% add.comm_neutral
tff(fact_461_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_462_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(zero_zero_real),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_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_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_465_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(zero_z5237406670263579293d_enat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_466_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_467_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_right
tff(fact_468_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_469_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_470_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_left
tff(fact_471_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_472_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_473_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C)) ) ).

% add_strict_right_mono
tff(fact_474_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_475_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_476_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B)) ) ).

% add_strict_left_mono
tff(fact_477_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_478_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_479_add__strict__mono,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),D)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B),D)) ) ) ).

% add_strict_mono
tff(fact_480_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) ) ) ).

% add_strict_mono
tff(fact_481_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_482_add__mono__thms__linordered__field_I1_J,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),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,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_483_add__mono__thms__linordered__field_I1_J,axiom,
    ! [Ia: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ia),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(Ia),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_484_add__mono__thms__linordered__field_I1_J,axiom,
    ! [Ia: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ia),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,Ia),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_485_add__mono__thms__linordered__field_I2_J,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( ( ( Ia = 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,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_486_add__mono__thms__linordered__field_I2_J,axiom,
    ! [Ia: real,J: real,K: real,L: real] :
      ( ( ( Ia = 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(plus_plus_real(Ia),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_487_add__mono__thms__linordered__field_I2_J,axiom,
    ! [Ia: int,J: int,K: int,L: int] :
      ( ( ( Ia = 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,Ia),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_488_add__mono__thms__linordered__field_I5_J,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),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,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_489_add__mono__thms__linordered__field_I5_J,axiom,
    ! [Ia: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ia),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(plus_plus_real(Ia),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_490_add__mono__thms__linordered__field_I5_J,axiom,
    ! [Ia: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ia),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,Ia),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_491_max__add__distrib__right,axiom,
    ! [Xb: real,Y: real,Z: real] : aa_real_real(plus_plus_real(Xb),aa_real_real(ord_max_real(Y),Z)) = aa_real_real(ord_max_real(aa_real_real(plus_plus_real(Xb),Y)),aa_real_real(plus_plus_real(Xb),Z)) ).

% max_add_distrib_right
tff(fact_492_max__add__distrib__right,axiom,
    ! [Xb: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Z)) ).

% max_add_distrib_right
tff(fact_493_max__add__distrib__right,axiom,
    ! [Xb: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),aa_int_int(aa_int_fun_int_int(ord_max_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Z)) ).

% max_add_distrib_right
tff(fact_494_max__add__distrib__left,axiom,
    ! [Xb: real,Y: real,Z: real] : aa_real_real(plus_plus_real(aa_real_real(ord_max_real(Xb),Y)),Z) = aa_real_real(ord_max_real(aa_real_real(plus_plus_real(Xb),Z)),aa_real_real(plus_plus_real(Y),Z)) ).

% max_add_distrib_left
tff(fact_495_max__add__distrib__left,axiom,
    ! [Xb: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z)) ).

% max_add_distrib_left
tff(fact_496_max__add__distrib__left,axiom,
    ! [Xb: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z)) ).

% max_add_distrib_left
tff(fact_497_add__nonpos__eq__0__iff,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xb),zero_z5237406670263579293d_enat)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Y),zero_z5237406670263579293d_enat)
       => ( ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Xb),Y) = zero_z5237406670263579293d_enat )
        <=> ( ( Xb = zero_z5237406670263579293d_enat )
            & ( Y = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_498_add__nonpos__eq__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( ( aa_real_real(plus_plus_real(Xb),Y) = zero_zero_real )
        <=> ( ( Xb = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_499_add__nonpos__eq__0__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),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,Xb),Y) = zero_zero_nat )
        <=> ( ( Xb = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_500_add__nonpos__eq__0__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),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,Xb),Y) = zero_zero_int )
        <=> ( ( Xb = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_501_add__nonneg__eq__0__iff,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),Xb)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),Y)
       => ( ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Xb),Y) = zero_z5237406670263579293d_enat )
        <=> ( ( Xb = zero_z5237406670263579293d_enat )
            & ( Y = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_502_add__nonneg__eq__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_real_real(plus_plus_real(Xb),Y) = zero_zero_real )
        <=> ( ( Xb = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_503_add__nonneg__eq__0__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
     => ( 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,Xb),Y) = zero_zero_nat )
        <=> ( ( Xb = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_504_add__nonneg__eq__0__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( 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,Xb),Y) = zero_zero_int )
        <=> ( ( Xb = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_505_add__nonpos__nonpos,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),zero_z5237406670263579293d_enat)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),zero_z5237406670263579293d_enat)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)),zero_z5237406670263579293d_enat) ) ) ).

% add_nonpos_nonpos
tff(fact_506_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),B)),zero_zero_real) ) ) ).

% add_nonpos_nonpos
tff(fact_507_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_508_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_509_add__nonneg__nonneg,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),A)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),B)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_510_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_511_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_512_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_513_add__increasing2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),C)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C)) ) ) ).

% add_increasing2
tff(fact_514_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(plus_plus_real(A),C)) ) ) ).

% add_increasing2
tff(fact_515_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_516_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_517_add__decreasing2,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),zero_z5237406670263579293d_enat)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C)),B) ) ) ).

% add_decreasing2
tff(fact_518_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),B) ) ) ).

% add_decreasing2
tff(fact_519_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_520_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_521_add__increasing,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),A)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),C)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C)) ) ) ).

% add_increasing
tff(fact_522_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(plus_plus_real(A),C)) ) ) ).

% add_increasing
tff(fact_523_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_524_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_525_add__decreasing,axiom,
    ! [A: extended_enat,C: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),zero_z5237406670263579293d_enat)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),B)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C)),B) ) ) ).

% add_decreasing
tff(fact_526_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),B) ) ) ).

% add_decreasing
tff(fact_527_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_528_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_529_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) ) ) ).

% add_less_le_mono
tff(fact_530_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_531_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_532_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) ) ) ).

% add_le_less_mono
tff(fact_533_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_534_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_535_add__mono__thms__linordered__field_I3_J,axiom,
    ! [Ia: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ia),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(plus_plus_real(Ia),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_536_add__mono__thms__linordered__field_I3_J,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),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,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_537_add__mono__thms__linordered__field_I3_J,axiom,
    ! [Ia: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ia),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,Ia),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_538_add__mono__thms__linordered__field_I4_J,axiom,
    ! [Ia: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ia),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(plus_plus_real(Ia),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_539_add__mono__thms__linordered__field_I4_J,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),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,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_540_add__mono__thms__linordered__field_I4_J,axiom,
    ! [Ia: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),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,Ia),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_541_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_542_pos__add__strict,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),A)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),C)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C)) ) ) ).

% pos_add_strict
tff(fact_543_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(plus_plus_real(A),C)) ) ) ).

% pos_add_strict
tff(fact_544_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_545_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ ! [C3: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_546_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ~ ! [C3: extended_enat] :
            ( ( B = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C3) )
           => ( C3 = zero_z5237406670263579293d_enat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_547_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_548_add__pos__pos,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),A)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),B)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)) ) ) ).

% add_pos_pos
tff(fact_549_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(A),B)) ) ) ).

% add_pos_pos
tff(fact_550_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_551_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_552_add__neg__neg,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),zero_z5237406670263579293d_enat)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),zero_z5237406670263579293d_enat)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)),zero_z5237406670263579293d_enat) ) ) ).

% add_neg_neg
tff(fact_553_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),B)),zero_zero_real) ) ) ).

% add_neg_neg
tff(fact_554_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_555_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(plus_plus_real(A),C)) ) ) ).

% add_strict_increasing2
tff(fact_556_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_557_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_558_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(plus_plus_real(A),C)) ) ) ).

% add_strict_increasing
tff(fact_559_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_560_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_561_add__pos__nonneg,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),A)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),B)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)) ) ) ).

% add_pos_nonneg
tff(fact_562_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(A),B)) ) ) ).

% add_pos_nonneg
tff(fact_563_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_564_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_565_add__nonpos__neg,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),zero_z5237406670263579293d_enat)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),zero_z5237406670263579293d_enat)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)),zero_z5237406670263579293d_enat) ) ) ).

% add_nonpos_neg
tff(fact_566_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),B)),zero_zero_real) ) ) ).

% add_nonpos_neg
tff(fact_567_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_568_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_569_add__nonneg__pos,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),A)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),B)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)) ) ) ).

% add_nonneg_pos
tff(fact_570_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(A),B)) ) ) ).

% add_nonneg_pos
tff(fact_571_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_572_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_573_add__neg__nonpos,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),zero_z5237406670263579293d_enat)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),zero_z5237406670263579293d_enat)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)),zero_z5237406670263579293d_enat) ) ) ).

% add_neg_nonpos
tff(fact_574_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),B)),zero_zero_real) ) ) ).

% add_neg_nonpos
tff(fact_575_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_576_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_577_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(plus_plus_real(A),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_578_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_579_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),Xb) ).

% vebt_member.simps(3)
tff(fact_580_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,Xb: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,vEBT_vebt_buildup(Nb)),Xb) ).

% buildup_nothing_in_leaf
tff(fact_581_field__le__epsilon,axiom,
    ! [Xb: 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,Xb),aa_real_real(plus_plus_real(Y),E)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% field_le_epsilon
tff(fact_582_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_583_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero_nat,Ts,S),Xb) = vEBT_Node(Info,zero_zero_nat,Ts,S) ).

% vebt_insert.simps(2)
tff(fact_584__C8_C,axiom,
    aa_nat_nat(suc,na) = m ).

% "8"
tff(fact_585_can__select__set__list__ex1,axiom,
    ! [P: fun_VEBT_VEBT_o,A2: list_VEBT_VEBT] :
      ( can_select_VEBT_VEBT(P,set_VEBT_VEBT2(A2))
    <=> aa_list_VEBT_VEBT_o(list_ex1_VEBT_VEBT(P),A2) ) ).

% can_select_set_list_ex1
tff(fact_586_can__select__set__list__ex1,axiom,
    ! [P: fun_int_o,A2: list_int] :
      ( aa_set_int_o(can_select_int(P),set_int2(A2))
    <=> aa_list_int_o(list_ex1_int(P),A2) ) ).

% can_select_set_list_ex1
tff(fact_587_can__select__set__list__ex1,axiom,
    ! [P: fun_nat_o,A2: list_nat] :
      ( aa_set_nat_o(can_select_nat(P),set_nat2(A2))
    <=> aa_list_nat_o(list_ex1_nat(P),A2) ) ).

% can_select_set_list_ex1
tff(fact_588_add__less__zeroD,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(Xb),Y)),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real) ) ) ).

% add_less_zeroD
tff(fact_589_add__less__zeroD,axiom,
    ! [Xb: 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,Xb),Y)),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_590_buildup__gives__empty,axiom,
    ! [Nb: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_591_even__odd__cases,axiom,
    ! [Xb: nat] :
      ( ! [N: nat] : Xb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)
     => ~ ! [N: nat] : Xb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(suc,N)) ) ).

% even_odd_cases
tff(fact_592_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tree),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
     => ? [Info2: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,Nb)),TreeList,S2) ) ).

% deg_SUcn_Node
tff(fact_593_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_594_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( aa_nat_nat(suc,X22) = aa_nat_nat(suc,Y22) )
    <=> ( X22 = Y22 ) ) ).

% nat.inject
tff(fact_595_lessI,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Nb)) ).

% lessI
tff(fact_596_Suc__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) ) ).

% Suc_mono
tff(fact_597_Suc__less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_less_eq
tff(fact_598_Suc__le__mono,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ).

% Suc_le_mono
tff(fact_599_add__Suc__right,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% add_Suc_right
tff(fact_600_max__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)) ).

% max_Suc_Suc
tff(fact_601_less__Suc0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_602_zero__less__Suc,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) ).

% zero_less_Suc
tff(fact_603_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa_nat_nat(suc,Nb) ).

% n_not_Suc_n
tff(fact_604_Suc__inject,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(suc,Xb) = aa_nat_nat(suc,Y) )
     => ( Xb = Y ) ) ).

% Suc_inject
tff(fact_605_can__select__def,axiom,
    ! [P: fun_real_o,A2: set_real] :
      ( aa_set_real_o(can_select_real(P),A2)
    <=> ? [X3: real] :
          ( aa_set_real_o(member_real2(X3),A2)
          & aa_real_o(P,X3)
          & ! [Y2: real] :
              ( ( aa_set_real_o(member_real2(Y2),A2)
                & aa_real_o(P,Y2) )
             => ( Y2 = X3 ) ) ) ) ).

% can_select_def
tff(fact_606_can__select__def,axiom,
    ! [P: fun_o_o,A2: set_o] :
      ( aa_set_o_o(can_select_o(P),A2)
    <=> ? [X3: $o] :
          ( aa_set_o_o(member_o2((X3)),A2)
          & aa_o_o(P,(X3))
          & ! [Y2: $o] :
              ( ( aa_set_o_o(member_o2((Y2)),A2)
                & aa_o_o(P,(Y2)) )
             => ( (Y2)
              <=> (X3) ) ) ) ) ).

% can_select_def
tff(fact_607_can__select__def,axiom,
    ! [P: fun_set_nat_o,A2: set_set_nat] :
      ( aa_set_set_nat_o(can_select_set_nat(P),A2)
    <=> ? [X3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat2(X3),A2)
          & aa_set_nat_o(P,X3)
          & ! [Y2: set_nat] :
              ( ( aa_set_set_nat_o(member_set_nat2(Y2),A2)
                & aa_set_nat_o(P,Y2) )
             => ( Y2 = X3 ) ) ) ) ).

% can_select_def
tff(fact_608_can__select__def,axiom,
    ! [P: fun_nat_o,A2: set_nat] :
      ( aa_set_nat_o(can_select_nat(P),A2)
    <=> ? [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),A2)
          & aa_nat_o(P,X3)
          & ! [Y2: nat] :
              ( ( aa_set_nat_o(member_nat2(Y2),A2)
                & aa_nat_o(P,Y2) )
             => ( Y2 = X3 ) ) ) ) ).

% can_select_def
tff(fact_609_can__select__def,axiom,
    ! [P: fun_int_o,A2: set_int] :
      ( aa_set_int_o(can_select_int(P),A2)
    <=> ? [X3: int] :
          ( aa_set_int_o(member_int2(X3),A2)
          & aa_int_o(P,X3)
          & ! [Y2: int] :
              ( ( aa_set_int_o(member_int2(Y2),A2)
                & aa_int_o(P,Y2) )
             => ( Y2 = X3 ) ) ) ) ).

% can_select_def
tff(fact_610_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero_nat != aa_nat_nat(suc,X22) ).

% nat.distinct(1)
tff(fact_611_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_612_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_613_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa_nat_nat(suc,X22) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_614_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_615_nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => aa_nat_o(P,aa_nat_nat(suc,N)) )
       => aa_nat_o(P,Nb) ) ) ).

% nat_induct
tff(fact_616_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Mb: nat,Nb: nat] :
      ( ! [X4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,X4),zero_zero_nat)
     => ( ! [Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,zero_zero_nat),aa_nat_nat(suc,Y3))
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(suc,X4)),aa_nat_nat(suc,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,Mb),Nb) ) ) ) ).

% diff_induct
tff(fact_617_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_618_Suc__neq__Zero,axiom,
    ! [Mb: nat] : aa_nat_nat(suc,Mb) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_619_Zero__neq__Suc,axiom,
    ! [Mb: nat] : zero_zero_nat != aa_nat_nat(suc,Mb) ).

% Zero_neq_Suc
tff(fact_620_Zero__not__Suc,axiom,
    ! [Mb: nat] : zero_zero_nat != aa_nat_nat(suc,Mb) ).

% Zero_not_Suc
tff(fact_621_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ? [M: nat] : Nb = aa_nat_nat(suc,M) ) ).

% not0_implies_Suc
tff(fact_622_Nat_OlessE,axiom,
    ! [Ia: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),K)
     => ( ( K != aa_nat_nat(suc,Ia) )
       => ~ ! [J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J2)
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_623_Suc__lessD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_lessD
tff(fact_624_Suc__lessE,axiom,
    ! [Ia: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Ia)),K)
     => ~ ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J2)
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_625_Suc__lessI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( ( aa_nat_nat(suc,Mb) != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),Nb) ) ) ).

% Suc_lessI
tff(fact_626_less__SucE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ( Mb = Nb ) ) ) ).

% less_SucE
tff(fact_627_less__SucI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% less_SucI
tff(fact_628_Ex__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Nb))
          & aa_nat_o(P,I4) )
    <=> ( aa_nat_o(P,Nb)
        | ? [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
            & aa_nat_o(P,I4) ) ) ) ).

% Ex_less_Suc
tff(fact_629_less__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) ) ) ).

% less_Suc_eq
tff(fact_630_not__less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb)) ) ).

% not_less_eq
tff(fact_631_All__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Nb))
         => aa_nat_o(P,I4) )
    <=> ( aa_nat_o(P,Nb)
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
           => aa_nat_o(P,I4) ) ) ) ).

% All_less_Suc
tff(fact_632_Suc__less__eq2,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb)
    <=> ? [M3: nat] :
          ( ( Mb = aa_nat_nat(suc,M3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M3) ) ) ).

% Suc_less_eq2
tff(fact_633_less__antisym,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
       => ( Mb = Nb ) ) ) ).

% less_antisym
tff(fact_634_Suc__less__SucD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_less_SucD
tff(fact_635_less__trans__Suc,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),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,Ia)),K) ) ) ).

% less_trans_Suc
tff(fact_636_less__Suc__induct,axiom,
    ! [Ia: nat,J: nat,P: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => ( ! [I2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I2),aa_nat_nat(suc,I2))
       => ( ! [I2: nat,J2: nat,K3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),K3)
               => ( aa_nat_o(aa_nat_fun_nat_o(P,I2),J2)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P,J2),K3)
                   => aa_nat_o(aa_nat_fun_nat_o(P,I2),K3) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,Ia),J) ) ) ) ).

% less_Suc_induct
tff(fact_637_strict__inc__induct,axiom,
    ! [Ia: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => ( ! [I2: nat] :
            ( ( J = aa_nat_nat(suc,I2) )
           => aa_nat_o(P,I2) )
       => ( ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
             => ( aa_nat_o(P,aa_nat_nat(suc,I2))
               => aa_nat_o(P,I2) ) )
         => aa_nat_o(P,Ia) ) ) ) ).

% strict_inc_induct
tff(fact_638_not__less__less__Suc__eq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
      <=> ( Nb = Mb ) ) ) ).

% not_less_less_Suc_eq
tff(fact_639_Suc__leD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% Suc_leD
tff(fact_640_le__SucE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( Mb = aa_nat_nat(suc,Nb) ) ) ) ).

% le_SucE
tff(fact_641_le__SucI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% le_SucI
tff(fact_642_Suc__le__D,axiom,
    ! [Nb: nat,M4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),M4)
     => ? [M: nat] : M4 = aa_nat_nat(suc,M) ) ).

% Suc_le_D
tff(fact_643_le__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        | ( Mb = aa_nat_nat(suc,Nb) ) ) ) ).

% le_Suc_eq
tff(fact_644_Suc__n__not__le__n,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Nb) ).

% Suc_n_not_le_n
tff(fact_645_not__less__eq__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Mb) ) ).

% not_less_eq_eq
tff(fact_646_full__nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M2)),N)
             => aa_nat_o(P,M2) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Nb) ) ).

% full_nat_induct
tff(fact_647_nat__induct__at__least,axiom,
    ! [Mb: nat,Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(P,Mb)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_at_least
tff(fact_648_transitive__stepwise__le,axiom,
    ! [Mb: nat,Nb: nat,R: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( ! [X4: nat] : aa_nat_o(aa_nat_fun_nat_o(R,X4),X4)
       => ( ! [X4: nat,Y3: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X4),Y3)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y3),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(R,X4),Z2) ) )
         => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(R,N),aa_nat_nat(suc,N))
           => aa_nat_o(aa_nat_fun_nat_o(R,Mb),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_649_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_650_add__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% add_Suc
tff(fact_651_add__Suc__shift,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(suc,Nb)) ).

% add_Suc_shift
tff(fact_652_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_653_lift__Suc__mono__less,axiom,
    ! [F: fun_na8797796813480665199d_enat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,N)),aa_nat_Extended_enat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,Nb)),aa_nat_Extended_enat(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_654_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_655_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_656_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_657_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_na8797796813480665199d_enat,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,N)),aa_nat_Extended_enat(F,aa_nat_nat(suc,N)))
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,Nb)),aa_nat_Extended_enat(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_658_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_659_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_660_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_filter_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : ord_le2510731241096832064er_nat(aa_nat_filter_nat(F,N),aa_nat_filter_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => ord_le2510731241096832064er_nat(aa_nat_filter_nat(F,Nb),aa_nat_filter_nat(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_661_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_real,Nb: nat,N2: nat] :
      ( ! [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)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,Nb)),aa_nat_real(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_662_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(F,N)),aa_nat_set_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(F,Nb)),aa_nat_set_nat(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_663_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_664_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Nb)),aa_nat_int(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_665_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_filter_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : ord_le2510731241096832064er_nat(aa_nat_filter_nat(F,aa_nat_nat(suc,N)),aa_nat_filter_nat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => ord_le2510731241096832064er_nat(aa_nat_filter_nat(F,N2),aa_nat_filter_nat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_666_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_real,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,aa_nat_nat(suc,N))),aa_nat_real(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_667_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(F,aa_nat_nat(suc,N))),aa_nat_set_nat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(F,N2)),aa_nat_set_nat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_668_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N))),aa_nat_nat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_669_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_670_Ex__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Nb))
          & aa_nat_o(P,I4) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        | ? [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
            & aa_nat_o(P,aa_nat_nat(suc,I4)) ) ) ) ).

% Ex_less_Suc2
tff(fact_671_gr0__conv__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ? [M5: nat] : Nb = aa_nat_nat(suc,M5) ) ).

% gr0_conv_Suc
tff(fact_672_All__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Nb))
         => aa_nat_o(P,I4) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
           => aa_nat_o(P,aa_nat_nat(suc,I4)) ) ) ) ).

% All_less_Suc2
tff(fact_673_gr0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ? [M: nat] : Nb = aa_nat_nat(suc,M) ) ).

% gr0_implies_Suc
tff(fact_674_less__Suc__eq__0__disj,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( ( Mb = zero_zero_nat )
        | ? [J3: nat] :
            ( ( Mb = aa_nat_nat(suc,J3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_675_Suc__leI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb) ) ).

% Suc_leI
tff(fact_676_Suc__le__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_le_eq
tff(fact_677_dec__induct,axiom,
    ! [Ia: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => ( aa_nat_o(P,Ia)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),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_678_inc__induct,axiom,
    ! [Ia: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => ( aa_nat_o(P,J)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),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,Ia) ) ) ) ).

% inc_induct
tff(fact_679_Suc__le__lessD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_le_lessD
tff(fact_680_le__less__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
      <=> ( Nb = Mb ) ) ) ).

% le_less_Suc_eq
tff(fact_681_less__Suc__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_Suc_eq_le
tff(fact_682_less__eq__Suc__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Mb) ) ).

% less_eq_Suc_le
tff(fact_683_le__imp__less__Suc,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% le_imp_less_Suc
tff(fact_684_one__is__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) )
    <=> ( ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( Mb = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_685_add__is__1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( Mb = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_686_less__imp__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ? [K3: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K3)) ) ).

% less_imp_Suc_add
tff(fact_687_less__iff__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> ? [K2: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)) ) ).

% less_iff_Suc_add
tff(fact_688_less__add__Suc2,axiom,
    ! [Ia: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Ia))) ).

% less_add_Suc2
tff(fact_689_less__add__Suc1,axiom,
    ! [Ia: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),Mb))) ).

% less_add_Suc1
tff(fact_690_less__natE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ~ ! [Q3: nat] : Nb != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q3)) ) ).

% less_natE
tff(fact_691_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S),Xb) = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S) ).

% vebt_insert.simps(3)
tff(fact_692_ex__least__nat__less,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),K3)
               => ~ aa_nat_o(P,I3) )
            & aa_nat_o(P,aa_nat_nat(suc,K3)) ) ) ) ).

% ex_least_nat_less
tff(fact_693_linordered__field__no__lb,axiom,
    ! [X: real] :
    ? [Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),X) ).

% linordered_field_no_lb
tff(fact_694_linordered__field__no__ub,axiom,
    ! [X: real] :
    ? [X_12: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X),X_12) ).

% linordered_field_no_ub
tff(fact_695_linorder__neqE__linordered__idom,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_696_linorder__neqE__linordered__idom,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_697_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),Xb) ).

% vebt_member.simps(4)
tff(fact_698_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tree),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tree),Xb)
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Tree),Xb)
          | aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Tree),Xb) ) ) ) ).

% member_valid_both_member_options
tff(fact_699_max__bot,axiom,
    ! [Xb: set_real] : ord_max_set_real(bot_bot_set_real,Xb) = Xb ).

% max_bot
tff(fact_700_max__bot,axiom,
    ! [Xb: set_o] : ord_max_set_o(bot_bot_set_o,Xb) = Xb ).

% max_bot
tff(fact_701_max__bot,axiom,
    ! [Xb: set_nat] : ord_max_set_nat(bot_bot_set_nat,Xb) = Xb ).

% max_bot
tff(fact_702_max__bot,axiom,
    ! [Xb: set_int] : ord_max_set_int(bot_bot_set_int,Xb) = Xb ).

% max_bot
tff(fact_703_max__bot,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),Xb) = Xb ).

% max_bot
tff(fact_704_max__bot,axiom,
    ! [Xb: extended_enat] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(bot_bo4199563552545308370d_enat),Xb) = Xb ).

% max_bot
tff(fact_705_max__bot2,axiom,
    ! [Xb: set_real] : ord_max_set_real(Xb,bot_bot_set_real) = Xb ).

% max_bot2
tff(fact_706_max__bot2,axiom,
    ! [Xb: set_o] : ord_max_set_o(Xb,bot_bot_set_o) = Xb ).

% max_bot2
tff(fact_707_max__bot2,axiom,
    ! [Xb: set_nat] : ord_max_set_nat(Xb,bot_bot_set_nat) = Xb ).

% max_bot2
tff(fact_708_max__bot2,axiom,
    ! [Xb: set_int] : ord_max_set_int(Xb,bot_bot_set_int) = Xb ).

% max_bot2
tff(fact_709_max__bot2,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),bot_bot_nat) = Xb ).

% max_bot2
tff(fact_710_max__bot2,axiom,
    ! [Xb: extended_enat] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(Xb),bot_bo4199563552545308370d_enat) = Xb ).

% max_bot2
tff(fact_711_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_712_subset__empty,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),bot_bot_set_o)
    <=> ( A2 = bot_bot_set_o ) ) ).

% subset_empty
tff(fact_713_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_714_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_715_empty__subsetI,axiom,
    ! [A2: set_real] : aa_set_real_o(ord_less_eq_set_real(bot_bot_set_real),A2) ).

% empty_subsetI
tff(fact_716_empty__subsetI,axiom,
    ! [A2: set_o] : aa_set_o_o(ord_less_eq_set_o(bot_bot_set_o),A2) ).

% empty_subsetI
tff(fact_717_empty__subsetI,axiom,
    ! [A2: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A2) ).

% empty_subsetI
tff(fact_718_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_719_both__member__options__def,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),Xb)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Ta),Xb)
        | aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Ta),Xb) ) ) ).

% both_member_options_def
tff(fact_720_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,vEBT_Node(Uu,zero_zero_nat,Uv,Uw)),Ux) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_721_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_722_option_Oinject,axiom,
    ! [X22: product_prod_nat_nat,Y22: product_prod_nat_nat] :
      ( ( some_P7363390416028606310at_nat(X22) = some_P7363390416028606310at_nat(Y22) )
    <=> ( X22 = Y22 ) ) ).

% option.inject
tff(fact_723_option_Oinject,axiom,
    ! [X22: num,Y22: num] :
      ( ( some_num(X22) = some_num(Y22) )
    <=> ( X22 = Y22 ) ) ).

% option.inject
tff(fact_724_subsetI,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ! [X4: real] :
          ( aa_set_real_o(member_real2(X4),A2)
         => aa_set_real_o(member_real2(X4),B2) )
     => aa_set_real_o(ord_less_eq_set_real(A2),B2) ) ).

% subsetI
tff(fact_725_subsetI,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ! [X4: $o] :
          ( aa_set_o_o(member_o2((X4)),A2)
         => aa_set_o_o(member_o2((X4)),B2) )
     => aa_set_o_o(ord_less_eq_set_o(A2),B2) ) ).

% subsetI
tff(fact_726_subsetI,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ! [X4: set_nat] :
          ( aa_set_set_nat_o(member_set_nat2(X4),A2)
         => aa_set_set_nat_o(member_set_nat2(X4),B2) )
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2) ) ).

% subsetI
tff(fact_727_subsetI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int2(X4),A2)
         => aa_set_int_o(member_int2(X4),B2) )
     => aa_set_int_o(ord_less_eq_set_int(A2),B2) ) ).

% subsetI
tff(fact_728_subsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat2(X4),A2)
         => aa_set_nat_o(member_nat2(X4),B2) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2) ) ).

% subsetI
tff(fact_729_psubsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( ( A2 != B2 )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2) ) ) ).

% psubsetI
tff(fact_730_buildup__nothing__in__min__max,axiom,
    ! [Nb: nat,Xb: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,vEBT_vebt_buildup(Nb)),Xb) ).

% buildup_nothing_in_min_max
tff(fact_731_dual__order_Orefl,axiom,
    ! [A: filter_nat] : ord_le2510731241096832064er_nat(A,A) ).

% dual_order.refl
tff(fact_732_dual__order_Orefl,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),A) ).

% dual_order.refl
tff(fact_733_dual__order_Orefl,axiom,
    ! [A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),A) ).

% dual_order.refl
tff(fact_734_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_735_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_736_order__refl,axiom,
    ! [Xb: filter_nat] : ord_le2510731241096832064er_nat(Xb,Xb) ).

% order_refl
tff(fact_737_order__refl,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Xb) ).

% order_refl
tff(fact_738_order__refl,axiom,
    ! [Xb: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Xb) ).

% order_refl
tff(fact_739_order__refl,axiom,
    ! [Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Xb) ).

% order_refl
tff(fact_740_order__refl,axiom,
    ! [Xb: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xb) ).

% order_refl
tff(fact_741_empty__Collect__eq,axiom,
    ! [P: fun_list_nat_o] :
      ( ( bot_bot_set_list_nat = collect_list_nat(P) )
    <=> ! [X3: list_nat] : ~ aa_list_nat_o(P,X3) ) ).

% empty_Collect_eq
tff(fact_742_empty__Collect__eq,axiom,
    ! [P: fun_set_nat_o] :
      ( ( bot_bot_set_set_nat = collect_set_nat(P) )
    <=> ! [X3: set_nat] : ~ aa_set_nat_o(P,X3) ) ).

% empty_Collect_eq
tff(fact_743_empty__Collect__eq,axiom,
    ! [P: fun_real_o] :
      ( ( bot_bot_set_real = collect_real(P) )
    <=> ! [X3: real] : ~ aa_real_o(P,X3) ) ).

% empty_Collect_eq
tff(fact_744_empty__Collect__eq,axiom,
    ! [P: fun_o_o] :
      ( ( bot_bot_set_o = collect_o(P) )
    <=> ! [X3: $o] : ~ aa_o_o(P,(X3)) ) ).

% empty_Collect_eq
tff(fact_745_empty__Collect__eq,axiom,
    ! [P: fun_nat_o] :
      ( ( bot_bot_set_nat = collect_nat(P) )
    <=> ! [X3: nat] : ~ aa_nat_o(P,X3) ) ).

% empty_Collect_eq
tff(fact_746_empty__Collect__eq,axiom,
    ! [P: fun_int_o] :
      ( ( bot_bot_set_int = collect_int(P) )
    <=> ! [X3: int] : ~ aa_int_o(P,X3) ) ).

% empty_Collect_eq
tff(fact_747_Collect__empty__eq,axiom,
    ! [P: fun_list_nat_o] :
      ( ( collect_list_nat(P) = bot_bot_set_list_nat )
    <=> ! [X3: list_nat] : ~ aa_list_nat_o(P,X3) ) ).

% Collect_empty_eq
tff(fact_748_Collect__empty__eq,axiom,
    ! [P: fun_set_nat_o] :
      ( ( collect_set_nat(P) = bot_bot_set_set_nat )
    <=> ! [X3: set_nat] : ~ aa_set_nat_o(P,X3) ) ).

% Collect_empty_eq
tff(fact_749_Collect__empty__eq,axiom,
    ! [P: fun_real_o] :
      ( ( collect_real(P) = bot_bot_set_real )
    <=> ! [X3: real] : ~ aa_real_o(P,X3) ) ).

% Collect_empty_eq
tff(fact_750_Collect__empty__eq,axiom,
    ! [P: fun_o_o] :
      ( ( collect_o(P) = bot_bot_set_o )
    <=> ! [X3: $o] : ~ aa_o_o(P,(X3)) ) ).

% Collect_empty_eq
tff(fact_751_Collect__empty__eq,axiom,
    ! [P: fun_nat_o] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ! [X3: nat] : ~ aa_nat_o(P,X3) ) ).

% Collect_empty_eq
tff(fact_752_Collect__empty__eq,axiom,
    ! [P: fun_int_o] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ! [X3: int] : ~ aa_int_o(P,X3) ) ).

% Collect_empty_eq
tff(fact_753_all__not__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ! [X3: set_nat] : ~ aa_set_set_nat_o(member_set_nat2(X3),A2)
    <=> ( A2 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
tff(fact_754_all__not__in__conv,axiom,
    ! [A2: set_real] :
      ( ! [X3: real] : ~ aa_set_real_o(member_real2(X3),A2)
    <=> ( A2 = bot_bot_set_real ) ) ).

% all_not_in_conv
tff(fact_755_all__not__in__conv,axiom,
    ! [A2: set_o] :
      ( ! [X3: $o] : ~ aa_set_o_o(member_o2((X3)),A2)
    <=> ( A2 = bot_bot_set_o ) ) ).

% all_not_in_conv
tff(fact_756_all__not__in__conv,axiom,
    ! [A2: set_nat] :
      ( ! [X3: nat] : ~ aa_set_nat_o(member_nat2(X3),A2)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_757_all__not__in__conv,axiom,
    ! [A2: set_int] :
      ( ! [X3: int] : ~ aa_set_int_o(member_int2(X3),A2)
    <=> ( A2 = bot_bot_set_int ) ) ).

% all_not_in_conv
tff(fact_758_empty__iff,axiom,
    ! [C: set_nat] : ~ aa_set_set_nat_o(member_set_nat2(C),bot_bot_set_set_nat) ).

% empty_iff
tff(fact_759_empty__iff,axiom,
    ! [C: real] : ~ aa_set_real_o(member_real2(C),bot_bot_set_real) ).

% empty_iff
tff(fact_760_empty__iff,axiom,
    ! [C: $o] : ~ aa_set_o_o(member_o2((C)),bot_bot_set_o) ).

% empty_iff
tff(fact_761_empty__iff,axiom,
    ! [C: nat] : ~ aa_set_nat_o(member_nat2(C),bot_bot_set_nat) ).

% empty_iff
tff(fact_762_empty__iff,axiom,
    ! [C: int] : ~ aa_set_int_o(member_int2(C),bot_bot_set_int) ).

% empty_iff
tff(fact_763_subset__antisym,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(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
       => ( A2 = B2 ) ) ) ).

% subset_antisym
tff(fact_764_bot__set__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(bot_bo8828579575001852285_nat_o) ).

% bot_set_def
tff(fact_765_bot__set__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(bot_bo3967814874310145751_nat_o) ).

% bot_set_def
tff(fact_766_bot__set__def,axiom,
    bot_bot_set_real = collect_real(bot_bot_fun_real_o) ).

% bot_set_def
tff(fact_767_bot__set__def,axiom,
    bot_bot_set_o = collect_o(bot_bot_fun_o_o) ).

% bot_set_def
tff(fact_768_bot__set__def,axiom,
    bot_bot_set_nat = collect_nat(bot_bot_fun_nat_o) ).

% bot_set_def
tff(fact_769_bot__set__def,axiom,
    bot_bot_set_int = collect_int(bot_bot_fun_int_o) ).

% bot_set_def
tff(fact_770_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_771_order__antisym__conv,axiom,
    ! [Y: filter_nat,Xb: filter_nat] :
      ( ord_le2510731241096832064er_nat(Y,Xb)
     => ( ord_le2510731241096832064er_nat(Xb,Y)
      <=> ( Xb = Y ) ) ) ).

% order_antisym_conv
tff(fact_772_order__antisym__conv,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% order_antisym_conv
tff(fact_773_order__antisym__conv,axiom,
    ! [Y: set_nat,Xb: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Xb)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% order_antisym_conv
tff(fact_774_order__antisym__conv,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% order_antisym_conv
tff(fact_775_order__antisym__conv,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% order_antisym_conv
tff(fact_776_linorder__le__cases,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ).

% linorder_le_cases
tff(fact_777_linorder__le__cases,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ).

% linorder_le_cases
tff(fact_778_linorder__le__cases,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ).

% linorder_le_cases
tff(fact_779_ord__le__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ( aa_real_real(F,B) = C )
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X4)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_780_ord__le__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ( aa_real_nat(F,B) = C )
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_781_ord__le__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ( aa_real_int(F,B) = C )
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,X4)),aa_real_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_782_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_real(F,B) = C )
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_783_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_nat(F,B) = C )
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_784_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_int,C: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_int(F,B) = C )
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X4)),aa_nat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_785_ord__le__eq__subst,axiom,
    ! [A: int,B: int,F: fun_int_real,C: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( aa_int_real(F,B) = C )
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_786_ord__le__eq__subst,axiom,
    ! [A: int,B: int,F: fun_int_nat,C: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( aa_int_nat(F,B) = C )
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X4)),aa_int_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_787_ord__le__eq__subst,axiom,
    ! [A: int,B: int,F: fun_int_int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( aa_int_int(F,B) = C )
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X4)),aa_int_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_788_ord__le__eq__subst,axiom,
    ! [A: filter_nat,B: filter_nat,F: fun_filter_nat_real,C: real] :
      ( ord_le2510731241096832064er_nat(A,B)
     => ( ( aa_filter_nat_real(F,B) = C )
       => ( ! [X4: filter_nat,Y3: filter_nat] :
              ( ord_le2510731241096832064er_nat(X4,Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_filter_nat_real(F,X4)),aa_filter_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_filter_nat_real(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_789_ord__eq__le__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_eq_real,B),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X4)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_790_ord__eq__le__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_eq_real,B),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_real_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_791_ord__eq__le__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_eq_real,B),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,X4)),aa_real_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_real_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_792_ord__eq__le__subst,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( ( A = aa_nat_real(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_793_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_nat_nat,B: nat,C: nat] :
      ( ( A = aa_nat_nat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_794_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_nat_int,B: nat,C: nat] :
      ( ( A = aa_nat_int(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X4)),aa_nat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_795_ord__eq__le__subst,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( ( A = aa_int_real(F,B) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_int_real(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_796_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_int_nat,B: int,C: int] :
      ( ( A = aa_int_nat(F,B) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X4)),aa_int_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_int_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_797_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_int_int,B: int,C: int] :
      ( ( A = aa_int_int(F,B) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X4)),aa_int_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_798_ord__eq__le__subst,axiom,
    ! [A: real,F: fun_filter_nat_real,B: filter_nat,C: filter_nat] :
      ( ( A = aa_filter_nat_real(F,B) )
     => ( ord_le2510731241096832064er_nat(B,C)
       => ( ! [X4: filter_nat,Y3: filter_nat] :
              ( ord_le2510731241096832064er_nat(X4,Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_filter_nat_real(F,X4)),aa_filter_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_filter_nat_real(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_799_linorder__linear,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
      | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ).

% linorder_linear
tff(fact_800_linorder__linear,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ).

% linorder_linear
tff(fact_801_linorder__linear,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ).

% linorder_linear
tff(fact_802_order__eq__refl,axiom,
    ! [Xb: filter_nat,Y: filter_nat] :
      ( ( Xb = Y )
     => ord_le2510731241096832064er_nat(Xb,Y) ) ).

% order_eq_refl
tff(fact_803_order__eq__refl,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb = Y )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% order_eq_refl
tff(fact_804_order__eq__refl,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ( Xb = Y )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Y) ) ).

% order_eq_refl
tff(fact_805_order__eq__refl,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb = Y )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).

% order_eq_refl
tff(fact_806_order__eq__refl,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb = Y )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).

% order_eq_refl
tff(fact_807_order__subst2,axiom,
    ! [A: real,B: real,F: fun_real_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(F,B)),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X4)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_808_order__subst2,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,B)),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_809_order__subst2,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,B)),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,X4)),aa_real_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_810_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,B)),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_811_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_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(F,B)),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_812_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_int,C: int] :
      ( 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(F,B)),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X4)),aa_nat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_813_order__subst2,axiom,
    ! [A: int,B: int,F: fun_int_real,C: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,B)),C)
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_814_order__subst2,axiom,
    ! [A: int,B: int,F: fun_int_nat,C: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,B)),C)
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X4)),aa_int_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_815_order__subst2,axiom,
    ! [A: int,B: int,F: fun_int_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(F,B)),C)
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X4)),aa_int_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_816_order__subst2,axiom,
    ! [A: filter_nat,B: filter_nat,F: fun_filter_nat_real,C: real] :
      ( ord_le2510731241096832064er_nat(A,B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_filter_nat_real(F,B)),C)
       => ( ! [X4: filter_nat,Y3: filter_nat] :
              ( ord_le2510731241096832064er_nat(X4,Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_filter_nat_real(F,X4)),aa_filter_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_filter_nat_real(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_817_order__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_eq_real,B),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X4)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(F,C)) ) ) ) ).

% order_subst1
tff(fact_818_order__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_eq_nat,B),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(F,C)) ) ) ) ).

% order_subst1
tff(fact_819_order__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_eq_int,B),C)
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_int_real(F,C)) ) ) ) ).

% order_subst1
tff(fact_820_order__subst1,axiom,
    ! [A: nat,F: fun_real_nat,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_real_nat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_real_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_821_order__subst1,axiom,
    ! [A: nat,F: fun_nat_nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_822_order__subst1,axiom,
    ! [A: nat,F: fun_int_nat,B: int,C: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_int_nat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X4)),aa_int_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_int_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_823_order__subst1,axiom,
    ! [A: int,F: fun_real_int,B: real,C: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_real_int(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,X4)),aa_real_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_real_int(F,C)) ) ) ) ).

% order_subst1
tff(fact_824_order__subst1,axiom,
    ! [A: int,F: fun_nat_int,B: nat,C: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X4)),aa_nat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(F,C)) ) ) ) ).

% order_subst1
tff(fact_825_order__subst1,axiom,
    ! [A: int,F: fun_int_int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X4)),aa_int_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(F,C)) ) ) ) ).

% order_subst1
tff(fact_826_order__subst1,axiom,
    ! [A: filter_nat,F: fun_real_filter_nat,B: real,C: real] :
      ( ord_le2510731241096832064er_nat(A,aa_real_filter_nat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => ord_le2510731241096832064er_nat(aa_real_filter_nat(F,X4),aa_real_filter_nat(F,Y3)) )
         => ord_le2510731241096832064er_nat(A,aa_real_filter_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_827_Orderings_Oorder__eq__iff,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( A = B )
    <=> ( ord_le2510731241096832064er_nat(A,B)
        & ord_le2510731241096832064er_nat(B,A) ) ) ).

% Orderings.order_eq_iff
tff(fact_828_Orderings_Oorder__eq__iff,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_eq_real,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_829_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_830_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_831_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_832_antisym,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ord_le2510731241096832064er_nat(A,B)
     => ( ord_le2510731241096832064er_nat(B,A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_833_antisym,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,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_834_antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_835_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_836_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_837_dual__order_Otrans,axiom,
    ! [B: filter_nat,A: filter_nat,C: filter_nat] :
      ( ord_le2510731241096832064er_nat(B,A)
     => ( ord_le2510731241096832064er_nat(C,B)
       => ord_le2510731241096832064er_nat(C,A) ) ) ).

% dual_order.trans
tff(fact_838_dual__order_Otrans,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),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A) ) ) ).

% dual_order.trans
tff(fact_839_dual__order_Otrans,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A) ) ) ).

% dual_order.trans
tff(fact_840_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_841_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_842_dual__order_Oantisym,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( ord_le2510731241096832064er_nat(B,A)
     => ( ord_le2510731241096832064er_nat(A,B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_843_dual__order_Oantisym,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_eq_real,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_844_dual__order_Oantisym,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_845_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_846_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_847_dual__order_Oeq__iff,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( A = B )
    <=> ( ord_le2510731241096832064er_nat(B,A)
        & ord_le2510731241096832064er_nat(A,B) ) ) ).

% dual_order.eq_iff
tff(fact_848_dual__order_Oeq__iff,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
    <=> ( 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.eq_iff
tff(fact_849_dual__order_Oeq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_850_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_851_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_852_linorder__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_eq_real,A3),B3)
         => aa_real_o(aa_real_fun_real_o(P,A3),B3) )
     => ( ! [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_wlog
tff(fact_853_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_854_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_855_order__trans,axiom,
    ! [Xb: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( ord_le2510731241096832064er_nat(Xb,Y)
     => ( ord_le2510731241096832064er_nat(Y,Z)
       => ord_le2510731241096832064er_nat(Xb,Z) ) ) ).

% order_trans
tff(fact_856_order__trans,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),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_eq_real,Xb),Z) ) ) ).

% order_trans
tff(fact_857_order__trans,axiom,
    ! [Xb: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Z)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Z) ) ) ).

% order_trans
tff(fact_858_order__trans,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),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,Xb),Z) ) ) ).

% order_trans
tff(fact_859_order__trans,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),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,Xb),Z) ) ) ).

% order_trans
tff(fact_860_order_Otrans,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat] :
      ( ord_le2510731241096832064er_nat(A,B)
     => ( ord_le2510731241096832064er_nat(B,C)
       => ord_le2510731241096832064er_nat(A,C) ) ) ).

% order.trans
tff(fact_861_order_Otrans,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,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C) ) ) ).

% order.trans
tff(fact_862_order_Otrans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C) ) ) ).

% order.trans
tff(fact_863_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_864_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_865_order__antisym,axiom,
    ! [Xb: filter_nat,Y: filter_nat] :
      ( ord_le2510731241096832064er_nat(Xb,Y)
     => ( ord_le2510731241096832064er_nat(Y,Xb)
       => ( Xb = Y ) ) ) ).

% order_antisym
tff(fact_866_order__antisym,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
       => ( Xb = Y ) ) ) ).

% order_antisym
tff(fact_867_order__antisym,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Xb)
       => ( Xb = Y ) ) ) ).

% order_antisym
tff(fact_868_order__antisym,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
       => ( Xb = Y ) ) ) ).

% order_antisym
tff(fact_869_order__antisym,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
       => ( Xb = Y ) ) ) ).

% order_antisym
tff(fact_870_ord__le__eq__trans,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat] :
      ( ord_le2510731241096832064er_nat(A,B)
     => ( ( B = C )
       => ord_le2510731241096832064er_nat(A,C) ) ) ).

% ord_le_eq_trans
tff(fact_871_ord__le__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ( B = C )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_872_ord__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( ( B = C )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_873_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_874_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_875_ord__eq__le__trans,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat] :
      ( ( A = B )
     => ( ord_le2510731241096832064er_nat(B,C)
       => ord_le2510731241096832064er_nat(A,C) ) ) ).

% ord_eq_le_trans
tff(fact_876_ord__eq__le__trans,axiom,
    ! [A: real,B: real,C: 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_eq_real,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_877_ord__eq__le__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( A = B )
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_878_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_879_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_880_order__class_Oorder__eq__iff,axiom,
    ! [Xb: filter_nat,Y: filter_nat] :
      ( ( Xb = Y )
    <=> ( ord_le2510731241096832064er_nat(Xb,Y)
        & ord_le2510731241096832064er_nat(Y,Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_881_order__class_Oorder__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb = Y )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_882_order__class_Oorder__eq__iff,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ( Xb = Y )
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Y)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_883_order__class_Oorder__eq__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb = Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_884_order__class_Oorder__eq__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb = Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_885_le__cases3,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),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_eq_real,Y),Xb)
         => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Z) )
       => ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Z)
           => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),Y) )
         => ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),Y)
             => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) )
           => ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z)
               => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),Xb) )
             => ~ ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),Xb)
                 => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_886_le__cases3,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),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),Xb)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Z) )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),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),Xb) )
           => ( ( 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),Xb) )
             => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xb)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_887_le__cases3,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),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),Xb)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Z) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),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),Xb) )
           => ( ( 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),Xb) )
             => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xb)
                 => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_888_nle__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,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_889_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_890_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_891_order__less__imp__not__less,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% order_less_imp_not_less
tff(fact_892_order__less__imp__not__less,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xb) ) ).

% order_less_imp_not_less
tff(fact_893_order__less__imp__not__less,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% order_less_imp_not_less
tff(fact_894_order__less__imp__not__less,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% order_less_imp_not_less
tff(fact_895_order__less__imp__not__eq2,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( Y != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_896_order__less__imp__not__eq2,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
     => ( Y != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_897_order__less__imp__not__eq2,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( Y != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_898_order__less__imp__not__eq2,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( Y != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_899_order__less__imp__not__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( Xb != Y ) ) ).

% order_less_imp_not_eq
tff(fact_900_order__less__imp__not__eq,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
     => ( Xb != Y ) ) ).

% order_less_imp_not_eq
tff(fact_901_order__less__imp__not__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( Xb != Y ) ) ).

% order_less_imp_not_eq
tff(fact_902_order__less__imp__not__eq,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( Xb != Y ) ) ).

% order_less_imp_not_eq
tff(fact_903_linorder__less__linear,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
      | ( Xb = Y )
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% linorder_less_linear
tff(fact_904_linorder__less__linear,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
      | ( Xb = Y )
      | aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xb) ) ).

% linorder_less_linear
tff(fact_905_linorder__less__linear,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
      | ( Xb = Y )
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% linorder_less_linear
tff(fact_906_linorder__less__linear,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
      | ( Xb = Y )
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% linorder_less_linear
tff(fact_907_order__less__imp__triv,axiom,
    ! [Xb: nat,Y: nat,P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_908_order__less__imp__triv,axiom,
    ! [Xb: extended_enat,Y: extended_enat,P: $o] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_909_order__less__imp__triv,axiom,
    ! [Xb: real,Y: real,P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_910_order__less__imp__triv,axiom,
    ! [Xb: int,Y: int,P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_911_order__less__not__sym,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% order_less_not_sym
tff(fact_912_order__less__not__sym,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xb) ) ).

% order_less_not_sym
tff(fact_913_order__less__not__sym,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% order_less_not_sym
tff(fact_914_order__less__not__sym,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% order_less_not_sym
tff(fact_915_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_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(F,B)),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_916_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: fun_na8797796813480665199d_enat,C: extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,B)),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,X4)),aa_nat_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_917_order__less__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_real,aa_nat_real(F,B)),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_918_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_int,C: int] :
      ( 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(F,B)),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X4)),aa_nat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_919_order__less__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: fun_Ex7660298398204468809at_nat,C: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Extended_enat_nat(F,B)),C)
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Extended_enat_nat(F,X4)),aa_Extended_enat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Extended_enat_nat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_920_order__less__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: fun_Ex5852502975177633731d_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,B)),C)
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,X4)),aa_Ext7969882492699152700d_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_921_order__less__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: fun_Ex1451620480498224165t_real,C: real] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Ext2541572767299617110t_real(F,B)),C)
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Ext2541572767299617110t_real(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_922_order__less__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: fun_Ex3482447378695272101at_int,C: int] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Extended_enat_int(F,B)),C)
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Extended_enat_int(F,X4)),aa_Extended_enat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Extended_enat_int(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_923_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)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X4)),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_924_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_re6065469302365115155d_enat,C: extended_enat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,B)),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X4)),aa_rea1194391308913012860d_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_925_order__less__subst1,axiom,
    ! [A: nat,F: fun_nat_nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_926_order__less__subst1,axiom,
    ! [A: nat,F: fun_Ex7660298398204468809at_nat,B: extended_enat,C: extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_Extended_enat_nat(F,B))
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),C)
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Extended_enat_nat(F,X4)),aa_Extended_enat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_Extended_enat_nat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_927_order__less__subst1,axiom,
    ! [A: nat,F: fun_real_nat,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_real_nat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_real_nat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_928_order__less__subst1,axiom,
    ! [A: nat,F: fun_int_nat,B: int,C: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_int_nat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X4)),aa_int_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_int_nat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_929_order__less__subst1,axiom,
    ! [A: extended_enat,F: fun_na8797796813480665199d_enat,B: nat,C: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_nat_Extended_enat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,X4)),aa_nat_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_nat_Extended_enat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_930_order__less__subst1,axiom,
    ! [A: extended_enat,F: fun_Ex5852502975177633731d_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_Ext7969882492699152700d_enat(F,B))
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),C)
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,X4)),aa_Ext7969882492699152700d_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_Ext7969882492699152700d_enat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_931_order__less__subst1,axiom,
    ! [A: extended_enat,F: fun_re6065469302365115155d_enat,B: real,C: real] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_rea1194391308913012860d_enat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X4)),aa_rea1194391308913012860d_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_rea1194391308913012860d_enat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_932_order__less__subst1,axiom,
    ! [A: extended_enat,F: fun_in5914218760682845459d_enat,B: int,C: int] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_int_Extended_enat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_int_Extended_enat(F,X4)),aa_int_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_int_Extended_enat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_933_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)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X4)),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_934_order__less__subst1,axiom,
    ! [A: real,F: fun_Ex1451620480498224165t_real,B: extended_enat,C: extended_enat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_Ext2541572767299617110t_real(F,B))
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),C)
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_Ext2541572767299617110t_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_935_order__less__irrefl,axiom,
    ! [Xb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xb) ).

% order_less_irrefl
tff(fact_936_order__less__irrefl,axiom,
    ! [Xb: extended_enat] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Xb) ).

% order_less_irrefl
tff(fact_937_order__less__irrefl,axiom,
    ! [Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Xb) ).

% order_less_irrefl
tff(fact_938_order__less__irrefl,axiom,
    ! [Xb: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Xb) ).

% order_less_irrefl
tff(fact_939_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( ( aa_nat_nat(F,B) = C )
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_940_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_na8797796813480665199d_enat,C: extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( ( aa_nat_Extended_enat(F,B) = C )
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,X4)),aa_nat_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_941_ord__less__eq__subst,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_nat_real(F,B) = C )
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_942_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_int,C: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( ( aa_nat_int(F,B) = C )
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X4)),aa_nat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_943_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B: extended_enat,F: fun_Ex7660298398204468809at_nat,C: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( ( aa_Extended_enat_nat(F,B) = C )
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Extended_enat_nat(F,X4)),aa_Extended_enat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Extended_enat_nat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_944_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B: extended_enat,F: fun_Ex5852502975177633731d_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( ( aa_Ext7969882492699152700d_enat(F,B) = C )
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,X4)),aa_Ext7969882492699152700d_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_945_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B: extended_enat,F: fun_Ex1451620480498224165t_real,C: real] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( ( aa_Ext2541572767299617110t_real(F,B) = C )
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Ext2541572767299617110t_real(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_946_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B: extended_enat,F: fun_Ex3482447378695272101at_int,C: int] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( ( aa_Extended_enat_int(F,B) = C )
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Extended_enat_int(F,X4)),aa_Extended_enat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Extended_enat_int(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_947_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 )
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X4)),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_948_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_re6065469302365115155d_enat,C: extended_enat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_rea1194391308913012860d_enat(F,B) = C )
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X4)),aa_rea1194391308913012860d_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_949_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_nat_nat,B: nat,C: nat] :
      ( ( A = aa_nat_nat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_950_ord__eq__less__subst,axiom,
    ! [A: extended_enat,F: fun_na8797796813480665199d_enat,B: nat,C: nat] :
      ( ( A = aa_nat_Extended_enat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,X4)),aa_nat_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_nat_Extended_enat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_951_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( ( A = aa_nat_real(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_952_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_nat_int,B: nat,C: nat] :
      ( ( A = aa_nat_int(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X4)),aa_nat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_953_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_Ex7660298398204468809at_nat,B: extended_enat,C: extended_enat] :
      ( ( A = aa_Extended_enat_nat(F,B) )
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),C)
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Extended_enat_nat(F,X4)),aa_Extended_enat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_Extended_enat_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_954_ord__eq__less__subst,axiom,
    ! [A: extended_enat,F: fun_Ex5852502975177633731d_enat,B: extended_enat,C: extended_enat] :
      ( ( A = aa_Ext7969882492699152700d_enat(F,B) )
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),C)
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,X4)),aa_Ext7969882492699152700d_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_Ext7969882492699152700d_enat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_955_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_Ex1451620480498224165t_real,B: extended_enat,C: extended_enat] :
      ( ( A = aa_Ext2541572767299617110t_real(F,B) )
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),C)
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_Ext2541572767299617110t_real(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_956_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_Ex3482447378695272101at_int,B: extended_enat,C: extended_enat] :
      ( ( A = aa_Extended_enat_int(F,B) )
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),C)
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Extended_enat_int(F,X4)),aa_Extended_enat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_Extended_enat_int(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_957_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)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X4)),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_958_ord__eq__less__subst,axiom,
    ! [A: extended_enat,F: fun_re6065469302365115155d_enat,B: real,C: real] :
      ( ( A = aa_rea1194391308913012860d_enat(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X4)),aa_rea1194391308913012860d_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_rea1194391308913012860d_enat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_959_order__less__trans,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),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,Xb),Z) ) ) ).

% order_less_trans
tff(fact_960_order__less__trans,axiom,
    ! [Xb: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Z)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Z) ) ) ).

% order_less_trans
tff(fact_961_order__less__trans,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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,Xb),Z) ) ) ).

% order_less_trans
tff(fact_962_order__less__trans,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),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,Xb),Z) ) ) ).

% order_less_trans
tff(fact_963_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_964_order__less__asym_H,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A) ) ).

% order_less_asym'
tff(fact_965_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_966_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_967_linorder__neq__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb != Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).

% linorder_neq_iff
tff(fact_968_linorder__neq__iff,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( ( Xb != Y )
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
        | aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xb) ) ) ).

% linorder_neq_iff
tff(fact_969_linorder__neq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != Y )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).

% linorder_neq_iff
tff(fact_970_linorder__neq__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb != Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).

% linorder_neq_iff
tff(fact_971_order__less__asym,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% order_less_asym
tff(fact_972_order__less__asym,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xb) ) ).

% order_less_asym
tff(fact_973_order__less__asym,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% order_less_asym
tff(fact_974_order__less__asym,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% order_less_asym
tff(fact_975_linorder__neqE,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).

% linorder_neqE
tff(fact_976_linorder__neqE,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( ( Xb != Y )
     => ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xb) ) ) ).

% linorder_neqE
tff(fact_977_linorder__neqE,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).

% linorder_neqE
tff(fact_978_linorder__neqE,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).

% linorder_neqE
tff(fact_979_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_980_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_981_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_982_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_983_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_984_order_Ostrict__implies__not__eq,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_985_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_986_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_987_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_988_dual__order_Ostrict__trans,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),B)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_989_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_990_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_991_not__less__iff__gr__or__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
        | ( Xb = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_992_not__less__iff__gr__or__eq,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xb)
        | ( Xb = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_993_not__less__iff__gr__or__eq,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
        | ( Xb = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_994_not__less__iff__gr__or__eq,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb)
        | ( Xb = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_995_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_996_order_Ostrict__trans,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),C)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),C) ) ) ).

% order.strict_trans
tff(fact_997_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_998_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_999_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_1000_linorder__less__wlog,axiom,
    ! [P: fun_Ex239764483819389678enat_o,A: extended_enat,B: extended_enat] :
      ( ! [A3: extended_enat,B3: extended_enat] :
          ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A3),B3)
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(P,A3),B3) )
     => ( ! [A3: extended_enat] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(P,A3),A3)
       => ( ! [A3: extended_enat,B3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(P,B3),A3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(P,A3),B3) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1001_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_1002_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_1003_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_1004_exists__least__iff,axiom,
    ! [P: fun_Extended_enat_o] :
      ( ? [X_13: extended_enat] : aa_Extended_enat_o(P,X_13)
    <=> ? [N3: extended_enat] :
          ( aa_Extended_enat_o(P,N3)
          & ! [M5: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,M5),N3)
             => ~ aa_Extended_enat_o(P,M5) ) ) ) ).

% exists_least_iff
tff(fact_1005_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% dual_order.irrefl
tff(fact_1006_dual__order_Oirrefl,axiom,
    ! [A: extended_enat] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),A) ).

% dual_order.irrefl
tff(fact_1007_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% dual_order.irrefl
tff(fact_1008_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% dual_order.irrefl
tff(fact_1009_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_1010_dual__order_Oasym,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B) ) ).

% dual_order.asym
tff(fact_1011_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_1012_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_1013_linorder__cases,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( ( Xb != Y )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).

% linorder_cases
tff(fact_1014_linorder__cases,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
     => ( ( Xb != Y )
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xb) ) ) ).

% linorder_cases
tff(fact_1015_linorder__cases,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( ( Xb != Y )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).

% linorder_cases
tff(fact_1016_linorder__cases,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( ( Xb != Y )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).

% linorder_cases
tff(fact_1017_antisym__conv3,axiom,
    ! [Y: nat,Xb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv3
tff(fact_1018_antisym__conv3,axiom,
    ! [Y: extended_enat,Xb: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xb)
     => ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv3
tff(fact_1019_antisym__conv3,axiom,
    ! [Y: real,Xb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv3
tff(fact_1020_antisym__conv3,axiom,
    ! [Y: int,Xb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv3
tff(fact_1021_less__induct,axiom,
    ! [P: fun_nat_o,A: nat] :
      ( ! [X4: nat] :
          ( ! [Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),X4)
             => aa_nat_o(P,Y4) )
         => aa_nat_o(P,X4) )
     => aa_nat_o(P,A) ) ).

% less_induct
tff(fact_1022_less__induct,axiom,
    ! [P: fun_Extended_enat_o,A: extended_enat] :
      ( ! [X4: extended_enat] :
          ( ! [Y4: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y4),X4)
             => aa_Extended_enat_o(P,Y4) )
         => aa_Extended_enat_o(P,X4) )
     => aa_Extended_enat_o(P,A) ) ).

% less_induct
tff(fact_1023_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_1024_ord__less__eq__trans,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( ( B = C )
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_1025_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_1026_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_1027_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_1028_ord__eq__less__trans,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( A = B )
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),C)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_1029_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_1030_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_1031_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_1032_order_Oasym,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A) ) ).

% order.asym
tff(fact_1033_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_1034_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_1035_less__imp__neq,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( Xb != Y ) ) ).

% less_imp_neq
tff(fact_1036_less__imp__neq,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
     => ( Xb != Y ) ) ).

% less_imp_neq
tff(fact_1037_less__imp__neq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( Xb != Y ) ) ).

% less_imp_neq
tff(fact_1038_less__imp__neq,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( Xb != Y ) ) ).

% less_imp_neq
tff(fact_1039_dense,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ? [Z2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),Y) ) ) ).

% dense
tff(fact_1040_gt__ex,axiom,
    ! [Xb: nat] :
    ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),X_12) ).

% gt_ex
tff(fact_1041_gt__ex,axiom,
    ! [Xb: real] :
    ? [X_12: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),X_12) ).

% gt_ex
tff(fact_1042_gt__ex,axiom,
    ! [Xb: int] :
    ? [X_12: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),X_12) ).

% gt_ex
tff(fact_1043_lt__ex,axiom,
    ! [Xb: real] :
    ? [Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb) ).

% lt_ex
tff(fact_1044_lt__ex,axiom,
    ! [Xb: int] :
    ? [Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Xb) ).

% lt_ex
tff(fact_1045_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_1046_not__psubset__empty,axiom,
    ! [A2: set_o] : ~ aa_set_o_o(ord_less_set_o(A2),bot_bot_set_o) ).

% not_psubset_empty
tff(fact_1047_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_1048_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_1049_ex__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ? [X3: set_nat] : aa_set_set_nat_o(member_set_nat2(X3),A2)
    <=> ( A2 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
tff(fact_1050_ex__in__conv,axiom,
    ! [A2: set_real] :
      ( ? [X3: real] : aa_set_real_o(member_real2(X3),A2)
    <=> ( A2 != bot_bot_set_real ) ) ).

% ex_in_conv
tff(fact_1051_ex__in__conv,axiom,
    ! [A2: set_o] :
      ( ? [X3: $o] : aa_set_o_o(member_o2((X3)),A2)
    <=> ( A2 != bot_bot_set_o ) ) ).

% ex_in_conv
tff(fact_1052_ex__in__conv,axiom,
    ! [A2: set_nat] :
      ( ? [X3: nat] : aa_set_nat_o(member_nat2(X3),A2)
    <=> ( A2 != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_1053_ex__in__conv,axiom,
    ! [A2: set_int] :
      ( ? [X3: int] : aa_set_int_o(member_int2(X3),A2)
    <=> ( A2 != bot_bot_set_int ) ) ).

% ex_in_conv
tff(fact_1054_equals0I,axiom,
    ! [A2: set_set_nat] :
      ( ! [Y3: set_nat] : ~ aa_set_set_nat_o(member_set_nat2(Y3),A2)
     => ( A2 = bot_bot_set_set_nat ) ) ).

% equals0I
tff(fact_1055_equals0I,axiom,
    ! [A2: set_real] :
      ( ! [Y3: real] : ~ aa_set_real_o(member_real2(Y3),A2)
     => ( A2 = bot_bot_set_real ) ) ).

% equals0I
tff(fact_1056_equals0I,axiom,
    ! [A2: set_o] :
      ( ! [Y3: $o] : ~ aa_set_o_o(member_o2((Y3)),A2)
     => ( A2 = bot_bot_set_o ) ) ).

% equals0I
tff(fact_1057_equals0I,axiom,
    ! [A2: set_nat] :
      ( ! [Y3: nat] : ~ aa_set_nat_o(member_nat2(Y3),A2)
     => ( A2 = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_1058_equals0I,axiom,
    ! [A2: set_int] :
      ( ! [Y3: int] : ~ aa_set_int_o(member_int2(Y3),A2)
     => ( A2 = bot_bot_set_int ) ) ).

% equals0I
tff(fact_1059_equals0D,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( A2 = bot_bot_set_set_nat )
     => ~ aa_set_set_nat_o(member_set_nat2(A),A2) ) ).

% equals0D
tff(fact_1060_equals0D,axiom,
    ! [A2: set_real,A: real] :
      ( ( A2 = bot_bot_set_real )
     => ~ aa_set_real_o(member_real2(A),A2) ) ).

% equals0D
tff(fact_1061_equals0D,axiom,
    ! [A2: set_o,A: $o] :
      ( ( A2 = bot_bot_set_o )
     => ~ aa_set_o_o(member_o2((A)),A2) ) ).

% equals0D
tff(fact_1062_equals0D,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( A2 = bot_bot_set_nat )
     => ~ aa_set_nat_o(member_nat2(A),A2) ) ).

% equals0D
tff(fact_1063_equals0D,axiom,
    ! [A2: set_int,A: int] :
      ( ( A2 = bot_bot_set_int )
     => ~ aa_set_int_o(member_int2(A),A2) ) ).

% equals0D
tff(fact_1064_emptyE,axiom,
    ! [A: set_nat] : ~ aa_set_set_nat_o(member_set_nat2(A),bot_bot_set_set_nat) ).

% emptyE
tff(fact_1065_emptyE,axiom,
    ! [A: real] : ~ aa_set_real_o(member_real2(A),bot_bot_set_real) ).

% emptyE
tff(fact_1066_emptyE,axiom,
    ! [A: $o] : ~ aa_set_o_o(member_o2((A)),bot_bot_set_o) ).

% emptyE
tff(fact_1067_emptyE,axiom,
    ! [A: nat] : ~ aa_set_nat_o(member_nat2(A),bot_bot_set_nat) ).

% emptyE
tff(fact_1068_emptyE,axiom,
    ! [A: int] : ~ aa_set_int_o(member_int2(A),bot_bot_set_int) ).

% emptyE
tff(fact_1069_subset__iff__psubset__eq,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(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2)
        | ( A2 = B2 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1070_subset__psubset__trans,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_set_nat,B2),C4)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),C4) ) ) ).

% subset_psubset_trans
tff(fact_1071_subset__not__subset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2)
    <=> ( 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),A2) ) ) ).

% subset_not_subset_eq
tff(fact_1072_psubset__subset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C4)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),C4) ) ) ).

% psubset_subset_trans
tff(fact_1073_psubset__imp__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2) ) ).

% psubset_imp_subset
tff(fact_1074_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))
    <=> ! [X3: real] :
          ( aa_real_o(P,X3)
         => aa_real_o(Q,X3) ) ) ).

% Collect_mono_iff
tff(fact_1075_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))
    <=> ! [X3: list_nat] :
          ( aa_list_nat_o(P,X3)
         => aa_list_nat_o(Q,X3) ) ) ).

% Collect_mono_iff
tff(fact_1076_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))
    <=> ! [X3: set_nat] :
          ( aa_set_nat_o(P,X3)
         => aa_set_nat_o(Q,X3) ) ) ).

% Collect_mono_iff
tff(fact_1077_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))
    <=> ! [X3: int] :
          ( aa_int_o(P,X3)
         => aa_int_o(Q,X3) ) ) ).

% Collect_mono_iff
tff(fact_1078_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))
    <=> ! [X3: nat] :
          ( aa_nat_o(P,X3)
         => aa_nat_o(Q,X3) ) ) ).

% Collect_mono_iff
tff(fact_1079_set__eq__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
    <=> ( 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),A2) ) ) ).

% set_eq_subset
tff(fact_1080_subset__trans,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)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C4) ) ) ).

% subset_trans
tff(fact_1081_Collect__mono,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( ! [X4: real] :
          ( aa_real_o(P,X4)
         => aa_real_o(Q,X4) )
     => aa_set_real_o(ord_less_eq_set_real(collect_real(P)),collect_real(Q)) ) ).

% Collect_mono
tff(fact_1082_Collect__mono,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ! [X4: list_nat] :
          ( aa_list_nat_o(P,X4)
         => aa_list_nat_o(Q,X4) )
     => ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q)) ) ).

% Collect_mono
tff(fact_1083_Collect__mono,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X4: set_nat] :
          ( aa_set_nat_o(P,X4)
         => aa_set_nat_o(Q,X4) )
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(P)),collect_set_nat(Q)) ) ).

% Collect_mono
tff(fact_1084_Collect__mono,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X4: int] :
          ( aa_int_o(P,X4)
         => aa_int_o(Q,X4) )
     => aa_set_int_o(ord_less_eq_set_int(collect_int(P)),collect_int(Q)) ) ).

% Collect_mono
tff(fact_1085_Collect__mono,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_nat_o(P,X4)
         => aa_nat_o(Q,X4) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(P)),collect_nat(Q)) ) ).

% Collect_mono
tff(fact_1086_subset__refl,axiom,
    ! [A2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),A2) ).

% subset_refl
tff(fact_1087_subset__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
    <=> ! [T: real] :
          ( aa_set_real_o(member_real2(T),A2)
         => aa_set_real_o(member_real2(T),B2) ) ) ).

% subset_iff
tff(fact_1088_subset__iff,axiom,
    ! [A2: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
    <=> ! [T: $o] :
          ( aa_set_o_o(member_o2((T)),A2)
         => aa_set_o_o(member_o2((T)),B2) ) ) ).

% subset_iff
tff(fact_1089_subset__iff,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
    <=> ! [T: set_nat] :
          ( aa_set_set_nat_o(member_set_nat2(T),A2)
         => aa_set_set_nat_o(member_set_nat2(T),B2) ) ) ).

% subset_iff
tff(fact_1090_subset__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
    <=> ! [T: int] :
          ( aa_set_int_o(member_int2(T),A2)
         => aa_set_int_o(member_int2(T),B2) ) ) ).

% subset_iff
tff(fact_1091_subset__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
    <=> ! [T: nat] :
          ( aa_set_nat_o(member_nat2(T),A2)
         => aa_set_nat_o(member_nat2(T),B2) ) ) ).

% subset_iff
tff(fact_1092_psubset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
        & ( A2 != B2 ) ) ) ).

% psubset_eq
tff(fact_1093_equalityD2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2) ) ).

% equalityD2
tff(fact_1094_equalityD1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2) ) ).

% equalityD1
tff(fact_1095_subset__eq,axiom,
    ! [A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
    <=> ! [X3: real] :
          ( aa_set_real_o(member_real2(X3),A2)
         => aa_set_real_o(member_real2(X3),B2) ) ) ).

% subset_eq
tff(fact_1096_subset__eq,axiom,
    ! [A2: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
    <=> ! [X3: $o] :
          ( aa_set_o_o(member_o2((X3)),A2)
         => aa_set_o_o(member_o2((X3)),B2) ) ) ).

% subset_eq
tff(fact_1097_subset__eq,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
    <=> ! [X3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat2(X3),A2)
         => aa_set_set_nat_o(member_set_nat2(X3),B2) ) ) ).

% subset_eq
tff(fact_1098_subset__eq,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
    <=> ! [X3: int] :
          ( aa_set_int_o(member_int2(X3),A2)
         => aa_set_int_o(member_int2(X3),B2) ) ) ).

% subset_eq
tff(fact_1099_subset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),A2)
         => aa_set_nat_o(member_nat2(X3),B2) ) ) ).

% subset_eq
tff(fact_1100_equalityE,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => ~ ( 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),A2) ) ) ).

% equalityE
tff(fact_1101_psubsetE,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2)
     => ~ ( 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),A2) ) ) ).

% psubsetE
tff(fact_1102_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_real2(C),A2)
       => aa_set_real_o(member_real2(C),B2) ) ) ).

% subsetD
tff(fact_1103_subsetD,axiom,
    ! [A2: set_o,B2: set_o,C: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
     => ( aa_set_o_o(member_o2((C)),A2)
       => aa_set_o_o(member_o2((C)),B2) ) ) ).

% subsetD
tff(fact_1104_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_nat2(C),A2)
       => aa_set_set_nat_o(member_set_nat2(C),B2) ) ) ).

% subsetD
tff(fact_1105_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_int2(C),A2)
       => aa_set_int_o(member_int2(C),B2) ) ) ).

% subsetD
tff(fact_1106_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_nat2(C),A2)
       => aa_set_nat_o(member_nat2(C),B2) ) ) ).

% subsetD
tff(fact_1107_in__mono,axiom,
    ! [A2: set_real,B2: set_real,Xb: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
     => ( aa_set_real_o(member_real2(Xb),A2)
       => aa_set_real_o(member_real2(Xb),B2) ) ) ).

% in_mono
tff(fact_1108_in__mono,axiom,
    ! [A2: set_o,B2: set_o,Xb: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
     => ( aa_set_o_o(member_o2((Xb)),A2)
       => aa_set_o_o(member_o2((Xb)),B2) ) ) ).

% in_mono
tff(fact_1109_in__mono,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,Xb: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
     => ( aa_set_set_nat_o(member_set_nat2(Xb),A2)
       => aa_set_set_nat_o(member_set_nat2(Xb),B2) ) ) ).

% in_mono
tff(fact_1110_in__mono,axiom,
    ! [A2: set_int,B2: set_int,Xb: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( aa_set_int_o(member_int2(Xb),A2)
       => aa_set_int_o(member_int2(Xb),B2) ) ) ).

% in_mono
tff(fact_1111_in__mono,axiom,
    ! [A2: set_nat,B2: set_nat,Xb: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( aa_set_nat_o(member_nat2(Xb),A2)
       => aa_set_nat_o(member_nat2(Xb),B2) ) ) ).

% in_mono
tff(fact_1112_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),zero_zero_nat,Va,Vb)),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_1113_order__le__imp__less__or__eq,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xb),Y)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1114_order__le__imp__less__or__eq,axiom,
    ! [Xb: filter_nat,Y: filter_nat] :
      ( ord_le2510731241096832064er_nat(Xb,Y)
     => ( ord_less_filter_nat(Xb,Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1115_order__le__imp__less__or__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1116_order__le__imp__less__or__eq,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1117_order__le__imp__less__or__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1118_order__le__imp__less__or__eq,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1119_linorder__le__less__linear,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xb),Y)
      | aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xb) ) ).

% linorder_le_less_linear
tff(fact_1120_linorder__le__less__linear,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% linorder_le_less_linear
tff(fact_1121_linorder__le__less__linear,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% linorder_le_less_linear
tff(fact_1122_linorder__le__less__linear,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% linorder_le_less_linear
tff(fact_1123_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_na8797796813480665199d_enat,C: extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,B)),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,X4)),aa_nat_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1124_order__less__le__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: fun_Ex5852502975177633731d_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(F,B)),C)
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,X4)),aa_Ext7969882492699152700d_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1125_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_re6065469302365115155d_enat,C: extended_enat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_rea1194391308913012860d_enat(F,B)),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X4)),aa_rea1194391308913012860d_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1126_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_in5914218760682845459d_enat,C: extended_enat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_int_Extended_enat(F,B)),C)
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_int_Extended_enat(F,X4)),aa_int_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_int_Extended_enat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1127_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)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X4)),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_1128_order__less__le__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: fun_Ex1451620480498224165t_real,C: real] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,B)),C)
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Ext2541572767299617110t_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1129_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)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X4)),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_1130_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)
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X4)),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_1131_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_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,aa_nat_nat(F,B)),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1132_order__less__le__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: fun_Ex7660298398204468809at_nat,C: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Extended_enat_nat(F,B)),C)
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Extended_enat_nat(F,X4)),aa_Extended_enat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Extended_enat_nat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1133_order__less__le__subst1,axiom,
    ! [A: extended_enat,F: fun_re6065469302365115155d_enat,B: real,C: real] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_rea1194391308913012860d_enat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_rea1194391308913012860d_enat(F,X4)),aa_rea1194391308913012860d_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_rea1194391308913012860d_enat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1134_order__less__le__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_eq_real,B),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X4)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1135_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_real_nat,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_real_nat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_real_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1136_order__less__le__subst1,axiom,
    ! [A: int,F: fun_real_int,B: real,C: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_real_int(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,X4)),aa_real_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_real_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1137_order__less__le__subst1,axiom,
    ! [A: extended_enat,F: fun_na8797796813480665199d_enat,B: nat,C: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_nat_Extended_enat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,X4)),aa_nat_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_nat_Extended_enat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1138_order__less__le__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_eq_nat,B),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1139_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_nat_nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1140_order__less__le__subst1,axiom,
    ! [A: int,F: fun_nat_int,B: nat,C: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X4)),aa_nat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1141_order__less__le__subst1,axiom,
    ! [A: extended_enat,F: fun_in5914218760682845459d_enat,B: int,C: int] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_int_Extended_enat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_int_Extended_enat(F,X4)),aa_int_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_int_Extended_enat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1142_order__less__le__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_eq_int,B),C)
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1143_order__le__less__subst2,axiom,
    ! [A: real,B: real,F: fun_re6065469302365115155d_enat,C: extended_enat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,B)),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_rea1194391308913012860d_enat(F,X4)),aa_rea1194391308913012860d_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1144_order__le__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_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,aa_real_real(F,B)),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X4)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1145_order__le__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,B)),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1146_order__le__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,B)),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,X4)),aa_real_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1147_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: fun_na8797796813480665199d_enat,C: extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,B)),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,X4)),aa_nat_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1148_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,B)),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1149_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_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,aa_nat_nat(F,B)),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1150_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_int,C: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,B)),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X4)),aa_nat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1151_order__le__less__subst2,axiom,
    ! [A: int,B: int,F: fun_in5914218760682845459d_enat,C: extended_enat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_int_Extended_enat(F,B)),C)
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_int_Extended_enat(F,X4)),aa_int_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_int_Extended_enat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1152_order__le__less__subst2,axiom,
    ! [A: int,B: int,F: fun_int_real,C: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,B)),C)
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1153_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: fun_na8797796813480665199d_enat,B: nat,C: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),aa_nat_Extended_enat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,X4)),aa_nat_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_nat_Extended_enat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1154_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: fun_Ex5852502975177633731d_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),aa_Ext7969882492699152700d_enat(F,B))
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),C)
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,X4)),aa_Ext7969882492699152700d_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_Ext7969882492699152700d_enat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1155_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: fun_re6065469302365115155d_enat,B: real,C: real] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),aa_rea1194391308913012860d_enat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X4)),aa_rea1194391308913012860d_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_rea1194391308913012860d_enat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1156_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: fun_in5914218760682845459d_enat,B: int,C: int] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),aa_int_Extended_enat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_int_Extended_enat(F,X4)),aa_int_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_int_Extended_enat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1157_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)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X4)),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_1158_order__le__less__subst1,axiom,
    ! [A: real,F: fun_Ex1451620480498224165t_real,B: extended_enat,C: extended_enat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_Ext2541572767299617110t_real(F,B))
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),C)
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_Ext2541572767299617110t_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1159_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)
       => ( ! [X4: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X4)),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_1160_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)
       => ( ! [X4: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X4)),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_1161_order__le__less__subst1,axiom,
    ! [A: nat,F: fun_nat_nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1162_order__le__less__subst1,axiom,
    ! [A: nat,F: fun_Ex7660298398204468809at_nat,B: extended_enat,C: extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_Extended_enat_nat(F,B))
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),C)
       => ( ! [X4: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Extended_enat_nat(F,X4)),aa_Extended_enat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_Extended_enat_nat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1163_order__less__le__trans,axiom,
    ! [Xb: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Y),Z)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1164_order__less__le__trans,axiom,
    ! [Xb: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( ord_less_filter_nat(Xb,Y)
     => ( ord_le2510731241096832064er_nat(Y,Z)
       => ord_less_filter_nat(Xb,Z) ) ) ).

% order_less_le_trans
tff(fact_1165_order__less__le__trans,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1166_order__less__le__trans,axiom,
    ! [Xb: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Xb),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Z)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1167_order__less__le__trans,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),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,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1168_order__less__le__trans,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),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,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1169_order__le__less__trans,axiom,
    ! [Xb: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xb),Y)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Z)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1170_order__le__less__trans,axiom,
    ! [Xb: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( ord_le2510731241096832064er_nat(Xb,Y)
     => ( ord_less_filter_nat(Y,Z)
       => ord_less_filter_nat(Xb,Z) ) ) ).

% order_le_less_trans
tff(fact_1171_order__le__less__trans,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),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,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1172_order__le__less__trans,axiom,
    ! [Xb: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Y),Z)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1173_order__le__less__trans,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),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,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1174_order__le__less__trans,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),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,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1175_order__neq__le__trans,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A != B )
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_1176_order__neq__le__trans,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( A != B )
     => ( ord_le2510731241096832064er_nat(A,B)
       => ord_less_filter_nat(A,B) ) ) ).

% order_neq_le_trans
tff(fact_1177_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_1178_order__neq__le__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A != B )
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_1179_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_1180_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_1181_order__le__neq__trans,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
     => ( ( A != B )
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_1182_order__le__neq__trans,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ord_le2510731241096832064er_nat(A,B)
     => ( ( A != B )
       => ord_less_filter_nat(A,B) ) ) ).

% order_le_neq_trans
tff(fact_1183_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_1184_order__le__neq__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( ( A != B )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_1185_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_1186_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_1187_order__less__imp__le,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xb),Y) ) ).

% order_less_imp_le
tff(fact_1188_order__less__imp__le,axiom,
    ! [Xb: filter_nat,Y: filter_nat] :
      ( ord_less_filter_nat(Xb,Y)
     => ord_le2510731241096832064er_nat(Xb,Y) ) ).

% order_less_imp_le
tff(fact_1189_order__less__imp__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% order_less_imp_le
tff(fact_1190_order__less__imp__le,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Xb),Y)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Y) ) ).

% order_less_imp_le
tff(fact_1191_order__less__imp__le,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).

% order_less_imp_le
tff(fact_1192_order__less__imp__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).

% order_less_imp_le
tff(fact_1193_linorder__not__less,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
    <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Y),Xb) ) ).

% linorder_not_less
tff(fact_1194_linorder__not__less,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ).

% linorder_not_less
tff(fact_1195_linorder__not__less,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ).

% linorder_not_less
tff(fact_1196_linorder__not__less,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ).

% linorder_not_less
tff(fact_1197_linorder__not__le,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xb),Y)
    <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xb) ) ).

% linorder_not_le
tff(fact_1198_linorder__not__le,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% linorder_not_le
tff(fact_1199_linorder__not__le,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% linorder_not_le
tff(fact_1200_linorder__not__le,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% linorder_not_le
tff(fact_1201_order__less__le,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_1202_order__less__le,axiom,
    ! [Xb: filter_nat,Y: filter_nat] :
      ( ord_less_filter_nat(Xb,Y)
    <=> ( ord_le2510731241096832064er_nat(Xb,Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_1203_order__less__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_1204_order__less__le,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Xb),Y)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_1205_order__less__le,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_1206_order__less__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_1207_order__le__less,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xb),Y)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_1208_order__le__less,axiom,
    ! [Xb: filter_nat,Y: filter_nat] :
      ( ord_le2510731241096832064er_nat(Xb,Y)
    <=> ( ord_less_filter_nat(Xb,Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_1209_order__le__less,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_1210_order__le__less,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Y)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_1211_order__le__less,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_1212_order__le__less,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_1213_dual__order_Ostrict__implies__order,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1214_dual__order_Ostrict__implies__order,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( ord_less_filter_nat(B,A)
     => ord_le2510731241096832064er_nat(B,A) ) ).

% dual_order.strict_implies_order
tff(fact_1215_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_1216_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1217_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_1218_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_1219_order_Ostrict__implies__order,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B) ) ).

% order.strict_implies_order
tff(fact_1220_order_Ostrict__implies__order,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ord_less_filter_nat(A,B)
     => ord_le2510731241096832064er_nat(A,B) ) ).

% order.strict_implies_order
tff(fact_1221_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_1222_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ).

% order.strict_implies_order
tff(fact_1223_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_1224_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_1225_dual__order_Ostrict__iff__not,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A)
        & ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1226_dual__order_Ostrict__iff__not,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( ord_less_filter_nat(B,A)
    <=> ( ord_le2510731241096832064er_nat(B,A)
        & ~ ord_le2510731241096832064er_nat(A,B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1227_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_1228_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
        & ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1229_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_1230_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_1231_dual__order_Ostrict__trans2,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),B)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1232_dual__order_Ostrict__trans2,axiom,
    ! [B: filter_nat,A: filter_nat,C: filter_nat] :
      ( ord_less_filter_nat(B,A)
     => ( ord_le2510731241096832064er_nat(C,B)
       => ord_less_filter_nat(C,A) ) ) ).

% dual_order.strict_trans2
tff(fact_1233_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_1234_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1235_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_1236_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_1237_dual__order_Ostrict__trans1,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),B)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1238_dual__order_Ostrict__trans1,axiom,
    ! [B: filter_nat,A: filter_nat,C: filter_nat] :
      ( ord_le2510731241096832064er_nat(B,A)
     => ( ord_less_filter_nat(C,B)
       => ord_less_filter_nat(C,A) ) ) ).

% dual_order.strict_trans1
tff(fact_1239_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_1240_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1241_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_1242_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_1243_dual__order_Ostrict__iff__order,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1244_dual__order_Ostrict__iff__order,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( ord_less_filter_nat(B,A)
    <=> ( ord_le2510731241096832064er_nat(B,A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1245_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_1246_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1247_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_1248_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_1249_dual__order_Oorder__iff__strict,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1250_dual__order_Oorder__iff__strict,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( ord_le2510731241096832064er_nat(B,A)
    <=> ( ord_less_filter_nat(B,A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1251_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_1252_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1253_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_1254_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_1255_dense__le__bounded,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( ! [W: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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_1256_dense__ge__bounded,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xb)
     => ( ! [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),Xb)
             => 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_1257_order_Ostrict__iff__not,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
        & ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_1258_order_Ostrict__iff__not,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ord_less_filter_nat(A,B)
    <=> ( ord_le2510731241096832064er_nat(A,B)
        & ~ ord_le2510731241096832064er_nat(B,A) ) ) ).

% order.strict_iff_not
tff(fact_1259_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_1260_order_Ostrict__iff__not,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
        & ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_1261_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_1262_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_1263_order_Ostrict__trans2,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),C)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),C) ) ) ).

% order.strict_trans2
tff(fact_1264_order_Ostrict__trans2,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat] :
      ( ord_less_filter_nat(A,B)
     => ( ord_le2510731241096832064er_nat(B,C)
       => ord_less_filter_nat(A,C) ) ) ).

% order.strict_trans2
tff(fact_1265_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_1266_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),C) ) ) ).

% order.strict_trans2
tff(fact_1267_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_1268_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_1269_order_Ostrict__trans1,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),C)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),C) ) ) ).

% order.strict_trans1
tff(fact_1270_order_Ostrict__trans1,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat] :
      ( ord_le2510731241096832064er_nat(A,B)
     => ( ord_less_filter_nat(B,C)
       => ord_less_filter_nat(A,C) ) ) ).

% order.strict_trans1
tff(fact_1271_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_1272_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),C)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),C) ) ) ).

% order.strict_trans1
tff(fact_1273_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_1274_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_1275_order_Ostrict__iff__order,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1276_order_Ostrict__iff__order,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ord_less_filter_nat(A,B)
    <=> ( ord_le2510731241096832064er_nat(A,B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1277_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_1278_order_Ostrict__iff__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1279_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_1280_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_1281_order_Oorder__iff__strict,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1282_order_Oorder__iff__strict,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ord_le2510731241096832064er_nat(A,B)
    <=> ( ord_less_filter_nat(A,B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1283_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_1284_order_Oorder__iff__strict,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1285_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_1286_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_1287_not__le__imp__less,axiom,
    ! [Y: extended_enat,Xb: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Y),Xb)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y) ) ).

% not_le_imp_less
tff(fact_1288_not__le__imp__less,axiom,
    ! [Y: real,Xb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% not_le_imp_less
tff(fact_1289_not__le__imp__less,axiom,
    ! [Y: nat,Xb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ).

% not_le_imp_less
tff(fact_1290_not__le__imp__less,axiom,
    ! [Y: int,Xb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ).

% not_le_imp_less
tff(fact_1291_less__le__not__le,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xb),Y)
        & ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Y),Xb) ) ) ).

% less_le_not_le
tff(fact_1292_less__le__not__le,axiom,
    ! [Xb: filter_nat,Y: filter_nat] :
      ( ord_less_filter_nat(Xb,Y)
    <=> ( ord_le2510731241096832064er_nat(Xb,Y)
        & ~ ord_le2510731241096832064er_nat(Y,Xb) ) ) ).

% less_le_not_le
tff(fact_1293_less__le__not__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ).

% less_le_not_le
tff(fact_1294_less__le__not__le,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Xb),Y)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Y)
        & ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Xb) ) ) ).

% less_le_not_le
tff(fact_1295_less__le__not__le,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ) ).

% less_le_not_le
tff(fact_1296_less__le__not__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ) ).

% less_le_not_le
tff(fact_1297_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Z) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ).

% dense_le
tff(fact_1298_dense__ge,axiom,
    ! [Z: real,Y: real] :
      ( ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X4) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ).

% dense_ge
tff(fact_1299_antisym__conv2,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xb),Y)
     => ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_1300_antisym__conv2,axiom,
    ! [Xb: filter_nat,Y: filter_nat] :
      ( ord_le2510731241096832064er_nat(Xb,Y)
     => ( ~ ord_less_filter_nat(Xb,Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_1301_antisym__conv2,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_1302_antisym__conv2,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Y)
     => ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_1303_antisym__conv2,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_1304_antisym__conv2,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_1305_antisym__conv1,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_1306_antisym__conv1,axiom,
    ! [Xb: filter_nat,Y: filter_nat] :
      ( ~ ord_less_filter_nat(Xb,Y)
     => ( ord_le2510731241096832064er_nat(Xb,Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_1307_antisym__conv1,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_1308_antisym__conv1,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Xb),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_1309_antisym__conv1,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_1310_antisym__conv1,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_1311_nless__le,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
    <=> ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1312_nless__le,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ~ ord_less_filter_nat(A,B)
    <=> ( ~ ord_le2510731241096832064er_nat(A,B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1313_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_1314_nless__le,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
    <=> ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1315_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_1316_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_1317_leI,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Y),Xb) ) ).

% leI
tff(fact_1318_leI,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ).

% leI
tff(fact_1319_leI,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ).

% leI
tff(fact_1320_leI,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ).

% leI
tff(fact_1321_leD,axiom,
    ! [Y: extended_enat,Xb: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Y),Xb)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xb),Y) ) ).

% leD
tff(fact_1322_leD,axiom,
    ! [Y: filter_nat,Xb: filter_nat] :
      ( ord_le2510731241096832064er_nat(Y,Xb)
     => ~ ord_less_filter_nat(Xb,Y) ) ).

% leD
tff(fact_1323_leD,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% leD
tff(fact_1324_leD,axiom,
    ! [Y: set_nat,Xb: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Xb)
     => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Xb),Y) ) ).

% leD
tff(fact_1325_leD,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ).

% leD
tff(fact_1326_leD,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ).

% leD
tff(fact_1327_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_1328_bot_Oextremum__uniqueI,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),bot_bot_set_o)
     => ( A = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
tff(fact_1329_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_1330_bot_Oextremum__uniqueI,axiom,
    ! [A: filter_nat] :
      ( ord_le2510731241096832064er_nat(A,bot_bot_filter_nat)
     => ( A = bot_bot_filter_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1331_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_1332_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_1333_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_1334_bot_Oextremum__unique,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),bot_bot_set_o)
    <=> ( A = bot_bot_set_o ) ) ).

% bot.extremum_unique
tff(fact_1335_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_1336_bot_Oextremum__unique,axiom,
    ! [A: filter_nat] :
      ( ord_le2510731241096832064er_nat(A,bot_bot_filter_nat)
    <=> ( A = bot_bot_filter_nat ) ) ).

% bot.extremum_unique
tff(fact_1337_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_1338_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_1339_bot_Oextremum,axiom,
    ! [A: set_real] : aa_set_real_o(ord_less_eq_set_real(bot_bot_set_real),A) ).

% bot.extremum
tff(fact_1340_bot_Oextremum,axiom,
    ! [A: set_o] : aa_set_o_o(ord_less_eq_set_o(bot_bot_set_o),A) ).

% bot.extremum
tff(fact_1341_bot_Oextremum,axiom,
    ! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A) ).

% bot.extremum
tff(fact_1342_bot_Oextremum,axiom,
    ! [A: filter_nat] : ord_le2510731241096832064er_nat(bot_bot_filter_nat,A) ).

% bot.extremum
tff(fact_1343_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_1344_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_1345_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_1346_bot_Onot__eq__extremum,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
    <=> aa_set_o_o(ord_less_set_o(bot_bot_set_o),A) ) ).

% bot.not_eq_extremum
tff(fact_1347_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_1348_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_1349_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_1350_bot_Onot__eq__extremum,axiom,
    ! [A: extended_enat] :
      ( ( A != bot_bo4199563552545308370d_enat )
    <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,bot_bo4199563552545308370d_enat),A) ) ).

% bot.not_eq_extremum
tff(fact_1351_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_1352_bot_Oextremum__strict,axiom,
    ! [A: set_o] : ~ aa_set_o_o(ord_less_set_o(A),bot_bot_set_o) ).

% bot.extremum_strict
tff(fact_1353_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_1354_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_1355_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_1356_bot_Oextremum__strict,axiom,
    ! [A: extended_enat] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),bot_bo4199563552545308370d_enat) ).

% bot.extremum_strict
tff(fact_1357_vebt__buildup_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero_nat )
     => ( ( Xb != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va2: nat] : Xb != aa_nat_nat(suc,aa_nat_nat(suc,Va2)) ) ) ).

% vebt_buildup.cases
tff(fact_1358_max__absorb2,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xb),Y)
     => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(Xb),Y) = Y ) ) ).

% max_absorb2
tff(fact_1359_max__absorb2,axiom,
    ! [Xb: filter_nat,Y: filter_nat] :
      ( ord_le2510731241096832064er_nat(Xb,Y)
     => ( ord_max_filter_nat(Xb,Y) = Y ) ) ).

% max_absorb2
tff(fact_1360_max__absorb2,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => ( aa_real_real(ord_max_real(Xb),Y) = Y ) ) ).

% max_absorb2
tff(fact_1361_max__absorb2,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Y)
     => ( ord_max_set_nat(Xb,Y) = Y ) ) ).

% max_absorb2
tff(fact_1362_max__absorb2,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y) = Y ) ) ).

% max_absorb2
tff(fact_1363_max__absorb2,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y) = Y ) ) ).

% max_absorb2
tff(fact_1364_max__absorb1,axiom,
    ! [Y: extended_enat,Xb: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Y),Xb)
     => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(Xb),Y) = Xb ) ) ).

% max_absorb1
tff(fact_1365_max__absorb1,axiom,
    ! [Y: filter_nat,Xb: filter_nat] :
      ( ord_le2510731241096832064er_nat(Y,Xb)
     => ( ord_max_filter_nat(Xb,Y) = Xb ) ) ).

% max_absorb1
tff(fact_1366_max__absorb1,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
     => ( aa_real_real(ord_max_real(Xb),Y) = Xb ) ) ).

% max_absorb1
tff(fact_1367_max__absorb1,axiom,
    ! [Y: set_nat,Xb: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Xb)
     => ( ord_max_set_nat(Xb,Y) = Xb ) ) ).

% max_absorb1
tff(fact_1368_max__absorb1,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y) = Xb ) ) ).

% max_absorb1
tff(fact_1369_max__absorb1,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y) = Xb ) ) ).

% max_absorb1
tff(fact_1370_max__def,axiom,
    ! [A: extended_enat,B: extended_enat] :
      aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = $ite(aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B),B,A) ).

% max_def
tff(fact_1371_max__def,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ord_max_filter_nat(A,B) = $ite(ord_le2510731241096832064er_nat(A,B),B,A) ).

% max_def
tff(fact_1372_max__def,axiom,
    ! [A: real,B: real] :
      aa_real_real(ord_max_real(A),B) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B),B,A) ).

% max_def
tff(fact_1373_max__def,axiom,
    ! [A: set_nat,B: set_nat] :
      ord_max_set_nat(A,B) = $ite(aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B),B,A) ).

% max_def
tff(fact_1374_max__def,axiom,
    ! [A: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),B,A) ).

% max_def
tff(fact_1375_max__def,axiom,
    ! [A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B),B,A) ).

% max_def
tff(fact_1376_option_Osize__gen_I2_J,axiom,
    ! [Xb: fun_Pr2602117396471556196at_nat,X22: product_prod_nat_nat] : size_o8335143837870341156at_nat(Xb,some_P7363390416028606310at_nat(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(Xb,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_1377_option_Osize__gen_I2_J,axiom,
    ! [Xb: fun_num_nat,X22: num] : size_option_num(Xb,some_num(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(Xb,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_1378_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A4: nat,B4: nat] :
      ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A4),B4) )
    <=> ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
tff(fact_1379_old_Oprod_Oinject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
      ( ( aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,A),B) = aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,A4),B4) )
    <=> ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
tff(fact_1380_old_Oprod_Oinject,axiom,
    ! [A: vEBT_VEBT,B: nat,A4: vEBT_VEBT,B4: nat] :
      ( ( aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,A),B) = aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,A4),B4) )
    <=> ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
tff(fact_1381_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_1382_old_Oprod_Oinject,axiom,
    ! [A: vEBT_VEBT,B: extended_enat,A4: vEBT_VEBT,B4: extended_enat] :
      ( ( aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,A),B) = aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,A4),B4) )
    <=> ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
tff(fact_1383_prod_Oinject,axiom,
    ! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
      ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X22) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_1384_prod_Oinject,axiom,
    ! [X1: product_prod_nat_nat,X22: product_prod_nat_nat,Y1: product_prod_nat_nat,Y22: product_prod_nat_nat] :
      ( ( aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,X1),X22) = aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_1385_prod_Oinject,axiom,
    ! [X1: vEBT_VEBT,X22: nat,Y1: vEBT_VEBT,Y22: nat] :
      ( ( aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,X1),X22) = aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_1386_prod_Oinject,axiom,
    ! [X1: int,X22: int,Y1: int,Y22: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_1387_prod_Oinject,axiom,
    ! [X1: vEBT_VEBT,X22: extended_enat,Y1: vEBT_VEBT,Y22: extended_enat] :
      ( ( aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,X1),X22) = aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_1388_inthall,axiom,
    ! [Xs: list_real,P: fun_real_o,Nb: nat] :
      ( ! [X4: real] :
          ( aa_set_real_o(member_real2(X4),set_real2(Xs))
         => aa_real_o(P,X4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
       => aa_real_o(P,aa_nat_real(nth_real(Xs),Nb)) ) ) ).

% inthall
tff(fact_1389_inthall,axiom,
    ! [Xs: list_o,P: fun_o_o,Nb: nat] :
      ( ! [X4: $o] :
          ( aa_set_o_o(member_o2((X4)),set_o2(Xs))
         => aa_o_o(P,(X4)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% inthall
tff(fact_1390_inthall,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_o,Nb: nat] :
      ( ! [X4: set_nat] :
          ( aa_set_set_nat_o(member_set_nat2(X4),set_set_nat2(Xs))
         => aa_set_nat_o(P,X4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3254054031482475050et_nat(Xs))
       => aa_set_nat_o(P,aa_nat_set_nat(nth_set_nat(Xs),Nb)) ) ) ).

% inthall
tff(fact_1391_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Nb: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT2(X4,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% inthall
tff(fact_1392_inthall,axiom,
    ! [Xs: list_int,P: fun_int_o,Nb: nat] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int2(X4),set_int2(Xs))
         => aa_int_o(P,X4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% inthall
tff(fact_1393_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Nb: nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat2(X4),set_nat2(Xs))
         => aa_nat_o(P,X4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% inthall
tff(fact_1394_set__vebt__finite,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_set_nat_o(finite_finite_nat,vEBT_VEBT_set_vebt(Ta)) ) ).

% set_vebt_finite
tff(fact_1395_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_1396_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_1397_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_1398_nat__descend__induct,axiom,
    ! [Nb: nat,P: fun_nat_o,Mb: nat] :
      ( ! [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K3)
         => aa_nat_o(P,K3) )
     => ( ! [K3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),Nb)
           => ( ! [I3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),I3)
                 => aa_nat_o(P,I3) )
             => aa_nat_o(P,K3) ) )
       => aa_nat_o(P,Mb) ) ) ).

% nat_descend_induct
tff(fact_1399_subset__emptyI,axiom,
    ! [A2: set_set_nat] :
      ( ! [X4: set_nat] : ~ aa_set_set_nat_o(member_set_nat2(X4),A2)
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),bot_bot_set_set_nat) ) ).

% subset_emptyI
tff(fact_1400_subset__emptyI,axiom,
    ! [A2: set_real] :
      ( ! [X4: real] : ~ aa_set_real_o(member_real2(X4),A2)
     => aa_set_real_o(ord_less_eq_set_real(A2),bot_bot_set_real) ) ).

% subset_emptyI
tff(fact_1401_subset__emptyI,axiom,
    ! [A2: set_o] :
      ( ! [X4: $o] : ~ aa_set_o_o(member_o2((X4)),A2)
     => aa_set_o_o(ord_less_eq_set_o(A2),bot_bot_set_o) ) ).

% subset_emptyI
tff(fact_1402_subset__emptyI,axiom,
    ! [A2: set_int] :
      ( ! [X4: int] : ~ aa_set_int_o(member_int2(X4),A2)
     => aa_set_int_o(ord_less_eq_set_int(A2),bot_bot_set_int) ) ).

% subset_emptyI
tff(fact_1403_subset__emptyI,axiom,
    ! [A2: set_nat] :
      ( ! [X4: nat] : ~ aa_set_nat_o(member_nat2(X4),A2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),bot_bot_set_nat) ) ).

% subset_emptyI
tff(fact_1404_exists__least__lemma,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_nat_o(P,zero_zero_nat)
     => ( ? [X_1: nat] : aa_nat_o(P,X_1)
       => ? [N: nat] :
            ( ~ aa_nat_o(P,N)
            & aa_nat_o(P,aa_nat_nat(suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_1405_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : finite5795047828879050333T_VEBT(set_VEBT_VEBT2(Xs)) ).

% List.finite_set
tff(fact_1406_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : aa_set_nat_o(finite_finite_nat,set_nat2(Xs)) ).

% List.finite_set
tff(fact_1407_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : aa_set_complex_o(finite3207457112153483333omplex,set_complex2(Xs)) ).

% List.finite_set
tff(fact_1408_List_Ofinite__set,axiom,
    ! [Xs: list_int] : aa_set_int_o(finite_finite_int,set_int2(Xs)) ).

% List.finite_set
tff(fact_1409_List_Ofinite__set,axiom,
    ! [Xs: list_Extended_enat] : aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,set_Extended_enat2(Xs)) ).

% List.finite_set
tff(fact_1410_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,Ia: nat,Xb: vEBT_VEBT] : size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(Xs,Ia,Xb)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_list_update
tff(fact_1411_length__list__update,axiom,
    ! [Xs: list_int,Ia: nat,Xb: int] : size_size_list_int(list_update_int(Xs,Ia,Xb)) = size_size_list_int(Xs) ).

% length_list_update
tff(fact_1412_length__list__update,axiom,
    ! [Xs: list_nat,Ia: nat,Xb: nat] : size_size_list_nat(list_update_nat(Xs,Ia,Xb)) = size_size_list_nat(Xs) ).

% length_list_update
tff(fact_1413_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,Ia: nat,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),Ia)
     => ( list_u1324408373059187874T_VEBT(Xs,Ia,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_1414_list__update__beyond,axiom,
    ! [Xs: list_int,Ia: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),Ia)
     => ( list_update_int(Xs,Ia,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_1415_list__update__beyond,axiom,
    ! [Xs: list_nat,Ia: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),Ia)
     => ( list_update_nat(Xs,Ia,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_1416_nth__list__update__eq,axiom,
    ! [Ia: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,Ia,Xb),Ia) = Xb ) ) ).

% nth_list_update_eq
tff(fact_1417_nth__list__update__eq,axiom,
    ! [Ia: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,Ia,Xb)),Ia) = Xb ) ) ).

% nth_list_update_eq
tff(fact_1418_nth__list__update__eq,axiom,
    ! [Ia: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,Ia,Xb)),Ia) = Xb ) ) ).

% nth_list_update_eq
tff(fact_1419_set__swap,axiom,
    ! [Ia: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs))
       => ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,Ia,nth_VEBT_VEBT(Xs,J)),J,nth_VEBT_VEBT(Xs,Ia))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_1420_set__swap,axiom,
    ! [Ia: nat,Xs: list_int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(Xs))
       => ( set_int2(list_update_int(list_update_int(Xs,Ia,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),Ia))) = set_int2(Xs) ) ) ) ).

% set_swap
tff(fact_1421_set__swap,axiom,
    ! [Ia: nat,Xs: list_nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_nat(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_nat(Xs))
       => ( set_nat2(list_update_nat(list_update_nat(Xs,Ia,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),Ia))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_1422_neq__if__length__neq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) != size_s6755466524823107622T_VEBT(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1423_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) != size_size_list_int(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1424_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) != size_size_list_nat(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1425_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_1426_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_int] : size_size_list_int(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_1427_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_nat] : size_size_list_nat(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_1428_size__neq__size__imp__neq,axiom,
    ! [Xb: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xb) != size_s6755466524823107622T_VEBT(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1429_size__neq__size__imp__neq,axiom,
    ! [Xb: num,Y: num] :
      ( ( aa_num_nat(size_size_num,Xb) != aa_num_nat(size_size_num,Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1430_size__neq__size__imp__neq,axiom,
    ! [Xb: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,Xb) != aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1431_size__neq__size__imp__neq,axiom,
    ! [Xb: list_int,Y: list_int] :
      ( ( size_size_list_int(Xb) != size_size_list_int(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1432_size__neq__size__imp__neq,axiom,
    ! [Xb: list_nat,Y: list_nat] :
      ( ( size_size_list_nat(Xb) != size_size_list_nat(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1433_length__induct,axiom,
    ! [P: fun_list_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(Ys2)),size_s6755466524823107622T_VEBT(Xs2))
             => aa_list_VEBT_VEBT_o(P,Ys2) )
         => aa_list_VEBT_VEBT_o(P,Xs2) )
     => aa_list_VEBT_VEBT_o(P,Xs) ) ).

% length_induct
tff(fact_1434_length__induct,axiom,
    ! [P: fun_list_int_o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys2: list_int] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(Ys2)),size_size_list_int(Xs2))
             => aa_list_int_o(P,Ys2) )
         => aa_list_int_o(P,Xs2) )
     => aa_list_int_o(P,Xs) ) ).

% length_induct
tff(fact_1435_length__induct,axiom,
    ! [P: fun_list_nat_o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(Ys2)),size_size_list_nat(Xs2))
             => aa_list_nat_o(P,Ys2) )
         => aa_list_nat_o(P,Xs2) )
     => aa_list_nat_o(P,Xs) ) ).

% length_induct
tff(fact_1436_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A2)
     => ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A2 ) ).

% finite_list
tff(fact_1437_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_1438_finite__list,axiom,
    ! [A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ? [Xs2: list_complex] : set_complex2(Xs2) = A2 ) ).

% finite_list
tff(fact_1439_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_1440_finite__list,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ? [Xs2: list_Extended_enat] : set_Extended_enat2(Xs2) = A2 ) ).

% finite_list
tff(fact_1441_infinite__growing,axiom,
    ! [X5: set_o] :
      ( ( X5 != bot_bot_set_o )
     => ( ! [X4: $o] :
            ( aa_set_o_o(member_o2((X4)),X5)
           => ? [Xa: $o] :
                ( aa_set_o_o(member_o2((Xa)),X5)
                & aa_o_o(aa_o_fun_o_o(ord_less_o,(X4)),(Xa)) ) )
       => ~ aa_set_o_o(finite_finite_o,X5) ) ) ).

% infinite_growing
tff(fact_1442_infinite__growing,axiom,
    ! [X5: set_nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),X5)
           => ? [Xa: nat] :
                ( aa_set_nat_o(member_nat2(Xa),X5)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Xa) ) )
       => ~ aa_set_nat_o(finite_finite_nat,X5) ) ) ).

% infinite_growing
tff(fact_1443_infinite__growing,axiom,
    ! [X5: set_Extended_enat] :
      ( ( X5 != bot_bo7653980558646680370d_enat )
     => ( ! [X4: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),X5)
           => ? [Xa: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),X5)
                & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Xa) ) )
       => ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,X5) ) ) ).

% infinite_growing
tff(fact_1444_infinite__growing,axiom,
    ! [X5: set_real] :
      ( ( X5 != bot_bot_set_real )
     => ( ! [X4: real] :
            ( aa_set_real_o(member_real2(X4),X5)
           => ? [Xa: real] :
                ( aa_set_real_o(member_real2(Xa),X5)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Xa) ) )
       => ~ aa_set_real_o(finite_finite_real,X5) ) ) ).

% infinite_growing
tff(fact_1445_infinite__growing,axiom,
    ! [X5: set_int] :
      ( ( X5 != bot_bot_set_int )
     => ( ! [X4: int] :
            ( aa_set_int_o(member_int2(X4),X5)
           => ? [Xa: int] :
                ( aa_set_int_o(member_int2(Xa),X5)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Xa) ) )
       => ~ aa_set_int_o(finite_finite_int,X5) ) ) ).

% infinite_growing
tff(fact_1446_ex__min__if__finite,axiom,
    ! [S3: set_o] :
      ( aa_set_o_o(finite_finite_o,S3)
     => ( ( S3 != bot_bot_set_o )
       => ? [X4: $o] :
            ( aa_set_o_o(member_o2((X4)),S3)
            & ~ ? [Xa: $o] :
                  ( aa_set_o_o(member_o2((Xa)),S3)
                  & aa_o_o(aa_o_fun_o_o(ord_less_o,(Xa)),(X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1447_ex__min__if__finite,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( ( S3 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),S3)
            & ~ ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat2(Xa),S3)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X4) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1448_ex__min__if__finite,axiom,
    ! [S3: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ? [X4: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),S3)
            & ~ ? [Xa: extended_enat] :
                  ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),S3)
                  & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),X4) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1449_ex__min__if__finite,axiom,
    ! [S3: set_real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( ( S3 != bot_bot_set_real )
       => ? [X4: real] :
            ( aa_set_real_o(member_real2(X4),S3)
            & ~ ? [Xa: real] :
                  ( aa_set_real_o(member_real2(Xa),S3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X4) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1450_ex__min__if__finite,axiom,
    ! [S3: set_int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ( S3 != bot_bot_set_int )
       => ? [X4: int] :
            ( aa_set_int_o(member_int2(X4),S3)
            & ~ ? [Xa: int] :
                  ( aa_set_int_o(member_int2(Xa),S3)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X4) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1451_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I2) = nth_VEBT_VEBT(Ys,I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_1452_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_1453_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_1454_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na6887940067898170868VEBT_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
         => ? [X_13: vEBT_VEBT] : aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I4),X_13) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K )
          & ! [I4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
             => aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I4),nth_VEBT_VEBT(Xs3,I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_1455_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_int_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
         => ? [X_13: int] : aa_int_o(aa_nat_fun_int_o(P,I4),X_13) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K )
          & ! [I4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
             => aa_int_o(aa_nat_fun_int_o(P,I4),aa_nat_int(nth_int(Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_1456_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
         => ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I4),X_13) )
    <=> ? [Xs3: list_nat] :
          ( ( size_size_list_nat(Xs3) = K )
          & ! [I4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
             => aa_nat_o(aa_nat_fun_nat_o(P,I4),aa_nat_nat(nth_nat(Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_1457_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Xs = Ys )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I4) = nth_VEBT_VEBT(Ys,I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_1458_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I4) = aa_nat_int(nth_int(Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_1459_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I4) = aa_nat_nat(nth_nat(Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_1460_length__pos__if__in__set,axiom,
    ! [Xb: real,Xs: list_real] :
      ( aa_set_real_o(member_real2(Xb),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_1461_length__pos__if__in__set,axiom,
    ! [Xb: $o,Xs: list_o] :
      ( aa_set_o_o(member_o2((Xb)),set_o2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_o(Xs)) ) ).

% length_pos_if_in_set
tff(fact_1462_length__pos__if__in__set,axiom,
    ! [Xb: set_nat,Xs: list_set_nat] :
      ( aa_set_set_nat_o(member_set_nat2(Xb),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_1463_length__pos__if__in__set,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)) ) ).

% length_pos_if_in_set
tff(fact_1464_length__pos__if__in__set,axiom,
    ! [Xb: int,Xs: list_int] :
      ( aa_set_int_o(member_int2(Xb),set_int2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_int(Xs)) ) ).

% length_pos_if_in_set
tff(fact_1465_length__pos__if__in__set,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat2(Xb),set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_1466_nth__mem,axiom,
    ! [Nb: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => aa_set_real_o(member_real2(aa_nat_real(nth_real(Xs),Nb)),set_real2(Xs)) ) ).

% nth_mem
tff(fact_1467_nth__mem,axiom,
    ! [Nb: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => aa_set_o_o(member_o2(aa_nat_o(nth_o(Xs),Nb)),set_o2(Xs)) ) ).

% nth_mem
tff(fact_1468_nth__mem,axiom,
    ! [Nb: nat,Xs: list_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3254054031482475050et_nat(Xs))
     => aa_set_set_nat_o(member_set_nat2(aa_nat_set_nat(nth_set_nat(Xs),Nb)),set_set_nat2(Xs)) ) ).

% nth_mem
tff(fact_1469_nth__mem,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => member_VEBT_VEBT2(nth_VEBT_VEBT(Xs,Nb),set_VEBT_VEBT2(Xs)) ) ).

% nth_mem
tff(fact_1470_nth__mem,axiom,
    ! [Nb: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => aa_set_int_o(member_int2(aa_nat_int(nth_int(Xs),Nb)),set_int2(Xs)) ) ).

% nth_mem
tff(fact_1471_nth__mem,axiom,
    ! [Nb: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat2(aa_nat_nat(nth_nat(Xs),Nb)),set_nat2(Xs)) ) ).

% nth_mem
tff(fact_1472_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( ! [X4: vEBT_VEBT] :
            ( member_VEBT_VEBT2(X4,set_VEBT_VEBT2(Xs))
           => aa_VEBT_VEBT_o(P,X4) )
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% list_ball_nth
tff(fact_1473_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_int,P: fun_int_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => ( ! [X4: int] :
            ( aa_set_int_o(member_int2(X4),set_int2(Xs))
           => aa_int_o(P,X4) )
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_1474_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),set_nat2(Xs))
           => aa_nat_o(P,X4) )
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_1475_in__set__conv__nth,axiom,
    ! [Xb: real,Xs: list_real] :
      ( aa_set_real_o(member_real2(Xb),set_real2(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_real(Xs))
          & ( aa_nat_real(nth_real(Xs),I4) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_1476_in__set__conv__nth,axiom,
    ! [Xb: $o,Xs: list_o] :
      ( aa_set_o_o(member_o2((Xb)),set_o2(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_o(Xs))
          & ( aa_nat_o(nth_o(Xs),I4)
          <=> (Xb) ) ) ) ).

% in_set_conv_nth
tff(fact_1477_in__set__conv__nth,axiom,
    ! [Xb: set_nat,Xs: list_set_nat] :
      ( aa_set_set_nat_o(member_set_nat2(Xb),set_set_nat2(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s3254054031482475050et_nat(Xs))
          & ( aa_nat_set_nat(nth_set_nat(Xs),I4) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_1478_in__set__conv__nth,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs))
          & ( nth_VEBT_VEBT(Xs,I4) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_1479_in__set__conv__nth,axiom,
    ! [Xb: int,Xs: list_int] :
      ( aa_set_int_o(member_int2(Xb),set_int2(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_int(Xs))
          & ( aa_nat_int(nth_int(Xs),I4) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_1480_in__set__conv__nth,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat2(Xb),set_nat2(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_nat(Xs))
          & ( aa_nat_nat(nth_nat(Xs),I4) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_1481_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_o,Xb: real] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Xs))
         => aa_real_o(P,aa_nat_real(nth_real(Xs),I2)) )
     => ( aa_set_real_o(member_real2(Xb),set_real2(Xs))
       => aa_real_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_1482_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: fun_o_o,Xb: $o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I2)) )
     => ( aa_set_o_o(member_o2((Xb)),set_o2(Xs))
       => aa_o_o(P,(Xb)) ) ) ).

% all_nth_imp_all_set
tff(fact_1483_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_o,Xb: set_nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3254054031482475050et_nat(Xs))
         => aa_set_nat_o(P,aa_nat_set_nat(nth_set_nat(Xs),I2)) )
     => ( aa_set_set_nat_o(member_set_nat2(Xb),set_set_nat2(Xs))
       => aa_set_nat_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_1484_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Xb: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I2)) )
     => ( member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(Xs))
       => aa_VEBT_VEBT_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_1485_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_o,Xb: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I2)) )
     => ( aa_set_int_o(member_int2(Xb),set_int2(Xs))
       => aa_int_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_1486_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Xb: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I2)) )
     => ( aa_set_nat_o(member_nat2(Xb),set_nat2(Xs))
       => aa_nat_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_1487_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X3) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_1488_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_o] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int2(X3),set_int2(Xs))
         => aa_int_o(P,X3) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_1489_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),set_nat2(Xs))
         => aa_nat_o(P,X3) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_1490_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => aa_set_real_o(member_real2(Xb),set_real2(list_update_real(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_1491_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => aa_set_o_o(member_o2((Xb)),set_o2(list_update_o(Xs,Nb,(Xb)))) ) ).

% set_update_memI
tff(fact_1492_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_set_nat,Xb: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3254054031482475050et_nat(Xs))
     => aa_set_set_nat_o(member_set_nat2(Xb),set_set_nat2(list_update_set_nat(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_1493_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_1494_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => aa_set_int_o(member_int2(Xb),set_int2(list_update_int(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_1495_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat2(Xb),set_nat2(list_update_nat(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_1496_list__update__same__conv,axiom,
    ! [Ia: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_s6755466524823107622T_VEBT(Xs))
     => ( ( list_u1324408373059187874T_VEBT(Xs,Ia,Xb) = Xs )
      <=> ( nth_VEBT_VEBT(Xs,Ia) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_1497_list__update__same__conv,axiom,
    ! [Ia: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_int(Xs))
     => ( ( list_update_int(Xs,Ia,Xb) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),Ia) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_1498_list__update__same__conv,axiom,
    ! [Ia: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_nat(Xs))
     => ( ( list_update_nat(Xs,Ia,Xb) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),Ia) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_1499_nth__list__update,axiom,
    ! [Ia: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,Ia,Xb),J) = $ite(Ia = J,Xb,nth_VEBT_VEBT(Xs,J)) ) ) ).

% nth_list_update
tff(fact_1500_nth__list__update,axiom,
    ! [Ia: nat,Xs: list_int,Xb: int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,Ia,Xb)),J) = $ite(Ia = J,Xb,aa_nat_int(nth_int(Xs),J)) ) ) ).

% nth_list_update
tff(fact_1501_nth__list__update,axiom,
    ! [Ia: nat,Xs: list_nat,Xb: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,Ia,Xb)),J) = $ite(Ia = J,Xb,aa_nat_nat(nth_nat(Xs),J)) ) ) ).

% nth_list_update
tff(fact_1502_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] : Y != aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A3),B3) ).

% old.prod.exhaust
tff(fact_1503_old_Oprod_Oexhaust,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] : Y != aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,A3),B3) ).

% old.prod.exhaust
tff(fact_1504_old_Oprod_Oexhaust,axiom,
    ! [Y: produc9072475918466114483BT_nat] :
      ~ ! [A3: vEBT_VEBT,B3: nat] : Y != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,A3),B3) ).

% old.prod.exhaust
tff(fact_1505_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_1506_old_Oprod_Oexhaust,axiom,
    ! [Y: produc7272778201969148633d_enat] :
      ~ ! [A3: vEBT_VEBT,B3: extended_enat] : Y != aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,A3),B3) ).

% old.prod.exhaust
tff(fact_1507_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X4: nat,Y3: nat] : P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3) ).

% surj_pair
tff(fact_1508_surj__pair,axiom,
    ! [P2: produc859450856879609959at_nat] :
    ? [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] : P2 = aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,X4),Y3) ).

% surj_pair
tff(fact_1509_surj__pair,axiom,
    ! [P2: produc9072475918466114483BT_nat] :
    ? [X4: vEBT_VEBT,Y3: nat] : P2 = aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,X4),Y3) ).

% surj_pair
tff(fact_1510_surj__pair,axiom,
    ! [P2: product_prod_int_int] :
    ? [X4: int,Y3: int] : P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3) ).

% surj_pair
tff(fact_1511_surj__pair,axiom,
    ! [P2: produc7272778201969148633d_enat] :
    ? [X4: vEBT_VEBT,Y3: extended_enat] : P2 = aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,X4),Y3) ).

% surj_pair
tff(fact_1512_prod__cases,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,P2: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A3),B3))
     => aa_Pro6150637604445964947_nat_o(P,P2) ) ).

% prod_cases
tff(fact_1513_prod__cases,axiom,
    ! [P: fun_Pr363642936170290612_nat_o,P2: produc859450856879609959at_nat] :
      ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] : aa_Pro602569247023830573_nat_o(P,aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,A3),B3))
     => aa_Pro602569247023830573_nat_o(P,P2) ) ).

% prod_cases
tff(fact_1514_prod__cases,axiom,
    ! [P: fun_Pr7546709192626087934_nat_o,P2: produc9072475918466114483BT_nat] :
      ( ! [A3: vEBT_VEBT,B3: nat] : aa_Pro632845113475883111_nat_o(P,aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,A3),B3))
     => aa_Pro632845113475883111_nat_o(P,P2) ) ).

% prod_cases
tff(fact_1515_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_1516_prod__cases,axiom,
    ! [P: fun_Pr5843119925985263810enat_o,P2: produc7272778201969148633d_enat] :
      ( ! [A3: vEBT_VEBT,B3: extended_enat] : aa_Pro1441479085436938555enat_o(P,aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,A3),B3))
     => aa_Pro1441479085436938555enat_o(P,P2) ) ).

% prod_cases
tff(fact_1517_Pair__inject,axiom,
    ! [A: nat,B: nat,A4: nat,B4: nat] :
      ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A4),B4) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
tff(fact_1518_Pair__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
      ( ( aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,A),B) = aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,A4),B4) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
tff(fact_1519_Pair__inject,axiom,
    ! [A: vEBT_VEBT,B: nat,A4: vEBT_VEBT,B4: nat] :
      ( ( aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,A),B) = aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,A4),B4) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
tff(fact_1520_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_1521_Pair__inject,axiom,
    ! [A: vEBT_VEBT,B: extended_enat,A4: vEBT_VEBT,B4: extended_enat] :
      ( ( aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,A),B) = aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,A4),B4) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
tff(fact_1522_prod__cases3,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A3: product_prod_nat_nat,B3: nat,C3: nat] : Y != aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,A3),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B3),C3)) ).

% prod_cases3
tff(fact_1523_prod__induct3,axiom,
    ! [P: fun_Pr363642936170290612_nat_o,Xb: produc859450856879609959at_nat] :
      ( ! [A3: product_prod_nat_nat,B3: nat,C3: nat] : aa_Pro602569247023830573_nat_o(P,aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,A3),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B3),C3)))
     => aa_Pro602569247023830573_nat_o(P,Xb) ) ).

% prod_induct3
tff(fact_1524_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S: nat,R: set_Pr1261947904930325089at_nat,S4: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,R2),S),R)
     => ( ( S4 = S )
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,R2),S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1525_ssubst__Pair__rhs,axiom,
    ! [R2: product_prod_nat_nat,S: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,S4: product_prod_nat_nat] :
      ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,R2),S),R)
     => ( ( S4 = S )
       => member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,R2),S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1526_ssubst__Pair__rhs,axiom,
    ! [R2: vEBT_VEBT,S: nat,R: set_Pr7556676689462069481BT_nat,S4: nat] :
      ( member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,R2),S),R)
     => ( ( S4 = S )
       => member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,R2),S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1527_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_1528_ssubst__Pair__rhs,axiom,
    ! [R2: vEBT_VEBT,S: extended_enat,R: set_Pr2457182780427864761d_enat,S4: extended_enat] :
      ( member38198578724832770d_enat(aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,R2),S),R)
     => ( ( S4 = S )
       => member38198578724832770d_enat(aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,R2),S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1529_finite__has__minimal,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ? [X4: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),A2)
            & ! [Xa: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
               => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1530_finite__has__minimal,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ? [X4: $o] :
            ( aa_set_o_o(member_o2((X4)),A2)
            & ! [Xa: $o] :
                ( aa_set_o_o(member_o2((Xa)),A2)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa)),(X4))
                 => ( (X4)
                  <=> (Xa) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1531_finite__has__minimal,axiom,
    ! [A2: set_filter_nat] :
      ( finite2119507909894593271er_nat(A2)
     => ( ( A2 != bot_bo498966703094740906er_nat )
       => ? [X4: filter_nat] :
            ( member_filter_nat(X4,A2)
            & ! [Xa: filter_nat] :
                ( member_filter_nat(Xa,A2)
               => ( ord_le2510731241096832064er_nat(Xa,X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1532_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ? [X4: real] :
            ( aa_set_real_o(member_real2(X4),A2)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real2(Xa),A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1533_finite__has__minimal,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X4: set_nat] :
            ( aa_set_set_nat_o(member_set_nat2(X4),A2)
            & ! [Xa: set_nat] :
                ( aa_set_set_nat_o(member_set_nat2(Xa),A2)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1534_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),A2)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat2(Xa),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1535_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ? [X4: int] :
            ( aa_set_int_o(member_int2(X4),A2)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int2(Xa),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1536_finite__has__maximal,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ? [X4: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),A2)
            & ! [Xa: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
               => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1537_finite__has__maximal,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ? [X4: $o] :
            ( aa_set_o_o(member_o2((X4)),A2)
            & ! [Xa: $o] :
                ( aa_set_o_o(member_o2((Xa)),A2)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X4)),(Xa))
                 => ( (X4)
                  <=> (Xa) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1538_finite__has__maximal,axiom,
    ! [A2: set_filter_nat] :
      ( finite2119507909894593271er_nat(A2)
     => ( ( A2 != bot_bo498966703094740906er_nat )
       => ? [X4: filter_nat] :
            ( member_filter_nat(X4,A2)
            & ! [Xa: filter_nat] :
                ( member_filter_nat(Xa,A2)
               => ( ord_le2510731241096832064er_nat(X4,Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1539_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ? [X4: real] :
            ( aa_set_real_o(member_real2(X4),A2)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real2(Xa),A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1540_finite__has__maximal,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X4: set_nat] :
            ( aa_set_set_nat_o(member_set_nat2(X4),A2)
            & ! [Xa: set_nat] :
                ( aa_set_set_nat_o(member_set_nat2(Xa),A2)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1541_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),A2)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat2(Xa),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1542_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ? [X4: int] :
            ( aa_set_int_o(member_int2(X4),A2)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int2(Xa),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1543_bot__empty__eq,axiom,
    ! [X: set_nat] :
      ( aa_set_nat_o(bot_bo3967814874310145751_nat_o,X)
    <=> aa_set_set_nat_o(member_set_nat2(X),bot_bot_set_set_nat) ) ).

% bot_empty_eq
tff(fact_1544_bot__empty__eq,axiom,
    ! [X: real] :
      ( aa_real_o(bot_bot_fun_real_o,X)
    <=> aa_set_real_o(member_real2(X),bot_bot_set_real) ) ).

% bot_empty_eq
tff(fact_1545_bot__empty__eq,axiom,
    ! [X: $o] :
      ( aa_o_o(bot_bot_fun_o_o,(X))
    <=> aa_set_o_o(member_o2((X)),bot_bot_set_o) ) ).

% bot_empty_eq
tff(fact_1546_bot__empty__eq,axiom,
    ! [X: nat] :
      ( aa_nat_o(bot_bot_fun_nat_o,X)
    <=> aa_set_nat_o(member_nat2(X),bot_bot_set_nat) ) ).

% bot_empty_eq
tff(fact_1547_bot__empty__eq,axiom,
    ! [X: int] :
      ( aa_int_o(bot_bot_fun_int_o,X)
    <=> aa_set_int_o(member_int2(X),bot_bot_set_int) ) ).

% bot_empty_eq
tff(fact_1548_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_1549_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_1550_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_1551_Collect__empty__eq__bot,axiom,
    ! [P: fun_o_o] :
      ( ( collect_o(P) = bot_bot_set_o )
    <=> ( P = bot_bot_fun_o_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1552_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_1553_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_1554_nth__enumerate__eq,axiom,
    ! [Mb: nat,Xs: list_VEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_Pr744662078594809490T_VEBT(enumerate_VEBT_VEBT(Nb,Xs),Mb) = produc599794634098209291T_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb),nth_VEBT_VEBT(Xs,Mb)) ) ) ).

% nth_enumerate_eq
tff(fact_1555_nth__enumerate__eq,axiom,
    ! [Mb: nat,Xs: list_int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),size_size_list_int(Xs))
     => ( nth_Pr3440142176431000676at_int(enumerate_int(Nb,Xs),Mb) = product_Pair_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb),aa_nat_int(nth_int(Xs),Mb)) ) ) ).

% nth_enumerate_eq
tff(fact_1556_nth__enumerate__eq,axiom,
    ! [Mb: nat,Xs: list_nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),size_size_list_nat(Xs))
     => ( nth_Pr7617993195940197384at_nat(enumerate_nat(Nb,Xs),Mb) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),aa_nat_nat(nth_nat(Xs),Mb)) ) ) ).

% nth_enumerate_eq
tff(fact_1557_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_nat2(N3),S3) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_1558_finite__nat__set__iff__bounded__le,axiom,
    ! [N4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N4)
    <=> ? [M5: nat] :
        ! [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),M5) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_1559_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_nat2(N5),S3) ) )
     => ~ aa_set_nat_o(finite_finite_nat,S3) ) ).

% unbounded_k_infinite
tff(fact_1560_bounded__nat__set__is__finite,axiom,
    ! [N4: set_nat,Nb: nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat2(X4),N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Nb) )
     => aa_set_nat_o(finite_finite_nat,N4) ) ).

% bounded_nat_set_is_finite
tff(fact_1561_length__enumerate,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT] : size_s4762443039079500285T_VEBT(enumerate_VEBT_VEBT(Nb,Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_enumerate
tff(fact_1562_length__enumerate,axiom,
    ! [Nb: nat,Xs: list_int] : size_s2970893825323803983at_int(enumerate_int(Nb,Xs)) = size_size_list_int(Xs) ).

% length_enumerate
tff(fact_1563_length__enumerate,axiom,
    ! [Nb: nat,Xs: list_nat] : size_s5460976970255530739at_nat(enumerate_nat(Nb,Xs)) = size_size_list_nat(Xs) ).

% length_enumerate
tff(fact_1564_finite__maxlen,axiom,
    ! [M6: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M6)
     => ? [N: nat] :
        ! [X: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(X)),N) ) ) ).

% finite_maxlen
tff(fact_1565_finite__maxlen,axiom,
    ! [M6: set_list_int] :
      ( finite3922522038869484883st_int(M6)
     => ? [N: nat] :
        ! [X: list_int] :
          ( member_list_int(X,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(X)),N) ) ) ).

% finite_maxlen
tff(fact_1566_finite__maxlen,axiom,
    ! [M6: set_list_nat] :
      ( finite8100373058378681591st_nat(M6)
     => ? [N: nat] :
        ! [X: list_nat] :
          ( member_list_nat(X,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(X)),N) ) ) ).

% finite_maxlen
tff(fact_1567_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3),R2)
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3),S) )
     => ord_le3146513528884898305at_nat(R2,S) ) ).

% subrelI
tff(fact_1568_subrelI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,X4),Y3),R2)
         => member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,X4),Y3),S) )
     => ord_le3000389064537975527at_nat(R2,S) ) ).

% subrelI
tff(fact_1569_subrelI,axiom,
    ! [R2: set_Pr7556676689462069481BT_nat,S: set_Pr7556676689462069481BT_nat] :
      ( ! [X4: vEBT_VEBT,Y3: nat] :
          ( member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,X4),Y3),R2)
         => member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,X4),Y3),S) )
     => ord_le3442269383143156041BT_nat(R2,S) ) ).

% subrelI
tff(fact_1570_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ! [X4: int,Y3: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3),R2)
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3),S) )
     => ord_le2843351958646193337nt_int(R2,S) ) ).

% subrelI
tff(fact_1571_subrelI,axiom,
    ! [R2: set_Pr2457182780427864761d_enat,S: set_Pr2457182780427864761d_enat] :
      ( ! [X4: vEBT_VEBT,Y3: extended_enat] :
          ( member38198578724832770d_enat(aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,X4),Y3),R2)
         => member38198578724832770d_enat(aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,X4),Y3),S) )
     => ord_le8566326065971749465d_enat(R2,S) ) ).

% subrelI
tff(fact_1572_bounded__Max__nat,axiom,
    ! [P: fun_nat_o,Xb: nat,M6: nat] :
      ( aa_nat_o(P,Xb)
     => ( ! [X4: nat] :
            ( aa_nat_o(P,X4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),M6) )
       => ~ ! [M: nat] :
              ( aa_nat_o(P,M)
             => ~ ! [X: nat] :
                    ( aa_nat_o(P,X)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),M) ) ) ) ) ).

% bounded_Max_nat
tff(fact_1573_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [Xb: produc9217583074051984497at_nat] :
      ~ ! [F2: fun_nat_fun_nat_nat,A3: nat,B3: nat,Acc: nat] : Xb != produc2818902965508115497at_nat(F2,produc487386426758144856at_nat(A3,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B3),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_1574_finite__has__minimal2,axiom,
    ! [A2: set_o,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o2((A)),A2)
       => ? [X4: $o] :
            ( aa_set_o_o(member_o2((X4)),A2)
            & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X4)),(A))
            & ! [Xa: $o] :
                ( aa_set_o_o(member_o2((Xa)),A2)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa)),(X4))
                 => ( (X4)
                  <=> (Xa) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1575_finite__has__minimal2,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(A),A2)
       => ? [X4: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),A2)
            & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,X4),A)
            & ! [Xa: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
               => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1576_finite__has__minimal2,axiom,
    ! [A2: set_filter_nat,A: filter_nat] :
      ( finite2119507909894593271er_nat(A2)
     => ( member_filter_nat(A,A2)
       => ? [X4: filter_nat] :
            ( member_filter_nat(X4,A2)
            & ord_le2510731241096832064er_nat(X4,A)
            & ! [Xa: filter_nat] :
                ( member_filter_nat(Xa,A2)
               => ( ord_le2510731241096832064er_nat(Xa,X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1577_finite__has__minimal2,axiom,
    ! [A2: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real2(A),A2)
       => ? [X4: real] :
            ( aa_set_real_o(member_real2(X4),A2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),A)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real2(Xa),A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1578_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_nat2(A),A2)
       => ? [X4: set_nat] :
            ( aa_set_set_nat_o(member_set_nat2(X4),A2)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X4),A)
            & ! [Xa: set_nat] :
                ( aa_set_set_nat_o(member_set_nat2(Xa),A2)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1579_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat2(A),A2)
       => ? [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),A2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),A)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat2(Xa),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1580_finite__has__minimal2,axiom,
    ! [A2: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int2(A),A2)
       => ? [X4: int] :
            ( aa_set_int_o(member_int2(X4),A2)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),A)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int2(Xa),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1581_finite__has__maximal2,axiom,
    ! [A2: set_o,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o2((A)),A2)
       => ? [X4: $o] :
            ( aa_set_o_o(member_o2((X4)),A2)
            & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A)),(X4))
            & ! [Xa: $o] :
                ( aa_set_o_o(member_o2((Xa)),A2)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X4)),(Xa))
                 => ( (X4)
                  <=> (Xa) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1582_finite__has__maximal2,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(A),A2)
       => ? [X4: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),A2)
            & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),X4)
            & ! [Xa: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),A2)
               => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1583_finite__has__maximal2,axiom,
    ! [A2: set_filter_nat,A: filter_nat] :
      ( finite2119507909894593271er_nat(A2)
     => ( member_filter_nat(A,A2)
       => ? [X4: filter_nat] :
            ( member_filter_nat(X4,A2)
            & ord_le2510731241096832064er_nat(A,X4)
            & ! [Xa: filter_nat] :
                ( member_filter_nat(Xa,A2)
               => ( ord_le2510731241096832064er_nat(X4,Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1584_finite__has__maximal2,axiom,
    ! [A2: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real2(A),A2)
       => ? [X4: real] :
            ( aa_set_real_o(member_real2(X4),A2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real2(Xa),A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1585_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_nat2(A),A2)
       => ? [X4: set_nat] :
            ( aa_set_set_nat_o(member_set_nat2(X4),A2)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X4)
            & ! [Xa: set_nat] :
                ( aa_set_set_nat_o(member_set_nat2(Xa),A2)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1586_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat2(A),A2)
       => ? [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),A2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X4)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat2(Xa),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1587_finite__has__maximal2,axiom,
    ! [A2: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int2(A),A2)
       => ? [X4: int] :
            ( aa_set_int_o(member_int2(X4),A2)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X4)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int2(Xa),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1588_finite__transitivity__chain,axiom,
    ! [A2: set_set_nat,R: fun_se2170385732985038782_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ! [X4: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(R,X4),X4)
       => ( ! [X4: set_nat,Y3: set_nat,Z2: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(R,X4),Y3)
             => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(R,Y3),Z2)
               => aa_set_nat_o(aa_set7861427066675100071_nat_o(R,X4),Z2) ) )
         => ( ! [X4: set_nat] :
                ( aa_set_set_nat_o(member_set_nat2(X4),A2)
               => ? [Y4: set_nat] :
                    ( aa_set_set_nat_o(member_set_nat2(Y4),A2)
                    & aa_set_nat_o(aa_set7861427066675100071_nat_o(R,X4),Y4) ) )
           => ( A2 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1589_finite__transitivity__chain,axiom,
    ! [A2: set_complex,R: fun_co1148497455491383662plex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X4: complex] : ~ aa_complex_o(aa_com5808801942499186519plex_o(R,X4),X4)
       => ( ! [X4: complex,Y3: complex,Z2: complex] :
              ( aa_complex_o(aa_com5808801942499186519plex_o(R,X4),Y3)
             => ( aa_complex_o(aa_com5808801942499186519plex_o(R,Y3),Z2)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,X4),Z2) ) )
         => ( ! [X4: complex] :
                ( aa_set_complex_o(member_complex(X4),A2)
               => ? [Y4: complex] :
                    ( aa_set_complex_o(member_complex(Y4),A2)
                    & aa_complex_o(aa_com5808801942499186519plex_o(R,X4),Y4) ) )
           => ( A2 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1590_finite__transitivity__chain,axiom,
    ! [A2: set_Extended_enat,R: fun_Ex239764483819389678enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ! [X4: extended_enat] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(R,X4),X4)
       => ( ! [X4: extended_enat,Y3: extended_enat,Z2: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(R,X4),Y3)
             => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(R,Y3),Z2)
               => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(R,X4),Z2) ) )
         => ( ! [X4: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),A2)
               => ? [Y4: extended_enat] :
                    ( aa_set3337841544853333262enat_o(member_Extended_enat(Y4),A2)
                    & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(R,X4),Y4) ) )
           => ( A2 = bot_bo7653980558646680370d_enat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1591_finite__transitivity__chain,axiom,
    ! [A2: set_real,R: fun_real_fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ! [X4: real] : ~ aa_real_o(aa_real_fun_real_o(R,X4),X4)
       => ( ! [X4: real,Y3: real,Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(R,X4),Y3)
             => ( aa_real_o(aa_real_fun_real_o(R,Y3),Z2)
               => aa_real_o(aa_real_fun_real_o(R,X4),Z2) ) )
         => ( ! [X4: real] :
                ( aa_set_real_o(member_real2(X4),A2)
               => ? [Y4: real] :
                    ( aa_set_real_o(member_real2(Y4),A2)
                    & aa_real_o(aa_real_fun_real_o(R,X4),Y4) ) )
           => ( A2 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1592_finite__transitivity__chain,axiom,
    ! [A2: set_o,R: fun_o_fun_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ! [X4: $o] : ~ aa_o_o(aa_o_fun_o_o(R,(X4)),(X4))
       => ( ! [X4: $o,Y3: $o,Z2: $o] :
              ( aa_o_o(aa_o_fun_o_o(R,(X4)),(Y3))
             => ( aa_o_o(aa_o_fun_o_o(R,(Y3)),(Z2))
               => aa_o_o(aa_o_fun_o_o(R,(X4)),(Z2)) ) )
         => ( ! [X4: $o] :
                ( aa_set_o_o(member_o2((X4)),A2)
               => ? [Y4: $o] :
                    ( aa_set_o_o(member_o2((Y4)),A2)
                    & aa_o_o(aa_o_fun_o_o(R,(X4)),(Y4)) ) )
           => ( A2 = bot_bot_set_o ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1593_finite__transitivity__chain,axiom,
    ! [A2: set_nat,R: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [X4: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(R,X4),X4)
       => ( ! [X4: nat,Y3: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X4),Y3)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y3),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(R,X4),Z2) ) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(member_nat2(X4),A2)
               => ? [Y4: nat] :
                    ( aa_set_nat_o(member_nat2(Y4),A2)
                    & aa_nat_o(aa_nat_fun_nat_o(R,X4),Y4) ) )
           => ( A2 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1594_finite__transitivity__chain,axiom,
    ! [A2: set_int,R: fun_int_fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X4: int] : ~ aa_int_o(aa_int_fun_int_o(R,X4),X4)
       => ( ! [X4: int,Y3: int,Z2: int] :
              ( aa_int_o(aa_int_fun_int_o(R,X4),Y3)
             => ( aa_int_o(aa_int_fun_int_o(R,Y3),Z2)
               => aa_int_o(aa_int_fun_int_o(R,X4),Z2) ) )
         => ( ! [X4: int] :
                ( aa_set_int_o(member_int2(X4),A2)
               => ? [Y4: int] :
                    ( aa_set_int_o(member_int2(Y4),A2)
                    & aa_int_o(aa_int_fun_int_o(R,X4),Y4) ) )
           => ( A2 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1595_infinite__imp__nonempty,axiom,
    ! [S3: set_complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( S3 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
tff(fact_1596_infinite__imp__nonempty,axiom,
    ! [S3: set_Extended_enat] :
      ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
     => ( S3 != bot_bo7653980558646680370d_enat ) ) ).

% infinite_imp_nonempty
tff(fact_1597_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_1598_infinite__imp__nonempty,axiom,
    ! [S3: set_o] :
      ( ~ aa_set_o_o(finite_finite_o,S3)
     => ( S3 != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
tff(fact_1599_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_1600_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_1601_finite_OemptyI,axiom,
    aa_set_complex_o(finite3207457112153483333omplex,bot_bot_set_complex) ).

% finite.emptyI
tff(fact_1602_finite_OemptyI,axiom,
    aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,bot_bo7653980558646680370d_enat) ).

% finite.emptyI
tff(fact_1603_finite_OemptyI,axiom,
    aa_set_real_o(finite_finite_real,bot_bot_set_real) ).

% finite.emptyI
tff(fact_1604_finite_OemptyI,axiom,
    aa_set_o_o(finite_finite_o,bot_bot_set_o) ).

% finite.emptyI
tff(fact_1605_finite_OemptyI,axiom,
    aa_set_nat_o(finite_finite_nat,bot_bot_set_nat) ).

% finite.emptyI
tff(fact_1606_finite_OemptyI,axiom,
    aa_set_int_o(finite_finite_int,bot_bot_set_int) ).

% finite.emptyI
tff(fact_1607_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_1608_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_1609_rev__finite__subset,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B2)
       => aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2) ) ) ).

% rev_finite_subset
tff(fact_1610_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_1611_infinite__super,axiom,
    ! [S3: set_complex,T2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
     => ( ~ aa_set_complex_o(finite3207457112153483333omplex,S3)
       => ~ aa_set_complex_o(finite3207457112153483333omplex,T2) ) ) ).

% infinite_super
tff(fact_1612_infinite__super,axiom,
    ! [S3: set_int,T2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
     => ( ~ aa_set_int_o(finite_finite_int,S3)
       => ~ aa_set_int_o(finite_finite_int,T2) ) ) ).

% infinite_super
tff(fact_1613_infinite__super,axiom,
    ! [S3: set_Extended_enat,T2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
     => ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
       => ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2) ) ) ).

% infinite_super
tff(fact_1614_infinite__super,axiom,
    ! [S3: set_nat,T2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),T2)
     => ( ~ aa_set_nat_o(finite_finite_nat,S3)
       => ~ aa_set_nat_o(finite_finite_nat,T2) ) ) ).

% infinite_super
tff(fact_1615_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_1616_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_1617_finite__subset,axiom,
    ! [A2: set_Extended_enat,B2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B2)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
       => aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2) ) ) ).

% finite_subset
tff(fact_1618_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_1619_finite__nat__set__iff__bounded,axiom,
    ! [N4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N4)
    <=> ? [M5: nat] :
        ! [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),M5) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_1620_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_nat2(N3),S3) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_1621_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ( S3 != bot_bot_set_complex )
       => ~ ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),S3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,X)),aa_complex_nat(F,lattic5364784637807008409ex_nat(F,S3))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1622_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ~ ? [X: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X),S3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Extended_enat_nat(F,X)),aa_Extended_enat_nat(F,lattic3845382081240766429at_nat(F,S3))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1623_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( ( S3 != bot_bot_set_real )
       => ~ ? [X: real] :
              ( aa_set_real_o(member_real2(X),S3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X)),aa_real_nat(F,lattic5055836439445974935al_nat(F,S3))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1624_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_o,F: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S3)
     => ( ( S3 != bot_bot_set_o )
       => ~ ? [X: $o] :
              ( aa_set_o_o(member_o2((X)),S3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(F,(X))),aa_o_nat(F,aa_set_o_o(lattic2775856028456453135_o_nat(F),S3))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1625_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( ( S3 != bot_bot_set_nat )
       => ~ ? [X: nat] :
              ( aa_set_nat_o(member_nat2(X),S3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X)),aa_nat_nat(F,aa_set_nat_nat(lattic7446932960582359483at_nat(F),S3))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1626_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ( S3 != bot_bot_set_int )
       => ~ ? [X: int] :
              ( aa_set_int_o(member_int2(X),S3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X)),aa_int_nat(F,lattic8446286672483414039nt_nat(F,S3))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1627_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_complex,F: fun_co6496365337113880337d_enat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ( S3 != bot_bot_set_complex )
       => ~ ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),S3)
              & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_com3683005661447327866d_enat(F,X)),aa_com3683005661447327866d_enat(F,lattic7796887085614042845d_enat(F,S3))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1628_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_Extended_enat,F: fun_Ex5852502975177633731d_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ~ ? [X: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X),S3)
              & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,X)),aa_Ext7969882492699152700d_enat(F,lattic1996716550891908761d_enat(F,S3))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1629_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_real,F: fun_re6065469302365115155d_enat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( ( S3 != bot_bot_set_real )
       => ~ ? [X: real] :
              ( aa_set_real_o(member_real2(X),S3)
              & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X)),aa_rea1194391308913012860d_enat(F,lattic9066027731366277983d_enat(F,S3))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1630_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_o,F: fun_o_Extended_enat] :
      ( aa_set_o_o(finite_finite_o,S3)
     => ( ( S3 != bot_bot_set_o )
       => ~ ? [X: $o] :
              ( aa_set_o_o(member_o2((X)),S3)
              & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_o_Extended_enat(F,(X))),aa_o_Extended_enat(F,aa_set_o_o(lattic7542187609180611815d_enat(F),S3))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1631_arg__min__least,axiom,
    ! [S3: set_complex,Y: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ( S3 != bot_bot_set_complex )
       => ( aa_set_complex_o(member_complex(Y),S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,lattic8794016678065449205x_real(F,S3))),aa_complex_real(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1632_arg__min__least,axiom,
    ! [S3: set_Extended_enat,Y: extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ( aa_set3337841544853333262enat_o(member_Extended_enat(Y),S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,lattic1189837152898106425t_real(F,S3))),aa_Ext2541572767299617110t_real(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1633_arg__min__least,axiom,
    ! [S3: set_real,Y: real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( ( S3 != bot_bot_set_real )
       => ( aa_set_real_o(member_real2(Y),S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,lattic8440615504127631091l_real(F,S3))),aa_real_real(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1634_arg__min__least,axiom,
    ! [S3: set_o,Y: $o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,S3)
     => ( ( S3 != bot_bot_set_o )
       => ( aa_set_o_o(member_o2((Y)),S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(F,aa_set_o_o(lattic8697145971487455083o_real(F),S3))),aa_o_real(F,(Y))) ) ) ) ).

% arg_min_least
tff(fact_1635_arg__min__least,axiom,
    ! [S3: set_nat,Y: nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( ( S3 != bot_bot_set_nat )
       => ( aa_set_nat_o(member_nat2(Y),S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,aa_set_nat_nat(lattic488527866317076247t_real(F),S3))),aa_nat_real(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1636_arg__min__least,axiom,
    ! [S3: set_int,Y: int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ( S3 != bot_bot_set_int )
       => ( aa_set_int_o(member_int2(Y),S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,lattic2675449441010098035t_real(F,S3))),aa_int_real(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1637_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_1638_arg__min__least,axiom,
    ! [S3: set_Extended_enat,Y: extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ( aa_set3337841544853333262enat_o(member_Extended_enat(Y),S3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Extended_enat_nat(F,lattic3845382081240766429at_nat(F,S3))),aa_Extended_enat_nat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1639_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_real2(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_1640_arg__min__least,axiom,
    ! [S3: set_o,Y: $o,F: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S3)
     => ( ( S3 != bot_bot_set_o )
       => ( aa_set_o_o(member_o2((Y)),S3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,aa_set_o_o(lattic2775856028456453135_o_nat(F),S3))),aa_o_nat(F,(Y))) ) ) ) ).

% arg_min_least
tff(fact_1641_list__decode_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero_nat )
     => ~ ! [N: nat] : Xb != aa_nat_nat(suc,N) ) ).

% list_decode.cases
tff(fact_1642_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_1643_verit__sum__simplify,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),zero_zero_real) = A ).

% verit_sum_simplify
tff(fact_1644_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_1645_verit__sum__simplify,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% verit_sum_simplify
tff(fact_1646_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_1647_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B = aa_real_real(plus_plus_real(B),A) )
    <=> ( A = zero_zero_real ) ) ).

% add_0_iff
tff(fact_1648_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_1649_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_1650_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_1651_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_1652_less__numeral__extra_I3_J,axiom,
    ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),zero_z5237406670263579293d_enat) ).

% less_numeral_extra(3)
tff(fact_1653_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_1654_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_1655_complete__interval,axiom,
    ! [A: extended_enat,B: extended_enat,P: fun_Extended_enat_o] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( aa_Extended_enat_o(P,A)
       => ( ~ aa_Extended_enat_o(P,B)
         => ? [C3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),C3)
              & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C3),B)
              & ! [X: extended_enat] :
                  ( ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),X)
                    & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),C3) )
                 => aa_Extended_enat_o(P,X) )
              & ! [D3: extended_enat] :
                  ( ! [X4: extended_enat] :
                      ( ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),X4)
                        & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),D3) )
                     => aa_Extended_enat_o(P,X4) )
                 => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,D3),C3) ) ) ) ) ) ).

% complete_interval
tff(fact_1656_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)
         => ? [C3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C3)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C3),B)
              & ! [X: 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_real,X),C3) )
                 => aa_real_o(P,X) )
              & ! [D3: real] :
                  ( ! [X4: real] :
                      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
                        & aa_real_o(aa_real_fun_real_o(ord_less_real,X4),D3) )
                     => aa_real_o(P,X4) )
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D3),C3) ) ) ) ) ) ).

% complete_interval
tff(fact_1657_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)
         => ? [C3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C3),B)
              & ! [X: nat] :
                  ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),C3) )
                 => aa_nat_o(P,X) )
              & ! [D3: nat] :
                  ( ! [X4: nat] :
                      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X4)
                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),D3) )
                     => aa_nat_o(P,X4) )
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D3),C3) ) ) ) ) ) ).

% complete_interval
tff(fact_1658_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)
         => ? [C3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C3)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C3),B)
              & ! [X: int] :
                  ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
                    & aa_int_o(aa_int_fun_int_o(ord_less_int,X),C3) )
                 => aa_int_o(P,X) )
              & ! [D3: int] :
                  ( ! [X4: int] :
                      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X4)
                        & aa_int_o(aa_int_fun_int_o(ord_less_int,X4),D3) )
                     => aa_int_o(P,X4) )
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D3),C3) ) ) ) ) ) ).

% complete_interval
tff(fact_1659_verit__comp__simplify1_I3_J,axiom,
    ! [B4: extended_enat,A4: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B4),A4)
    <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_1660_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_1661_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_1662_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_1663_pinf_I6_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z2),X)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,X),Ta) ) ).

% pinf(6)
tff(fact_1664_pinf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ta) ) ).

% pinf(6)
tff(fact_1665_pinf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ta) ) ).

% pinf(6)
tff(fact_1666_pinf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta) ) ).

% pinf(6)
tff(fact_1667_verit__comp__simplify1_I2_J,axiom,
    ! [A: filter_nat] : ord_le2510731241096832064er_nat(A,A) ).

% verit_comp_simplify1(2)
tff(fact_1668_verit__comp__simplify1_I2_J,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1669_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1670_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_1671_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_1672_verit__la__disequality,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_eq_real,B),A) ) ).

% verit_la_disequality
tff(fact_1673_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_1674_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_1675_minf_I7_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X) ) ).

% minf(7)
tff(fact_1676_minf_I7_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Z2)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Ta),X) ) ).

% minf(7)
tff(fact_1677_minf_I7_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),X) ) ).

% minf(7)
tff(fact_1678_minf_I7_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X) ) ).

% minf(7)
tff(fact_1679_minf_I5_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Ta) ) ).

% minf(5)
tff(fact_1680_minf_I5_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Z2)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Ta) ) ).

% minf(5)
tff(fact_1681_minf_I5_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ta) ) ).

% minf(5)
tff(fact_1682_minf_I5_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Ta) ) ).

% minf(5)
tff(fact_1683_minf_I4_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_1684_minf_I4_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Z2)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_1685_minf_I4_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_1686_minf_I4_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_1687_minf_I3_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_1688_minf_I3_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Z2)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_1689_minf_I3_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_1690_minf_I3_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_1691_minf_I2_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q4: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z3)
         => ( aa_nat_o(P,X4)
          <=> aa_nat_o(P3,X4) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z3)
           => ( aa_nat_o(Q,X4)
            <=> aa_nat_o(Q4,X4) ) )
       => ? [Z2: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
           => ( ( aa_nat_o(P,X)
                | aa_nat_o(Q,X) )
            <=> ( aa_nat_o(P3,X)
                | aa_nat_o(Q4,X) ) ) ) ) ) ).

% minf(2)
tff(fact_1692_minf_I2_J,axiom,
    ! [P: fun_Extended_enat_o,P3: fun_Extended_enat_o,Q: fun_Extended_enat_o,Q4: fun_Extended_enat_o] :
      ( ? [Z3: extended_enat] :
        ! [X4: extended_enat] :
          ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Z3)
         => ( aa_Extended_enat_o(P,X4)
          <=> aa_Extended_enat_o(P3,X4) ) )
     => ( ? [Z3: extended_enat] :
          ! [X4: extended_enat] :
            ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Z3)
           => ( aa_Extended_enat_o(Q,X4)
            <=> aa_Extended_enat_o(Q4,X4) ) )
       => ? [Z2: extended_enat] :
          ! [X: extended_enat] :
            ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Z2)
           => ( ( aa_Extended_enat_o(P,X)
                | aa_Extended_enat_o(Q,X) )
            <=> ( aa_Extended_enat_o(P3,X)
                | aa_Extended_enat_o(Q4,X) ) ) ) ) ) ).

% minf(2)
tff(fact_1693_minf_I2_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q4: fun_real_o] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z3)
         => ( aa_real_o(P,X4)
          <=> aa_real_o(P3,X4) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z3)
           => ( aa_real_o(Q,X4)
            <=> aa_real_o(Q4,X4) ) )
       => ? [Z2: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
           => ( ( aa_real_o(P,X)
                | aa_real_o(Q,X) )
            <=> ( aa_real_o(P3,X)
                | aa_real_o(Q4,X) ) ) ) ) ) ).

% minf(2)
tff(fact_1694_minf_I2_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q4: fun_int_o] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z3)
         => ( aa_int_o(P,X4)
          <=> aa_int_o(P3,X4) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z3)
           => ( aa_int_o(Q,X4)
            <=> aa_int_o(Q4,X4) ) )
       => ? [Z2: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
           => ( ( aa_int_o(P,X)
                | aa_int_o(Q,X) )
            <=> ( aa_int_o(P3,X)
                | aa_int_o(Q4,X) ) ) ) ) ) ).

% minf(2)
tff(fact_1695_minf_I1_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q4: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z3)
         => ( aa_nat_o(P,X4)
          <=> aa_nat_o(P3,X4) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z3)
           => ( aa_nat_o(Q,X4)
            <=> aa_nat_o(Q4,X4) ) )
       => ? [Z2: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
           => ( ( aa_nat_o(P,X)
                & aa_nat_o(Q,X) )
            <=> ( aa_nat_o(P3,X)
                & aa_nat_o(Q4,X) ) ) ) ) ) ).

% minf(1)
tff(fact_1696_minf_I1_J,axiom,
    ! [P: fun_Extended_enat_o,P3: fun_Extended_enat_o,Q: fun_Extended_enat_o,Q4: fun_Extended_enat_o] :
      ( ? [Z3: extended_enat] :
        ! [X4: extended_enat] :
          ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Z3)
         => ( aa_Extended_enat_o(P,X4)
          <=> aa_Extended_enat_o(P3,X4) ) )
     => ( ? [Z3: extended_enat] :
          ! [X4: extended_enat] :
            ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X4),Z3)
           => ( aa_Extended_enat_o(Q,X4)
            <=> aa_Extended_enat_o(Q4,X4) ) )
       => ? [Z2: extended_enat] :
          ! [X: extended_enat] :
            ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Z2)
           => ( ( aa_Extended_enat_o(P,X)
                & aa_Extended_enat_o(Q,X) )
            <=> ( aa_Extended_enat_o(P3,X)
                & aa_Extended_enat_o(Q4,X) ) ) ) ) ) ).

% minf(1)
tff(fact_1697_minf_I1_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q4: fun_real_o] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z3)
         => ( aa_real_o(P,X4)
          <=> aa_real_o(P3,X4) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z3)
           => ( aa_real_o(Q,X4)
            <=> aa_real_o(Q4,X4) ) )
       => ? [Z2: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
           => ( ( aa_real_o(P,X)
                & aa_real_o(Q,X) )
            <=> ( aa_real_o(P3,X)
                & aa_real_o(Q4,X) ) ) ) ) ) ).

% minf(1)
tff(fact_1698_minf_I1_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q4: fun_int_o] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z3)
         => ( aa_int_o(P,X4)
          <=> aa_int_o(P3,X4) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z3)
           => ( aa_int_o(Q,X4)
            <=> aa_int_o(Q4,X4) ) )
       => ? [Z2: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
           => ( ( aa_int_o(P,X)
                & aa_int_o(Q,X) )
            <=> ( aa_int_o(P3,X)
                & aa_int_o(Q4,X) ) ) ) ) ) ).

% minf(1)
tff(fact_1699_pinf_I7_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X) ) ).

% pinf(7)
tff(fact_1700_pinf_I7_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z2),X)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Ta),X) ) ).

% pinf(7)
tff(fact_1701_pinf_I7_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),X) ) ).

% pinf(7)
tff(fact_1702_pinf_I7_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X) ) ).

% pinf(7)
tff(fact_1703_pinf_I5_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Ta) ) ).

% pinf(5)
tff(fact_1704_pinf_I5_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z2),X)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Ta) ) ).

% pinf(5)
tff(fact_1705_pinf_I5_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ta) ) ).

% pinf(5)
tff(fact_1706_pinf_I5_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Ta) ) ).

% pinf(5)
tff(fact_1707_pinf_I4_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_1708_pinf_I4_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z2),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_1709_pinf_I4_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_1710_pinf_I4_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_1711_pinf_I3_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_1712_pinf_I3_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z2),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_1713_pinf_I3_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_1714_pinf_I3_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_1715_pinf_I2_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q4: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X4)
         => ( aa_nat_o(P,X4)
          <=> aa_nat_o(P3,X4) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X4)
           => ( aa_nat_o(Q,X4)
            <=> aa_nat_o(Q4,X4) ) )
       => ? [Z2: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
           => ( ( aa_nat_o(P,X)
                | aa_nat_o(Q,X) )
            <=> ( aa_nat_o(P3,X)
                | aa_nat_o(Q4,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_1716_pinf_I2_J,axiom,
    ! [P: fun_Extended_enat_o,P3: fun_Extended_enat_o,Q: fun_Extended_enat_o,Q4: fun_Extended_enat_o] :
      ( ? [Z3: extended_enat] :
        ! [X4: extended_enat] :
          ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z3),X4)
         => ( aa_Extended_enat_o(P,X4)
          <=> aa_Extended_enat_o(P3,X4) ) )
     => ( ? [Z3: extended_enat] :
          ! [X4: extended_enat] :
            ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z3),X4)
           => ( aa_Extended_enat_o(Q,X4)
            <=> aa_Extended_enat_o(Q4,X4) ) )
       => ? [Z2: extended_enat] :
          ! [X: extended_enat] :
            ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z2),X)
           => ( ( aa_Extended_enat_o(P,X)
                | aa_Extended_enat_o(Q,X) )
            <=> ( aa_Extended_enat_o(P3,X)
                | aa_Extended_enat_o(Q4,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_1717_pinf_I2_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q4: fun_real_o] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X4)
         => ( aa_real_o(P,X4)
          <=> aa_real_o(P3,X4) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X4)
           => ( aa_real_o(Q,X4)
            <=> aa_real_o(Q4,X4) ) )
       => ? [Z2: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
           => ( ( aa_real_o(P,X)
                | aa_real_o(Q,X) )
            <=> ( aa_real_o(P3,X)
                | aa_real_o(Q4,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_1718_pinf_I2_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q4: fun_int_o] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X4)
         => ( aa_int_o(P,X4)
          <=> aa_int_o(P3,X4) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X4)
           => ( aa_int_o(Q,X4)
            <=> aa_int_o(Q4,X4) ) )
       => ? [Z2: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
           => ( ( aa_int_o(P,X)
                | aa_int_o(Q,X) )
            <=> ( aa_int_o(P3,X)
                | aa_int_o(Q4,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_1719_pinf_I1_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q4: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X4)
         => ( aa_nat_o(P,X4)
          <=> aa_nat_o(P3,X4) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X4)
           => ( aa_nat_o(Q,X4)
            <=> aa_nat_o(Q4,X4) ) )
       => ? [Z2: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
           => ( ( aa_nat_o(P,X)
                & aa_nat_o(Q,X) )
            <=> ( aa_nat_o(P3,X)
                & aa_nat_o(Q4,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_1720_pinf_I1_J,axiom,
    ! [P: fun_Extended_enat_o,P3: fun_Extended_enat_o,Q: fun_Extended_enat_o,Q4: fun_Extended_enat_o] :
      ( ? [Z3: extended_enat] :
        ! [X4: extended_enat] :
          ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z3),X4)
         => ( aa_Extended_enat_o(P,X4)
          <=> aa_Extended_enat_o(P3,X4) ) )
     => ( ? [Z3: extended_enat] :
          ! [X4: extended_enat] :
            ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z3),X4)
           => ( aa_Extended_enat_o(Q,X4)
            <=> aa_Extended_enat_o(Q4,X4) ) )
       => ? [Z2: extended_enat] :
          ! [X: extended_enat] :
            ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z2),X)
           => ( ( aa_Extended_enat_o(P,X)
                & aa_Extended_enat_o(Q,X) )
            <=> ( aa_Extended_enat_o(P3,X)
                & aa_Extended_enat_o(Q4,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_1721_pinf_I1_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q4: fun_real_o] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X4)
         => ( aa_real_o(P,X4)
          <=> aa_real_o(P3,X4) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X4)
           => ( aa_real_o(Q,X4)
            <=> aa_real_o(Q4,X4) ) )
       => ? [Z2: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
           => ( ( aa_real_o(P,X)
                & aa_real_o(Q,X) )
            <=> ( aa_real_o(P3,X)
                & aa_real_o(Q4,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_1722_pinf_I1_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q4: fun_int_o] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X4)
         => ( aa_int_o(P,X4)
          <=> aa_int_o(P3,X4) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X4)
           => ( aa_int_o(Q,X4)
            <=> aa_int_o(Q4,X4) ) )
       => ? [Z2: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
           => ( ( aa_int_o(P,X)
                & aa_int_o(Q,X) )
            <=> ( aa_int_o(P3,X)
                & aa_int_o(Q4,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_1723_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_1724_verit__comp__simplify1_I1_J,axiom,
    ! [A: extended_enat] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1725_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_1726_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_1727_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_1728_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_1729_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% is_num_normalize(1)
tff(fact_1730_prod__decode__aux_Ocases,axiom,
    ! [Xb: product_prod_nat_nat] :
      ~ ! [K3: nat,M: nat] : Xb != aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,K3),M) ).

% prod_decode_aux.cases
tff(fact_1731_le__numeral__extra_I3_J,axiom,
    aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),zero_z5237406670263579293d_enat) ).

% le_numeral_extra(3)
tff(fact_1732_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_1733_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_1734_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_1735_minf_I8_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Z2)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Ta),X) ) ).

% minf(8)
tff(fact_1736_minf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X) ) ).

% minf(8)
tff(fact_1737_minf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X) ) ).

% minf(8)
tff(fact_1738_minf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X) ) ).

% minf(8)
tff(fact_1739_minf_I6_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Z2)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,X),Ta) ) ).

% minf(6)
tff(fact_1740_minf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ta) ) ).

% minf(6)
tff(fact_1741_minf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ta) ) ).

% minf(6)
tff(fact_1742_minf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta) ) ).

% minf(6)
tff(fact_1743_pinf_I8_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z2),X)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Ta),X) ) ).

% pinf(8)
tff(fact_1744_pinf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X) ) ).

% pinf(8)
tff(fact_1745_pinf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X) ) ).

% pinf(8)
tff(fact_1746_pinf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X) ) ).

% pinf(8)
tff(fact_1747_triangle__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,Nb)),aa_nat_nat(suc,Nb)) ).

% triangle_Suc
tff(fact_1748_find__Some__iff2,axiom,
    ! [Xb: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o,Xs: list_P6011104703257516679at_nat] :
      ( ( some_P7363390416028606310at_nat(Xb) = find_P8199882355184865565at_nat(P,Xs) )
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s5460976970255530739at_nat(Xs))
          & aa_Pro6150637604445964947_nat_o(P,nth_Pr7617993195940197384at_nat(Xs,I4))
          & ( Xb = nth_Pr7617993195940197384at_nat(Xs,I4) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I4)
             => ~ aa_Pro6150637604445964947_nat_o(P,nth_Pr7617993195940197384at_nat(Xs,J3)) ) ) ) ).

% find_Some_iff2
tff(fact_1749_find__Some__iff2,axiom,
    ! [Xb: num,P: fun_num_o,Xs: list_num] :
      ( ( some_num(Xb) = find_num(P,Xs) )
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_num(Xs))
          & aa_num_o(P,nth_num(Xs,I4))
          & ( Xb = nth_num(Xs,I4) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I4)
             => ~ aa_num_o(P,nth_num(Xs,J3)) ) ) ) ).

% find_Some_iff2
tff(fact_1750_find__Some__iff2,axiom,
    ! [Xb: vEBT_VEBT,P: fun_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ( some_VEBT_VEBT(Xb) = find_VEBT_VEBT(P,Xs) )
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs))
          & aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I4))
          & ( Xb = nth_VEBT_VEBT(Xs,I4) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I4)
             => ~ aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,J3)) ) ) ) ).

% find_Some_iff2
tff(fact_1751_find__Some__iff2,axiom,
    ! [Xb: int,P: fun_int_o,Xs: list_int] :
      ( ( some_int(Xb) = find_int(P,Xs) )
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_int(Xs))
          & aa_int_o(P,aa_nat_int(nth_int(Xs),I4))
          & ( Xb = aa_nat_int(nth_int(Xs),I4) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I4)
             => ~ aa_int_o(P,aa_nat_int(nth_int(Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_1752_find__Some__iff2,axiom,
    ! [Xb: nat,P: fun_nat_o,Xs: list_nat] :
      ( ( some_nat(Xb) = find_nat(P,Xs) )
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_nat(Xs))
          & aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I4))
          & ( Xb = aa_nat_nat(nth_nat(Xs),I4) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I4)
             => ~ aa_nat_o(P,aa_nat_nat(nth_nat(Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_1753_find__Some__iff,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Xs: list_P6011104703257516679at_nat,Xb: product_prod_nat_nat] :
      ( ( find_P8199882355184865565at_nat(P,Xs) = some_P7363390416028606310at_nat(Xb) )
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s5460976970255530739at_nat(Xs))
          & aa_Pro6150637604445964947_nat_o(P,nth_Pr7617993195940197384at_nat(Xs,I4))
          & ( Xb = nth_Pr7617993195940197384at_nat(Xs,I4) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I4)
             => ~ aa_Pro6150637604445964947_nat_o(P,nth_Pr7617993195940197384at_nat(Xs,J3)) ) ) ) ).

% find_Some_iff
tff(fact_1754_find__Some__iff,axiom,
    ! [P: fun_num_o,Xs: list_num,Xb: num] :
      ( ( find_num(P,Xs) = some_num(Xb) )
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_num(Xs))
          & aa_num_o(P,nth_num(Xs,I4))
          & ( Xb = nth_num(Xs,I4) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I4)
             => ~ aa_num_o(P,nth_num(Xs,J3)) ) ) ) ).

% find_Some_iff
tff(fact_1755_find__Some__iff,axiom,
    ! [P: fun_VEBT_VEBT_o,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( ( find_VEBT_VEBT(P,Xs) = some_VEBT_VEBT(Xb) )
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs))
          & aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I4))
          & ( Xb = nth_VEBT_VEBT(Xs,I4) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I4)
             => ~ aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,J3)) ) ) ) ).

% find_Some_iff
tff(fact_1756_find__Some__iff,axiom,
    ! [P: fun_int_o,Xs: list_int,Xb: int] :
      ( ( find_int(P,Xs) = some_int(Xb) )
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_int(Xs))
          & aa_int_o(P,aa_nat_int(nth_int(Xs),I4))
          & ( Xb = aa_nat_int(nth_int(Xs),I4) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I4)
             => ~ aa_int_o(P,aa_nat_int(nth_int(Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_1757_find__Some__iff,axiom,
    ! [P: fun_nat_o,Xs: list_nat,Xb: nat] :
      ( ( find_nat(P,Xs) = some_nat(Xb) )
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_nat(Xs))
          & aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I4))
          & ( Xb = aa_nat_nat(nth_nat(Xs),I4) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I4)
             => ~ aa_nat_o(P,aa_nat_nat(nth_nat(Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_1758_nth__zip,axiom,
    ! [Ia: nat,Xs: list_VEBT_VEBT,Ys: list_Extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_s3941691890525107288d_enat(Ys))
       => ( nth_Pr7509392720524132704d_enat(zip_VE7205001627739651817d_enat(Xs,Ys),Ia) = aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,nth_VEBT_VEBT(Xs,Ia)),aa_nat_Extended_enat(nth_Extended_enat(Ys),Ia)) ) ) ) ).

% nth_zip
tff(fact_1759_nth__zip,axiom,
    ! [Ia: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_s6755466524823107622T_VEBT(Ys))
       => ( nth_Pr4953567300277697838T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys),Ia) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,Ia),nth_VEBT_VEBT(Ys,Ia)) ) ) ) ).

% nth_zip
tff(fact_1760_nth__zip,axiom,
    ! [Ia: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_int(Ys))
       => ( nth_Pr6837108013167703752BT_int(zip_VEBT_VEBT_int(Xs,Ys),Ia) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,Ia),aa_nat_int(nth_int(Ys),Ia)) ) ) ) ).

% nth_zip
tff(fact_1761_nth__zip,axiom,
    ! [Ia: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_nat(Ys))
       => ( nth_Pr1791586995822124652BT_nat(zip_VEBT_VEBT_nat(Xs,Ys),Ia) = aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,nth_VEBT_VEBT(Xs,Ia)),aa_nat_nat(nth_nat(Ys),Ia)) ) ) ) ).

% nth_zip
tff(fact_1762_nth__zip,axiom,
    ! [Ia: nat,Xs: list_int,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_s6755466524823107622T_VEBT(Ys))
       => ( nth_Pr3474266648193625910T_VEBT(zip_int_VEBT_VEBT(Xs,Ys),Ia) = produc3329399203697025711T_VEBT(aa_nat_int(nth_int(Xs),Ia),nth_VEBT_VEBT(Ys,Ia)) ) ) ) ).

% nth_zip
tff(fact_1763_nth__zip,axiom,
    ! [Ia: nat,Xs: list_int,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_int(Ys))
       => ( nth_Pr4439495888332055232nt_int(zip_int_int(Xs,Ys),Ia) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(nth_int(Xs),Ia)),aa_nat_int(nth_int(Ys),Ia)) ) ) ) ).

% nth_zip
tff(fact_1764_nth__zip,axiom,
    ! [Ia: nat,Xs: list_int,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_nat(Ys))
       => ( nth_Pr8617346907841251940nt_nat(zip_int_nat(Xs,Ys),Ia) = product_Pair_int_nat(aa_nat_int(nth_int(Xs),Ia),aa_nat_nat(nth_nat(Ys),Ia)) ) ) ) ).

% nth_zip
tff(fact_1765_nth__zip,axiom,
    ! [Ia: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_nat(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_s6755466524823107622T_VEBT(Ys))
       => ( nth_Pr744662078594809490T_VEBT(zip_nat_VEBT_VEBT(Xs,Ys),Ia) = produc599794634098209291T_VEBT(aa_nat_nat(nth_nat(Xs),Ia),nth_VEBT_VEBT(Ys,Ia)) ) ) ) ).

% nth_zip
tff(fact_1766_nth__zip,axiom,
    ! [Ia: nat,Xs: list_nat,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_nat(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_int(Ys))
       => ( nth_Pr3440142176431000676at_int(zip_nat_int(Xs,Ys),Ia) = product_Pair_nat_int(aa_nat_nat(nth_nat(Xs),Ia),aa_nat_int(nth_int(Ys),Ia)) ) ) ) ).

% nth_zip
tff(fact_1767_nth__zip,axiom,
    ! [Ia: nat,Xs: list_nat,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_nat(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_nat(Ys))
       => ( nth_Pr7617993195940197384at_nat(zip_nat_nat(Xs,Ys),Ia) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(nth_nat(Xs),Ia)),aa_nat_nat(nth_nat(Ys),Ia)) ) ) ) ).

% nth_zip
tff(fact_1768__C7_C,axiom,
    ( ( mi != ma )
   => ! [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,bit0(one))),m))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),na) = I3 )
           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),na)) )
          & ! [Y4: nat] :
              ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Y4),na) = I3 )
                & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Y4),na)) )
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,mi),Y4)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),ma) ) ) ) ) ) ).

% "7"
tff(fact_1769_count__notin,axiom,
    ! [Xb: real,Xs: list_real] :
      ( ~ aa_set_real_o(member_real2(Xb),set_real2(Xs))
     => ( aa_real_nat(count_list_real(Xs),Xb) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1770_count__notin,axiom,
    ! [Xb: $o,Xs: list_o] :
      ( ~ aa_set_o_o(member_o2((Xb)),set_o2(Xs))
     => ( aa_o_nat(count_list_o(Xs),(Xb)) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1771_count__notin,axiom,
    ! [Xb: set_nat,Xs: list_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat2(Xb),set_set_nat2(Xs))
     => ( aa_set_nat_nat(count_list_set_nat(Xs),Xb) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1772_count__notin,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ~ member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(Xs))
     => ( aa_VEBT_VEBT_nat(count_list_VEBT_VEBT(Xs),Xb) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1773_count__notin,axiom,
    ! [Xb: int,Xs: list_int] :
      ( ~ aa_set_int_o(member_int2(Xb),set_int2(Xs))
     => ( aa_int_nat(count_list_int(Xs),Xb) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1774_count__notin,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( ~ aa_set_nat_o(member_nat2(Xb),set_nat2(Xs))
     => ( aa_nat_nat(count_list_nat(Xs),Xb) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1775_listrel1__iff__update,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( member6693912407220327184at_nat(produc5943733680697469783at_nat(Xs,Ys),listre4828114922151135584at_nat(R2))
    <=> ? [Y2: product_prod_nat_nat,N3: nat] :
          ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,nth_Pr7617993195940197384at_nat(Xs,N3)),Y2),R2)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_s5460976970255530739at_nat(Xs))
          & ( Ys = list_u6180841689913720943at_nat(Xs,N3,Y2) ) ) ) ).

% listrel1_iff_update
tff(fact_1776_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))
    <=> ? [Y2: vEBT_VEBT,N3: nat] :
          ( member568628332442017744T_VEBT(produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,N3),Y2),R2)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_s6755466524823107622T_VEBT(Xs))
          & ( Ys = list_u1324408373059187874T_VEBT(Xs,N3,Y2) ) ) ) ).

% listrel1_iff_update
tff(fact_1777_listrel1__iff__update,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),listrel1_int(R2))
    <=> ? [Y2: int,N3: nat] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(nth_int(Xs),N3)),Y2),R2)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_size_list_int(Xs))
          & ( Ys = list_update_int(Xs,N3,Y2) ) ) ) ).

% listrel1_iff_update
tff(fact_1778_listrel1__iff__update,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),listrel1_nat(R2))
    <=> ? [Y2: nat,N3: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(nth_nat(Xs),N3)),Y2),R2)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_size_list_nat(Xs))
          & ( Ys = list_update_nat(Xs,N3,Y2) ) ) ) ).

% listrel1_iff_update
tff(fact_1779_pair__lessI2,axiom,
    ! [A: nat,B: nat,S: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S),Ta)
       => member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),S)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B),Ta)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_1780_intind,axiom,
    ! [Ia: nat,Nb: nat,P: fun_nat_o,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),Nb)
     => ( aa_nat_o(P,Xb)
       => aa_nat_o(P,aa_nat_nat(nth_nat(replicate_nat(Nb,Xb)),Ia)) ) ) ).

% intind
tff(fact_1781_intind,axiom,
    ! [Ia: nat,Nb: nat,P: fun_int_o,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),Nb)
     => ( aa_int_o(P,Xb)
       => aa_int_o(P,aa_nat_int(nth_int(replicate_int(Nb,Xb)),Ia)) ) ) ).

% intind
tff(fact_1782_intind,axiom,
    ! [Ia: nat,Nb: nat,P: fun_VEBT_VEBT_o,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),Nb)
     => ( aa_VEBT_VEBT_o(P,Xb)
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,Xb),Ia)) ) ) ).

% intind
tff(fact_1783_pair__less__iff1,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Y)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Z)),fun_pair_less)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z) ) ).

% pair_less_iff1
tff(fact_1784__C2_C,axiom,
    size_s6755466524823107622T_VEBT(treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),m) ).

% "2"
tff(fact_1785__C5_Oprems_C,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),deg)) ).

% "5.prems"
tff(fact_1786__C5_Ohyps_C_I8_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,bit0(one))),deg)) ).

% "5.hyps"(8)
tff(fact_1787__092_060open_062i_A_060_A2_A_094_Am_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,i),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),m)) ).

% \<open>i < 2 ^ m\<close>
tff(fact_1788__C5_OIH_C_I1_J,axiom,
    ! [X: vEBT_VEBT] :
      ( member_VEBT_VEBT2(X,set_VEBT_VEBT2(treeList))
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),na)
        & ! [Xa: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),na))
           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_vebt_insert(X,Xa)),na) ) ) ) ).

% "5.IH"(1)
tff(fact_1789__C4_C,axiom,
    ! [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,bit0(one))),m))
     => ( ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I3)),X_13)
      <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,summary),I3) ) ) ).

% "4"
tff(fact_1790_high__bound__aux,axiom,
    ! [Maa: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)) ) ).

% high_bound_aux
tff(fact_1791_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tree),Xb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tree),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) ) ) ).

% member_bound
tff(fact_1792_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(Mb)),numera1916890842035813515d_enat(Nb))
    <=> aa_num_o(ord_less_eq_num(Mb),Nb) ) ).

% numeral_le_iff
tff(fact_1793_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Mb)),numeral_numeral_real(Nb))
    <=> aa_num_o(ord_less_eq_num(Mb),Nb) ) ).

% numeral_le_iff
tff(fact_1794_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(ord_less_eq_num(Mb),Nb) ) ).

% numeral_le_iff
tff(fact_1795_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(Mb)),numeral_numeral_int(Nb))
    <=> aa_num_o(ord_less_eq_num(Mb),Nb) ) ).

% numeral_le_iff
tff(fact_1796_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(ord_less_num(Mb),Nb) ) ).

% numeral_less_iff
tff(fact_1797_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,numera1916890842035813515d_enat(Mb)),numera1916890842035813515d_enat(Nb))
    <=> aa_num_o(ord_less_num(Mb),Nb) ) ).

% numeral_less_iff
tff(fact_1798_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,numeral_numeral_int(Mb)),numeral_numeral_int(Nb))
    <=> aa_num_o(ord_less_num(Mb),Nb) ) ).

% numeral_less_iff
tff(fact_1799_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Mb)),numeral_numeral_real(Nb))
    <=> aa_num_o(ord_less_num(Mb),Nb) ) ).

% numeral_less_iff
tff(fact_1800_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,plus_plus_num(V,W2))),Z) ).

% add_numeral_left
tff(fact_1801_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(V)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(W2)),Z)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(plus_plus_num(V,W2))),Z) ).

% add_numeral_left
tff(fact_1802_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(W2)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(plus_plus_num(V,W2))),Z) ).

% add_numeral_left
tff(fact_1803_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: real] : aa_real_real(plus_plus_real(numeral_numeral_real(V)),aa_real_real(plus_plus_real(numeral_numeral_real(W2)),Z)) = aa_real_real(plus_plus_real(numeral_numeral_real(plus_plus_num(V,W2))),Z) ).

% add_numeral_left
tff(fact_1804_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,plus_plus_num(Mb,Nb)) ).

% numeral_plus_numeral
tff(fact_1805_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Mb)),numera1916890842035813515d_enat(Nb)) = numera1916890842035813515d_enat(plus_plus_num(Mb,Nb)) ).

% numeral_plus_numeral
tff(fact_1806_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(Mb)),numeral_numeral_int(Nb)) = numeral_numeral_int(plus_plus_num(Mb,Nb)) ).

% numeral_plus_numeral
tff(fact_1807_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(plus_plus_real(numeral_numeral_real(Mb)),numeral_numeral_real(Nb)) = numeral_numeral_real(plus_plus_num(Mb,Nb)) ).

% numeral_plus_numeral
tff(fact_1808_insert__simp__mima,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        | ( Xb = Maa ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Dega)
       => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ).

% insert_simp_mima
tff(fact_1809_valid__insert__both__member__options__pres,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))
         => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),Xb)
           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(Ta,Y)),Xb) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_1810_valid__insert__both__member__options__add,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(Ta,Xb)),Xb) ) ) ).

% valid_insert_both_member_options_add
tff(fact_1811_post__member__pre__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))
         => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,vEBT_vebt_insert(Ta,Xb)),Y)
           => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Y)
              | ( Xb = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_1812_replicate__eq__replicate,axiom,
    ! [Mb: nat,Xb: vEBT_VEBT,Nb: nat,Y: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(Mb,Xb) = replicate_VEBT_VEBT(Nb,Y) )
    <=> ( ( Mb = Nb )
        & ( ( Mb != zero_zero_nat )
         => ( Xb = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_1813_length__replicate,axiom,
    ! [Nb: nat,Xb: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(Nb,Xb)) = Nb ).

% length_replicate
tff(fact_1814_length__replicate,axiom,
    ! [Nb: nat,Xb: int] : size_size_list_int(replicate_int(Nb,Xb)) = Nb ).

% length_replicate
tff(fact_1815_length__replicate,axiom,
    ! [Nb: nat,Xb: nat] : size_size_list_nat(replicate_nat(Nb,Xb)) = Nb ).

% length_replicate
tff(fact_1816_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_1817_mi__ma__2__deg,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Dega)) ) ) ).

% mi_ma_2_deg
tff(fact_1818__C5_OIH_C_I2_J,axiom,
    ! [Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),m))
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_vebt_insert(summary,Xb)),m) ) ).

% "5.IH"(2)
tff(fact_1819_Suc__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,plus_plus_num(Nb,one)) ).

% Suc_numeral
tff(fact_1820_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)) = $ite(aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)),numera1916890842035813515d_enat(V),numera1916890842035813515d_enat(U)) ).

% max_number_of(1)
tff(fact_1821_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(ord_max_real(numeral_numeral_real(U)),numeral_numeral_real(V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(U)),numeral_numeral_real(V)),numeral_numeral_real(V),numeral_numeral_real(U)) ).

% max_number_of(1)
tff(fact_1822_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V),aa_num_nat(numeral_numeral_nat,U)) ).

% max_number_of(1)
tff(fact_1823_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,numeral_numeral_int(U)),numeral_numeral_int(V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(U)),numeral_numeral_int(V)),numeral_numeral_int(V),numeral_numeral_int(U)) ).

% max_number_of(1)
tff(fact_1824_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(3)
tff(fact_1825_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(Xb)) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(3)
tff(fact_1826_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),numeral_numeral_int(Xb)) = numeral_numeral_int(Xb) ).

% max_0_1(3)
tff(fact_1827_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_real_real(ord_max_real(zero_zero_real),numeral_numeral_real(Xb)) = numeral_numeral_real(Xb) ).

% max_0_1(3)
tff(fact_1828_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xb)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(4)
tff(fact_1829_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(Xb)),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(4)
tff(fact_1830_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,numeral_numeral_int(Xb)),zero_zero_int) = numeral_numeral_int(Xb) ).

% max_0_1(4)
tff(fact_1831_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_real_real(ord_max_real(numeral_numeral_real(Xb)),zero_zero_real) = numeral_numeral_real(Xb) ).

% max_0_1(4)
tff(fact_1832_myIHs,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(treeList))
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Xb),na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),na))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_vebt_insert(Xb,Xaa)),na) ) ) ) ).

% myIHs
tff(fact_1833__C6_C,axiom,
    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,mi),ma)
    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),deg)) ) ).

% "6"
tff(fact_1834_Ball__set__replicate,axiom,
    ! [Nb: nat,A: int,P: fun_int_o] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int2(X3),set_int2(replicate_int(Nb,A)))
         => aa_int_o(P,X3) )
    <=> ( aa_int_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_1835_Ball__set__replicate,axiom,
    ! [Nb: nat,A: nat,P: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),set_nat2(replicate_nat(Nb,A)))
         => aa_nat_o(P,X3) )
    <=> ( aa_nat_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_1836_Ball__set__replicate,axiom,
    ! [Nb: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
         => aa_VEBT_VEBT_o(P,X3) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_1837_Bex__set__replicate,axiom,
    ! [Nb: nat,A: int,P: fun_int_o] :
      ( ? [X3: int] :
          ( aa_set_int_o(member_int2(X3),set_int2(replicate_int(Nb,A)))
          & aa_int_o(P,X3) )
    <=> ( aa_int_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_1838_Bex__set__replicate,axiom,
    ! [Nb: nat,A: nat,P: fun_nat_o] :
      ( ? [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),set_nat2(replicate_nat(Nb,A)))
          & aa_nat_o(P,X3) )
    <=> ( aa_nat_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_1839_Bex__set__replicate,axiom,
    ! [Nb: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ? [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
          & aa_VEBT_VEBT_o(P,X3) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_1840_in__set__replicate,axiom,
    ! [Xb: real,Nb: nat,Y: real] :
      ( aa_set_real_o(member_real2(Xb),set_real2(replicate_real(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1841_in__set__replicate,axiom,
    ! [Xb: $o,Nb: nat,Y: $o] :
      ( aa_set_o_o(member_o2((Xb)),set_o2(replicate_o(Nb,(Y))))
    <=> ( ( (Xb)
        <=> (Y) )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1842_in__set__replicate,axiom,
    ! [Xb: set_nat,Nb: nat,Y: set_nat] :
      ( aa_set_set_nat_o(member_set_nat2(Xb),set_set_nat2(replicate_set_nat(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1843_in__set__replicate,axiom,
    ! [Xb: int,Nb: nat,Y: int] :
      ( aa_set_int_o(member_int2(Xb),set_int2(replicate_int(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1844_in__set__replicate,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( aa_set_nat_o(member_nat2(Xb),set_nat2(replicate_nat(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1845_in__set__replicate,axiom,
    ! [Xb: vEBT_VEBT,Nb: nat,Y: vEBT_VEBT] :
      ( member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1846_nth__replicate,axiom,
    ! [Ia: nat,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),Nb)
     => ( aa_nat_nat(nth_nat(replicate_nat(Nb,Xb)),Ia) = Xb ) ) ).

% nth_replicate
tff(fact_1847_nth__replicate,axiom,
    ! [Ia: nat,Nb: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),Nb)
     => ( aa_nat_int(nth_int(replicate_int(Nb,Xb)),Ia) = Xb ) ) ).

% nth_replicate
tff(fact_1848_nth__replicate,axiom,
    ! [Ia: nat,Nb: nat,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),Nb)
     => ( nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,Xb),Ia) = Xb ) ) ).

% nth_replicate
tff(fact_1849_highlowprop,axiom,
    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),m))
    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),na)) ) ).

% highlowprop
tff(fact_1850_add__2__eq__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc
tff(fact_1851_add__2__eq__Suc_H,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc'
tff(fact_1852_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,bit0(Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% numeral_Bit0
tff(fact_1853_numeral__Bit0,axiom,
    ! [Nb: num] : numera1916890842035813515d_enat(bit0(Nb)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Nb)),numera1916890842035813515d_enat(Nb)) ).

% numeral_Bit0
tff(fact_1854_numeral__Bit0,axiom,
    ! [Nb: num] : numeral_numeral_int(bit0(Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(Nb)),numeral_numeral_int(Nb)) ).

% numeral_Bit0
tff(fact_1855_numeral__Bit0,axiom,
    ! [Nb: num] : numeral_numeral_real(bit0(Nb)) = aa_real_real(plus_plus_real(numeral_numeral_real(Nb)),numeral_numeral_real(Nb)) ).

% numeral_Bit0
tff(fact_1856_numeral__2__eq__2,axiom,
    aa_num_nat(numeral_numeral_nat,bit0(one)) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% numeral_2_eq_2
tff(fact_1857_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xb: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_1858_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xb: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_1859_less__2__cases,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one)))
     => ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_1860_less__2__cases__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one)))
    <=> ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_1861_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_1862_Suc__nat__number__of__add,axiom,
    ! [V: num,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,plus_plus_num(V,one))),Nb) ).

% Suc_nat_number_of_add
tff(fact_1863_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != numera6690914467698888265omplex(Nb) ).

% zero_neq_numeral
tff(fact_1864_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,Nb) ).

% zero_neq_numeral
tff(fact_1865_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_z5237406670263579293d_enat != numera1916890842035813515d_enat(Nb) ).

% zero_neq_numeral
tff(fact_1866_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_int != numeral_numeral_int(Nb) ).

% zero_neq_numeral
tff(fact_1867_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_real != numeral_numeral_real(Nb) ).

% zero_neq_numeral
tff(fact_1868_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_1869_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))
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% listrel1_eq_len
tff(fact_1870_listrel1__eq__len,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),listrel1_int(R2))
     => ( size_size_list_int(Xs) = size_size_list_int(Ys) ) ) ).

% listrel1_eq_len
tff(fact_1871_listrel1__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),listrel1_nat(R2))
     => ( size_size_list_nat(Xs) = size_size_list_nat(Ys) ) ) ).

% listrel1_eq_len
tff(fact_1872_find__cong,axiom,
    ! [Xs: list_real,Ys: list_real,P: fun_real_o,Q: fun_real_o] :
      ( ( Xs = Ys )
     => ( ! [X4: real] :
            ( aa_set_real_o(member_real2(X4),set_real2(Ys))
           => ( aa_real_o(P,X4)
            <=> aa_real_o(Q,X4) ) )
       => ( find_real(P,Xs) = find_real(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1873_find__cong,axiom,
    ! [Xs: list_o,Ys: list_o,P: fun_o_o,Q: fun_o_o] :
      ( ( Xs = Ys )
     => ( ! [X4: $o] :
            ( aa_set_o_o(member_o2((X4)),set_o2(Ys))
           => ( aa_o_o(P,(X4))
            <=> aa_o_o(Q,(X4)) ) )
       => ( find_o(P,Xs) = find_o(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1874_find__cong,axiom,
    ! [Xs: list_set_nat,Ys: list_set_nat,P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ( Xs = Ys )
     => ( ! [X4: set_nat] :
            ( aa_set_set_nat_o(member_set_nat2(X4),set_set_nat2(Ys))
           => ( aa_set_nat_o(P,X4)
            <=> aa_set_nat_o(Q,X4) ) )
       => ( find_set_nat(P,Xs) = find_set_nat(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1875_find__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Q: fun_VEBT_VEBT_o] :
      ( ( Xs = Ys )
     => ( ! [X4: vEBT_VEBT] :
            ( member_VEBT_VEBT2(X4,set_VEBT_VEBT2(Ys))
           => ( aa_VEBT_VEBT_o(P,X4)
            <=> aa_VEBT_VEBT_o(Q,X4) ) )
       => ( find_VEBT_VEBT(P,Xs) = find_VEBT_VEBT(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1876_find__cong,axiom,
    ! [Xs: list_int,Ys: list_int,P: fun_int_o,Q: fun_int_o] :
      ( ( Xs = Ys )
     => ( ! [X4: int] :
            ( aa_set_int_o(member_int2(X4),set_int2(Ys))
           => ( aa_int_o(P,X4)
            <=> aa_int_o(Q,X4) ) )
       => ( find_int(P,Xs) = find_int(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1877_find__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,P: fun_nat_o,Q: fun_nat_o] :
      ( ( Xs = Ys )
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),set_nat2(Ys))
           => ( aa_nat_o(P,X4)
            <=> aa_nat_o(Q,X4) ) )
       => ( find_nat(P,Xs) = find_nat(Q,Ys) ) ) ) ).

% find_cong
tff(fact_1878_set__zip__rightD,axiom,
    ! [Xb: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Y),set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Ys)))
     => aa_set_nat_o(member_nat2(Y),set_nat2(Ys)) ) ).

% set_zip_rightD
tff(fact_1879_set__zip__rightD,axiom,
    ! [Xb: product_prod_nat_nat,Y: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,Xb),Y),set_Pr5518436109238095868at_nat(zip_Pr4664179122662387191at_nat(Xs,Ys)))
     => member8440522571783428010at_nat(Y,set_Pr5648618587558075414at_nat(Ys)) ) ).

% set_zip_rightD
tff(fact_1880_set__zip__rightD,axiom,
    ! [Xb: vEBT_VEBT,Y: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Y),set_Pr7031586669278753246BT_nat(zip_VEBT_VEBT_nat(Xs,Ys)))
     => aa_set_nat_o(member_nat2(Y),set_nat2(Ys)) ) ).

% set_zip_rightD
tff(fact_1881_set__zip__rightD,axiom,
    ! [Xb: int,Y: int,Xs: list_int,Ys: list_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Y),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Ys)))
     => aa_set_int_o(member_int2(Y),set_int2(Ys)) ) ).

% set_zip_rightD
tff(fact_1882_set__zip__rightD,axiom,
    ! [Xb: vEBT_VEBT,Y: extended_enat,Xs: list_VEBT_VEBT,Ys: list_Extended_enat] :
      ( member38198578724832770d_enat(aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,Xb),Y),set_Pr4712219556205076590d_enat(zip_VE7205001627739651817d_enat(Xs,Ys)))
     => aa_set3337841544853333262enat_o(member_Extended_enat(Y),set_Extended_enat2(Ys)) ) ).

% set_zip_rightD
tff(fact_1883_set__zip__leftD,axiom,
    ! [Xb: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Y),set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Ys)))
     => aa_set_nat_o(member_nat2(Xb),set_nat2(Xs)) ) ).

% set_zip_leftD
tff(fact_1884_set__zip__leftD,axiom,
    ! [Xb: product_prod_nat_nat,Y: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,Xb),Y),set_Pr5518436109238095868at_nat(zip_Pr4664179122662387191at_nat(Xs,Ys)))
     => member8440522571783428010at_nat(Xb,set_Pr5648618587558075414at_nat(Xs)) ) ).

% set_zip_leftD
tff(fact_1885_set__zip__leftD,axiom,
    ! [Xb: vEBT_VEBT,Y: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Y),set_Pr7031586669278753246BT_nat(zip_VEBT_VEBT_nat(Xs,Ys)))
     => member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(Xs)) ) ).

% set_zip_leftD
tff(fact_1886_set__zip__leftD,axiom,
    ! [Xb: int,Y: int,Xs: list_int,Ys: list_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Y),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Ys)))
     => aa_set_int_o(member_int2(Xb),set_int2(Xs)) ) ).

% set_zip_leftD
tff(fact_1887_set__zip__leftD,axiom,
    ! [Xb: vEBT_VEBT,Y: extended_enat,Xs: list_VEBT_VEBT,Ys: list_Extended_enat] :
      ( member38198578724832770d_enat(aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,Xb),Y),set_Pr4712219556205076590d_enat(zip_VE7205001627739651817d_enat(Xs,Ys)))
     => member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(Xs)) ) ).

% set_zip_leftD
tff(fact_1888_in__set__zipE,axiom,
    ! [Xb: real,Y: real,Xs: list_real,Ys: list_real] :
      ( member7849222048561428706l_real(produc4511245868158468465l_real(Xb,Y),set_Pr5999470521830281550l_real(zip_real_real(Xs,Ys)))
     => ~ ( aa_set_real_o(member_real2(Xb),set_real2(Xs))
         => ~ aa_set_real_o(member_real2(Y),set_real2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1889_in__set__zipE,axiom,
    ! [Xb: real,Y: $o,Xs: list_real,Ys: list_o] :
      ( member772602641336174712real_o(product_Pair_real_o(Xb,(Y)),set_Pr5196769464307566348real_o(zip_real_o(Xs,Ys)))
     => ~ ( aa_set_real_o(member_real2(Xb),set_real2(Xs))
         => ~ aa_set_o_o(member_o2((Y)),set_o2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1890_in__set__zipE,axiom,
    ! [Xb: $o,Y: real,Xs: list_o,Ys: list_real] :
      ( member7400031367953476362o_real(product_Pair_o_real((Xb),Y),set_Pr2600826154070092190o_real(zip_o_real(Xs,Ys)))
     => ~ ( aa_set_o_o(member_o2((Xb)),set_o2(Xs))
         => ~ aa_set_real_o(member_real2(Y),set_real2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1891_in__set__zipE,axiom,
    ! [Xb: $o,Y: $o,Xs: list_o,Ys: list_o] :
      ( member7466972457876170832od_o_o(product_Pair_o_o((Xb),(Y)),set_Product_prod_o_o2(zip_o_o(Xs,Ys)))
     => ~ ( aa_set_o_o(member_o2((Xb)),set_o2(Xs))
         => ~ aa_set_o_o(member_o2((Y)),set_o2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1892_in__set__zipE,axiom,
    ! [Xb: real,Y: vEBT_VEBT,Xs: list_real,Ys: list_VEBT_VEBT] :
      ( member7262085504369356948T_VEBT(produc6931449550656315951T_VEBT(Xb,Y),set_Pr8897343066327330088T_VEBT(zip_real_VEBT_VEBT(Xs,Ys)))
     => ~ ( aa_set_real_o(member_real2(Xb),set_real2(Xs))
         => ~ member_VEBT_VEBT2(Y,set_VEBT_VEBT2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1893_in__set__zipE,axiom,
    ! [Xb: $o,Y: vEBT_VEBT,Xs: list_o,Ys: list_VEBT_VEBT] :
      ( member5477980866518848620T_VEBT(produc2982872950893828659T_VEBT((Xb),Y),set_Pr655345902815428824T_VEBT(zip_o_VEBT_VEBT(Xs,Ys)))
     => ~ ( aa_set_o_o(member_o2((Xb)),set_o2(Xs))
         => ~ member_VEBT_VEBT2(Y,set_VEBT_VEBT2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1894_in__set__zipE,axiom,
    ! [Xb: real,Y: int,Xs: list_real,Ys: list_int] :
      ( member1627681773268152802al_int(produc3179012173361985393al_int(Xb,Y),set_Pr8219819362198175822al_int(zip_real_int(Xs,Ys)))
     => ~ ( aa_set_real_o(member_real2(Xb),set_real2(Xs))
         => ~ aa_set_int_o(member_int2(Y),set_int2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1895_in__set__zipE,axiom,
    ! [Xb: $o,Y: int,Xs: list_o,Ys: list_int] :
      ( member7847949116333733898_o_int(product_Pair_o_int((Xb),Y),set_Pr2828948584524939422_o_int(zip_o_int(Xs,Ys)))
     => ~ ( aa_set_o_o(member_o2((Xb)),set_o2(Xs))
         => ~ aa_set_int_o(member_int2(Y),set_int2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1896_in__set__zipE,axiom,
    ! [Xb: real,Y: nat,Xs: list_real,Ys: list_nat] :
      ( member5805532792777349510al_nat(produc3181502643871035669al_nat(Xb,Y),set_Pr3174298344852596722al_nat(zip_real_nat(Xs,Ys)))
     => ~ ( aa_set_real_o(member_real2(Xb),set_real2(Xs))
         => ~ aa_set_nat_o(member_nat2(Y),set_nat2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1897_in__set__zipE,axiom,
    ! [Xb: $o,Y: nat,Xs: list_o,Ys: list_nat] :
      ( member2802428098988154798_o_nat(product_Pair_o_nat((Xb),Y),set_Pr7006799604034136130_o_nat(zip_o_nat(Xs,Ys)))
     => ~ ( aa_set_o_o(member_o2((Xb)),set_o2(Xs))
         => ~ aa_set_nat_o(member_nat2(Y),set_nat2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1898_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_real2(A),set_real2(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1899_zip__same,axiom,
    ! [A: $o,B: $o,Xs: list_o] :
      ( member7466972457876170832od_o_o(product_Pair_o_o((A),(B)),set_Product_prod_o_o2(zip_o_o(Xs,Xs)))
    <=> ( aa_set_o_o(member_o2((A)),set_o2(Xs))
        & ( (A)
        <=> (B) ) ) ) ).

% zip_same
tff(fact_1900_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_nat2(A),set_set_nat2(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1901_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_VEBT2(A,set_VEBT_VEBT2(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1902_zip__same,axiom,
    ! [A: nat,B: nat,Xs: list_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B),set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Xs)))
    <=> ( aa_set_nat_o(member_nat2(A),set_nat2(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1903_zip__same,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,A),B),set_Pr5518436109238095868at_nat(zip_Pr4664179122662387191at_nat(Xs,Xs)))
    <=> ( member8440522571783428010at_nat(A,set_Pr5648618587558075414at_nat(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1904_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_int2(A),set_int2(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_1905_zip__update,axiom,
    ! [Xs: list_VEBT_VEBT,Ia: nat,Xb: vEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] : zip_VE537291747668921783T_VEBT(list_u1324408373059187874T_VEBT(Xs,Ia,Xb),list_u1324408373059187874T_VEBT(Ys,Ia,Y)) = list_u6961636818849549845T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys),Ia,produc537772716801021591T_VEBT(Xb,Y)) ).

% zip_update
tff(fact_1906_zip__update,axiom,
    ! [Xs: list_nat,Ia: nat,Xb: nat,Ys: list_nat,Y: nat] : zip_nat_nat(list_update_nat(Xs,Ia,Xb),list_update_nat(Ys,Ia,Y)) = list_u6180841689913720943at_nat(zip_nat_nat(Xs,Ys),Ia,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Y)) ).

% zip_update
tff(fact_1907_zip__update,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ia: nat,Xb: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] : zip_Pr4664179122662387191at_nat(list_u6180841689913720943at_nat(Xs,Ia,Xb),list_u6180841689913720943at_nat(Ys,Ia,Y)) = list_u5003261594476800725at_nat(zip_Pr4664179122662387191at_nat(Xs,Ys),Ia,aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,Xb),Y)) ).

% zip_update
tff(fact_1908_zip__update,axiom,
    ! [Xs: list_VEBT_VEBT,Ia: nat,Xb: vEBT_VEBT,Ys: list_nat,Y: nat] : zip_VEBT_VEBT_nat(list_u1324408373059187874T_VEBT(Xs,Ia,Xb),list_update_nat(Ys,Ia,Y)) = list_u2459188882655168453BT_nat(zip_VEBT_VEBT_nat(Xs,Ys),Ia,aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Y)) ).

% zip_update
tff(fact_1909_zip__update,axiom,
    ! [Xs: list_int,Ia: nat,Xb: int,Ys: list_int,Y: int] : zip_int_int(list_update_int(Xs,Ia,Xb),list_update_int(Ys,Ia,Y)) = list_u3002344382305578791nt_int(zip_int_int(Xs,Ys),Ia,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Y)) ).

% zip_update
tff(fact_1910_zip__update,axiom,
    ! [Xs: list_VEBT_VEBT,Ia: nat,Xb: vEBT_VEBT,Ys: list_Extended_enat,Y: extended_enat] : zip_VE7205001627739651817d_enat(list_u1324408373059187874T_VEBT(Xs,Ia,Xb),list_u3071683517702156500d_enat(Ys,Ia,Y)) = list_u2241885204319820103d_enat(zip_VE7205001627739651817d_enat(Xs,Ys),Ia,aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,Xb),Y)) ).

% zip_update
tff(fact_1911_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(Nb)),zero_z5237406670263579293d_enat) ).

% not_numeral_le_zero
tff(fact_1912_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Nb)),zero_zero_real) ).

% not_numeral_le_zero
tff(fact_1913_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).

% not_numeral_le_zero
tff(fact_1914_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(Nb)),zero_zero_int) ).

% not_numeral_le_zero
tff(fact_1915_zero__le__numeral,axiom,
    ! [Nb: num] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(Nb)) ).

% zero_le_numeral
tff(fact_1916_zero__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),numeral_numeral_real(Nb)) ).

% zero_le_numeral
tff(fact_1917_zero__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% zero_le_numeral
tff(fact_1918_zero__le__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),numeral_numeral_int(Nb)) ).

% zero_le_numeral
tff(fact_1919_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).

% not_numeral_less_zero
tff(fact_1920_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,numera1916890842035813515d_enat(Nb)),zero_z5237406670263579293d_enat) ).

% not_numeral_less_zero
tff(fact_1921_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,numeral_numeral_int(Nb)),zero_zero_int) ).

% not_numeral_less_zero
tff(fact_1922_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Nb)),zero_zero_real) ).

% not_numeral_less_zero
tff(fact_1923_zero__less__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% zero_less_numeral
tff(fact_1924_zero__less__numeral,axiom,
    ! [Nb: num] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(Nb)) ).

% zero_less_numeral
tff(fact_1925_zero__less__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),numeral_numeral_int(Nb)) ).

% zero_less_numeral
tff(fact_1926_zero__less__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),numeral_numeral_real(Nb)) ).

% zero_less_numeral
tff(fact_1927_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT2(X4,set_VEBT_VEBT2(Xs))
         => ( X4 = Xb ) )
     => ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_1928_replicate__length__same,axiom,
    ! [Xs: list_int,Xb: int] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int2(X4),set_int2(Xs))
         => ( X4 = Xb ) )
     => ( replicate_int(size_size_list_int(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_1929_replicate__length__same,axiom,
    ! [Xs: list_nat,Xb: nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat2(X4),set_nat2(Xs))
         => ( X4 = Xb ) )
     => ( replicate_nat(size_size_list_nat(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_1930_replicate__eqI,axiom,
    ! [Xs: list_real,Nb: nat,Xb: real] :
      ( ( size_size_list_real(Xs) = Nb )
     => ( ! [Y3: real] :
            ( aa_set_real_o(member_real2(Y3),set_real2(Xs))
           => ( Y3 = Xb ) )
       => ( Xs = replicate_real(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_1931_replicate__eqI,axiom,
    ! [Xs: list_o,Nb: nat,Xb: $o] :
      ( ( size_size_list_o(Xs) = Nb )
     => ( ! [Y3: $o] :
            ( aa_set_o_o(member_o2((Y3)),set_o2(Xs))
           => ( (Y3)
            <=> (Xb) ) )
       => ( Xs = replicate_o(Nb,(Xb)) ) ) ) ).

% replicate_eqI
tff(fact_1932_replicate__eqI,axiom,
    ! [Xs: list_set_nat,Nb: nat,Xb: set_nat] :
      ( ( size_s3254054031482475050et_nat(Xs) = Nb )
     => ( ! [Y3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat2(Y3),set_set_nat2(Xs))
           => ( Y3 = Xb ) )
       => ( Xs = replicate_set_nat(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_1933_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,Nb: nat,Xb: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = Nb )
     => ( ! [Y3: vEBT_VEBT] :
            ( member_VEBT_VEBT2(Y3,set_VEBT_VEBT2(Xs))
           => ( Y3 = Xb ) )
       => ( Xs = replicate_VEBT_VEBT(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_1934_replicate__eqI,axiom,
    ! [Xs: list_int,Nb: nat,Xb: int] :
      ( ( size_size_list_int(Xs) = Nb )
     => ( ! [Y3: int] :
            ( aa_set_int_o(member_int2(Y3),set_int2(Xs))
           => ( Y3 = Xb ) )
       => ( Xs = replicate_int(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_1935_replicate__eqI,axiom,
    ! [Xs: list_nat,Nb: nat,Xb: nat] :
      ( ( size_size_list_nat(Xs) = Nb )
     => ( ! [Y3: nat] :
            ( aa_set_nat_o(member_nat2(Y3),set_nat2(Xs))
           => ( Y3 = Xb ) )
       => ( Xs = replicate_nat(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_1936_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_VEBT_VEBT,Xb: real] :
      ( ( size_size_list_real(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( aa_set_real_o(member_real2(Xb),set_real2(Xs))
       => ~ ! [Y3: vEBT_VEBT] : ~ member7262085504369356948T_VEBT(produc6931449550656315951T_VEBT(Xb,Y3),set_Pr8897343066327330088T_VEBT(zip_real_VEBT_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1937_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_o,Ys: list_VEBT_VEBT,Xb: $o] :
      ( ( size_size_list_o(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( aa_set_o_o(member_o2((Xb)),set_o2(Xs))
       => ~ ! [Y3: vEBT_VEBT] : ~ member5477980866518848620T_VEBT(produc2982872950893828659T_VEBT((Xb),Y3),set_Pr655345902815428824T_VEBT(zip_o_VEBT_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1938_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_int,Xb: real] :
      ( ( size_size_list_real(Xs) = size_size_list_int(Ys) )
     => ( aa_set_real_o(member_real2(Xb),set_real2(Xs))
       => ~ ! [Y3: int] : ~ member1627681773268152802al_int(produc3179012173361985393al_int(Xb,Y3),set_Pr8219819362198175822al_int(zip_real_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1939_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_o,Ys: list_int,Xb: $o] :
      ( ( size_size_list_o(Xs) = size_size_list_int(Ys) )
     => ( aa_set_o_o(member_o2((Xb)),set_o2(Xs))
       => ~ ! [Y3: int] : ~ member7847949116333733898_o_int(product_Pair_o_int((Xb),Y3),set_Pr2828948584524939422_o_int(zip_o_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1940_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_nat,Xb: real] :
      ( ( size_size_list_real(Xs) = size_size_list_nat(Ys) )
     => ( aa_set_real_o(member_real2(Xb),set_real2(Xs))
       => ~ ! [Y3: nat] : ~ member5805532792777349510al_nat(produc3181502643871035669al_nat(Xb,Y3),set_Pr3174298344852596722al_nat(zip_real_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1941_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_o,Ys: list_nat,Xb: $o] :
      ( ( size_size_list_o(Xs) = size_size_list_nat(Ys) )
     => ( aa_set_o_o(member_o2((Xb)),set_o2(Xs))
       => ~ ! [Y3: nat] : ~ member2802428098988154798_o_nat(product_Pair_o_nat((Xb),Y3),set_Pr7006799604034136130_o_nat(zip_o_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1942_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_Extended_enat,Xb: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s3941691890525107288d_enat(Ys) )
     => ( member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(Xs))
       => ~ ! [Y3: extended_enat] : ~ member38198578724832770d_enat(aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,Xb),Y3),set_Pr4712219556205076590d_enat(zip_VE7205001627739651817d_enat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1943_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(Xs))
       => ~ ! [Y3: vEBT_VEBT] : ~ member568628332442017744T_VEBT(produc537772716801021591T_VEBT(Xb,Y3),set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1944_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int,Xb: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Ys) )
     => ( member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(Xs))
       => ~ ! [Y3: int] : ~ member5419026705395827622BT_int(produc736041933913180425BT_int(Xb,Y3),set_Pr2853735649769556538BT_int(zip_VEBT_VEBT_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1945_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,Xb: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) )
     => ( member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(Xs))
       => ~ ! [Y3: nat] : ~ member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Y3),set_Pr7031586669278753246BT_nat(zip_VEBT_VEBT_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1946_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_real,Y: real] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Ys) )
     => ( aa_set_real_o(member_real2(Y),set_real2(Ys))
       => ~ ! [X4: vEBT_VEBT] : ~ member8675245146396747942T_real(produc8117437818029410057T_real(X4,Y),set_Pr1087130671499945274T_real(zip_VEBT_VEBT_real(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1947_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_o,Y: $o] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_o(Ys) )
     => ( aa_set_o_o(member_o2((Y)),set_o2(Ys))
       => ~ ! [X4: vEBT_VEBT] : ~ member3307348790968139188VEBT_o(produc8721562602347293563VEBT_o(X4,(Y)),set_Pr7708085864119495200VEBT_o(zip_VEBT_VEBT_o(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1948_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_Extended_enat,Y: extended_enat] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s3941691890525107288d_enat(Ys) )
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Y),set_Extended_enat2(Ys))
       => ~ ! [X4: vEBT_VEBT] : ~ member38198578724832770d_enat(aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,X4),Y),set_Pr4712219556205076590d_enat(zip_VE7205001627739651817d_enat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1949_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( member_VEBT_VEBT2(Y,set_VEBT_VEBT2(Ys))
       => ~ ! [X4: vEBT_VEBT] : ~ member568628332442017744T_VEBT(produc537772716801021591T_VEBT(X4,Y),set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1950_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int,Y: int] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Ys) )
     => ( aa_set_int_o(member_int2(Y),set_int2(Ys))
       => ~ ! [X4: vEBT_VEBT] : ~ member5419026705395827622BT_int(produc736041933913180425BT_int(X4,Y),set_Pr2853735649769556538BT_int(zip_VEBT_VEBT_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1951_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,Y: nat] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) )
     => ( aa_set_nat_o(member_nat2(Y),set_nat2(Ys))
       => ~ ! [X4: vEBT_VEBT] : ~ member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,X4),Y),set_Pr7031586669278753246BT_nat(zip_VEBT_VEBT_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1952_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_int,Ys: list_real,Y: real] :
      ( ( size_size_list_int(Xs) = size_size_list_real(Ys) )
     => ( aa_set_real_o(member_real2(Y),set_real2(Ys))
       => ~ ! [X4: int] : ~ member2744130022092475746t_real(produc801115645435158769t_real(X4,Y),set_Pr112895574167722958t_real(zip_int_real(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1953_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_int,Ys: list_o,Y: $o] :
      ( ( size_size_list_int(Xs) = size_size_list_o(Ys) )
     => ( aa_set_o_o(member_o2((Y)),set_o2(Ys))
       => ~ ! [X4: int] : ~ member4489920277610959864_int_o(product_Pair_int_o(X4,(Y)),set_Pr8694291782656941196_int_o(zip_int_o(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1954_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_int,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( size_size_list_int(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( member_VEBT_VEBT2(Y,set_VEBT_VEBT2(Ys))
       => ~ ! [X4: int] : ~ member2056185340421749780T_VEBT(produc3329399203697025711T_VEBT(X4,Y),set_Pr8714266321650254504T_VEBT(zip_int_VEBT_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1955_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_int,Ys: list_int,Y: int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( aa_set_int_o(member_int2(Y),set_int2(Ys))
       => ~ ! [X4: int] : ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1956_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT2(X4,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X4),Nb) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summarya),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb) )
         => ( ( Mb = Nb )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ! [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,bit0(one))),Mb))
                   => ( ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeLista,I2)),X_13)
                    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summarya),I2) ) )
               => ( ( ( Mia = Maa )
                   => ! [X4: vEBT_VEBT] :
                        ( member_VEBT_VEBT2(X4,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X4),X_12) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [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,bit0(one))),Mb))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I2 )
                                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X4: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),Nb) = I2 )
                                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X4)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Maa) ) ) ) ) )
                       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_1957_num_Osize_I5_J,axiom,
    ! [X22: num] : aa_num_nat(size_size_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_1958_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT2(X4,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X4),Nb) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summarya),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb) )
         => ( ( Mb = aa_nat_nat(suc,Nb) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ! [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,bit0(one))),Mb))
                   => ( ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeLista,I2)),X_13)
                    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summarya),I2) ) )
               => ( ( ( Mia = Maa )
                   => ! [X4: vEBT_VEBT] :
                        ( member_VEBT_VEBT2(X4,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X4),X_12) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [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,bit0(one))),Mb))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I2 )
                                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X4: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),Nb) = I2 )
                                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X4)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Maa) ) ) ) ) )
                       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_1959_pair__lessI1,axiom,
    ! [A: nat,B: nat,S: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),S)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B),Ta)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_1960_count__le__length,axiom,
    ! [Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(count_list_VEBT_VEBT(Xs),Xb)),size_s6755466524823107622T_VEBT(Xs)) ).

% count_le_length
tff(fact_1961_count__le__length,axiom,
    ! [Xs: list_int,Xb: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(count_list_int(Xs),Xb)),size_size_list_int(Xs)) ).

% count_le_length
tff(fact_1962_count__le__length,axiom,
    ! [Xs: list_nat,Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(count_list_nat(Xs),Xb)),size_size_list_nat(Xs)) ).

% count_le_length
tff(fact_1963_sum__power2__eq__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(plus_plus_real(aa_nat_real(power_power_real(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))) = zero_zero_real )
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1964_sum__power2__eq__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))) = zero_zero_int )
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1965_zero__less__power2,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,bit0(one))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_1966_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,bit0(one))))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_1967_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,bit0(one)))),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_1968_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,bit0(one)))),zero_zero_int)
    <=> ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_1969_power2__eq__iff__nonneg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(power_power_real(Xb),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1970_power2__eq__iff__nonneg,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
     => ( 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,Xb),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1971_power2__eq__iff__nonneg,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1972_zero__eq__power2,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat )
    <=> ( A = zero_zero_nat ) ) ).

% zero_eq_power2
tff(fact_1973_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_1974_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% zero_eq_power2
tff(fact_1975_zero__eq__power2,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% zero_eq_power2
tff(fact_1976_power__mono__iff,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),Nb)),aa_nat_real(power_power_real(B),Nb))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_1977_power__mono__iff,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_1978_power__mono__iff,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_1979_insert__simp__excp,axiom,
    ! [Mia: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Xb: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Dega)
         => ( ( Xb != Maa )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mia),Maa))),Dega,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_1980_insert__simp__norm,axiom,
    ! [Xb: 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,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Dega)
         => ( ( Xb != Maa )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Maa))),Dega,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_1981_power__eq__0__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_1982_power__eq__0__iff,axiom,
    ! [A: real,Nb: nat] :
      ( ( aa_nat_real(power_power_real(A),Nb) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_1983_power__eq__0__iff,axiom,
    ! [A: int,Nb: nat] :
      ( ( aa_nat_int(power_power_int(A),Nb) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_1984_power__eq__0__iff,axiom,
    ! [A: complex,Nb: nat] :
      ( ( aa_nat_complex(power_power_complex(A),Nb) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_1985_member__inv,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Dega)
        & ( ( Xb = Mia )
          | ( Xb = Maa )
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeLista))
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ) ) ) ).

% member_inv
tff(fact_1986_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,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,bit0(one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B) ).

% pow_sum
tff(fact_1987_high__def,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) ).

% high_def
tff(fact_1988__C9_C,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,bit0(one))) = na ).

% "9"
tff(fact_1989_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_1990_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_1991_div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_1992_div__0,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(zero_zero_real),A) = zero_zero_real ).

% div_0
tff(fact_1993_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_1994_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_1995_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_1996_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_1997_div__by__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_1998_div__by__0,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_1999_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_2000_divide__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(divide_divide_real(C),A) = aa_real_real(divide_divide_real(C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_2001_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_2002_divide__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),C) = aa_real_real(divide_divide_real(B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_2003_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_2004_division__ring__divide__zero,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_2005_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_Extended_enat(power_8040749407984259932d_enat(zero_z5237406670263579293d_enat),aa_nat_nat(suc,Nb)) = zero_z5237406670263579293d_enat ).

% power_0_Suc
tff(fact_2006_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) = zero_zero_nat ).

% power_0_Suc
tff(fact_2007_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(power_power_real(zero_zero_real),aa_nat_nat(suc,Nb)) = zero_zero_real ).

% power_0_Suc
tff(fact_2008_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,Nb)) = zero_zero_int ).

% power_0_Suc
tff(fact_2009_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,Nb)) = zero_zero_complex ).

% power_0_Suc
tff(fact_2010_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_Extended_enat(power_8040749407984259932d_enat(zero_z5237406670263579293d_enat),aa_num_nat(numeral_numeral_nat,K)) = zero_z5237406670263579293d_enat ).

% power_zero_numeral
tff(fact_2011_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_2012_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_real(power_power_real(zero_zero_real),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_real ).

% power_zero_numeral
tff(fact_2013_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_2014_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_2015_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_2016_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_2017_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_2018_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_2019_power__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_2020_nat__power__eq__Suc__0__iff,axiom,
    ! [Xb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Mb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Mb = zero_zero_nat )
        | ( Xb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_2021_nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_2022_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_Node(Info,Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Dega))
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one)))))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,vEBT_Node(Info,Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_ding
tff(fact_2023_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ).

% add_divide_distrib
tff(fact_2024_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% divide_le_0_iff
tff(fact_2025_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ) ) ).

% divide_right_mono
tff(fact_2026_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_divide_iff
tff(fact_2027_divide__nonneg__nonneg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_2028_divide__nonneg__nonpos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Y)),zero_zero_real) ) ) ).

% divide_nonneg_nonpos
tff(fact_2029_divide__nonpos__nonneg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Y)),zero_zero_real) ) ) ).

% divide_nonpos_nonneg
tff(fact_2030_divide__nonpos__nonpos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_2031_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),aa_real_real(divide_divide_real(A),C)) ) ) ).

% divide_right_mono_neg
tff(fact_2032_divide__neg__neg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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(divide_divide_real(Xb),Y)) ) ) ).

% divide_neg_neg
tff(fact_2033_divide__neg__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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(divide_divide_real(Xb),Y)),zero_zero_real) ) ) ).

% divide_neg_pos
tff(fact_2034_divide__pos__neg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( 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(divide_divide_real(Xb),Y)),zero_zero_real) ) ) ).

% divide_pos_neg
tff(fact_2035_divide__pos__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( 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(divide_divide_real(Xb),Y)) ) ) ).

% divide_pos_pos
tff(fact_2036_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_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_2037_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_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_2038_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(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_2039_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(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ) ) ).

% divide_strict_right_mono
tff(fact_2040_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(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2041_frac__le,axiom,
    ! [Y: real,Xb: 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,Xb),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(divide_divide_real(Xb),Z)),aa_real_real(divide_divide_real(Y),W2)) ) ) ) ) ).

% frac_le
tff(fact_2042_frac__less,axiom,
    ! [Xb: real,Y: real,W2: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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(divide_divide_real(Xb),Z)),aa_real_real(divide_divide_real(Y),W2)) ) ) ) ) ).

% frac_less
tff(fact_2043_frac__less2,axiom,
    ! [Xb: real,Y: real,W2: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),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(divide_divide_real(Xb),Z)),aa_real_real(divide_divide_real(Y),W2)) ) ) ) ) ).

% frac_less2
tff(fact_2044_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_2045_divide__nonneg__neg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Y)),zero_zero_real) ) ) ).

% divide_nonneg_neg
tff(fact_2046_divide__nonneg__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_2047_divide__nonpos__neg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_2048_divide__nonpos__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Y)),zero_zero_real) ) ) ).

% divide_nonpos_pos
tff(fact_2049_field__sum__of__halves,axiom,
    ! [Xb: real] : aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(Xb),numeral_numeral_real(bit0(one)))),aa_real_real(divide_divide_real(Xb),numeral_numeral_real(bit0(one)))) = Xb ).

% field_sum_of_halves
tff(fact_2050_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(A),numeral_numeral_real(bit0(one))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% half_gt_zero_iff
tff(fact_2051_half__gt__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(A),numeral_numeral_real(bit0(one)))) ) ).

% half_gt_zero
tff(fact_2052_field__less__half__sum,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Xb),Y)),numeral_numeral_real(bit0(one)))) ) ).

% field_less_half_sum
tff(fact_2053_power__not__zero,axiom,
    ! [A: nat,Nb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) != zero_zero_nat ) ) ).

% power_not_zero
tff(fact_2054_power__not__zero,axiom,
    ! [A: real,Nb: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(power_power_real(A),Nb) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_2055_power__not__zero,axiom,
    ! [A: int,Nb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),Nb) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_2056_power__not__zero,axiom,
    ! [A: complex,Nb: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),Nb) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_2057_power__mono,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),Nb)),aa_nat_real(power_power_real(B),Nb)) ) ) ).

% power_mono
tff(fact_2058_power__mono,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).

% power_mono
tff(fact_2059_power__mono,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono
tff(fact_2060_zero__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),Nb)) ) ).

% zero_le_power
tff(fact_2061_zero__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% zero_le_power
tff(fact_2062_zero__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb)) ) ).

% zero_le_power
tff(fact_2063_zero__less__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% zero_less_power
tff(fact_2064_zero__less__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),Nb)) ) ).

% zero_less_power
tff(fact_2065_zero__less__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb)) ) ).

% zero_less_power
tff(fact_2066_nat__power__less__imp__less,axiom,
    ! [Ia: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ia)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ia),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ia),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% nat_power_less_imp_less
tff(fact_2067_power__less__imp__less__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),Nb)),aa_nat_real(power_power_real(B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2068_power__less__imp__less__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2069_power__less__imp__less__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2070_power__inject__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( ( aa_nat_real(power_power_real(A),aa_nat_nat(suc,Nb)) = aa_nat_real(power_power_real(B),aa_nat_nat(suc,Nb)) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2071_power__inject__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2072_power__inject__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,Nb)) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2073_power__le__imp__le__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,Nb))),aa_nat_real(power_power_real(B),aa_nat_nat(suc,Nb)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_2074_power__le__imp__le__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_2075_power__le__imp__le__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,Nb)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_2076_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_Extended_enat(power_8040749407984259932d_enat(zero_z5237406670263579293d_enat),Nb) = zero_z5237406670263579293d_enat ) ) ).

% zero_power
tff(fact_2077_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = zero_zero_nat ) ) ).

% zero_power
tff(fact_2078_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(power_power_real(zero_zero_real),Nb) = zero_zero_real ) ) ).

% zero_power
tff(fact_2079_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_int(power_power_int(zero_zero_int),Nb) = zero_zero_int ) ) ).

% zero_power
tff(fact_2080_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),Nb) = zero_zero_complex ) ) ).

% zero_power
tff(fact_2081_power__gt__expt,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),K)) ) ).

% power_gt_expt
tff(fact_2082_nat__one__le__power,axiom,
    ! [Ia: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Ia)
     => 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,Ia),Nb)) ) ).

% nat_one_le_power
tff(fact_2083_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( ( aa_nat_real(power_power_real(A),Nb) = aa_nat_real(power_power_real(B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2084_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2085_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( ( aa_nat_int(power_power_int(A),Nb) = aa_nat_int(power_power_int(B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2086_power__eq__imp__eq__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( ( aa_nat_real(power_power_real(A),Nb) = aa_nat_real(power_power_real(B),Nb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2087_power__eq__imp__eq__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2088_power__eq__imp__eq__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),Nb) = aa_nat_int(power_power_int(B),Nb) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2089_zero__power2,axiom,
    aa_nat_Extended_enat(power_8040749407984259932d_enat(zero_z5237406670263579293d_enat),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_z5237406670263579293d_enat ).

% zero_power2
tff(fact_2090_zero__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ).

% zero_power2
tff(fact_2091_zero__power2,axiom,
    aa_nat_real(power_power_real(zero_zero_real),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_real ).

% zero_power2
tff(fact_2092_zero__power2,axiom,
    aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_int ).

% zero_power2
tff(fact_2093_zero__power2,axiom,
    aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_complex ).

% zero_power2
tff(fact_2094_less__exp,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) ).

% less_exp
tff(fact_2095_power2__nat__le__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),aa_num_nat(numeral_numeral_nat,bit0(one)))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% power2_nat_le_imp_le
tff(fact_2096_power2__nat__le__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% power2_nat_le_eq_le
tff(fact_2097_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,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_2098_power2__le__imp__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,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,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_2099_power2__le__imp__le,axiom,
    ! [Xb: 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,Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,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,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_2100_power2__le__imp__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,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,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_2101_power2__eq__imp__eq,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_nat_real(power_power_real(Xb),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2102_power2__eq__imp__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2103_power2__eq__imp__eq,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2104_zero__le__power2,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% zero_le_power2
tff(fact_2105_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,bit0(one)))) ).

% zero_le_power2
tff(fact_2106_power__strict__mono,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),Nb)),aa_nat_real(power_power_real(B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_2107_power__strict__mono,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_2108_power__strict__mono,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_2109_power2__less__0,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,bit0(one)))),zero_zero_real) ).

% power2_less_0
tff(fact_2110_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,bit0(one)))),zero_zero_int) ).

% power2_less_0
tff(fact_2111_power2__less__imp__less,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,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,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_2112_power2__less__imp__less,axiom,
    ! [Xb: 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,Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,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,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_2113_power2__less__imp__less,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,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,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_2114_sum__power2__ge__zero,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% sum_power2_ge_zero
tff(fact_2115_sum__power2__ge__zero,axiom,
    ! [Xb: 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(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% sum_power2_ge_zero
tff(fact_2116_sum__power2__le__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),zero_zero_real)
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2117_sum__power2__le__zero__iff,axiom,
    ! [Xb: 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(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),zero_zero_int)
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2118_not__sum__power2__lt__zero,axiom,
    ! [Xb: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_nat_real(power_power_real(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),zero_zero_real) ).

% not_sum_power2_lt_zero
tff(fact_2119_not__sum__power2__lt__zero,axiom,
    ! [Xb: 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(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),zero_zero_int) ).

% not_sum_power2_lt_zero
tff(fact_2120_sum__power2__gt__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
    <=> ( ( Xb != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2121_sum__power2__gt__zero__iff,axiom,
    ! [Xb: 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(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
    <=> ( ( Xb != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2122_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xb: 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,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Dega)
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one)))))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_2123_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,bit0(one))) = Mb ).

% add_self_div_2
tff(fact_2124_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,bit0(one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% div2_Suc_Suc
tff(fact_2125_both__member__options__from__complete__tree__to__child,axiom,
    ! [Dega: nat,Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Dega)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one)))))
          | ( Xb = Mia )
          | ( Xb = Maa ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_2126_div__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = zero_zero_nat ) ) ).

% div_less
tff(fact_2127_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_2128_set__n__deg__not__0,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Mb: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT2(X4,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X4),Nb) )
     => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb) ) ) ).

% set_n_deg_not_0
tff(fact_2129_Suc__n__div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_2130_div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ).

% div_2_gt_zero
tff(fact_2131_div__exp__eq,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ).

% div_exp_eq
tff(fact_2132_div__exp__eq,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Mb))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ).

% div_exp_eq
tff(fact_2133_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_2134_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_2135_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_2136_bits__div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_2137_div__by__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),one_one_complex) = A ).

% div_by_1
tff(fact_2138_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_2139_div__by__1,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),one_one_int) = A ).

% div_by_1
tff(fact_2140_div__by__1,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),one_one_real) = A ).

% div_by_1
tff(fact_2141_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(divide_divide_real(one_one_real),A) )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_2142_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(divide_divide_real(one_one_real),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_2143_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real = aa_real_real(divide_divide_real(B),A) )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_2144_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(divide_divide_real(B),A) = one_one_real )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_2145_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_2146_divide__self__if,axiom,
    ! [A: real] :
      aa_real_real(divide_divide_real(A),A) = $ite(A = zero_zero_real,zero_zero_real,one_one_real) ).

% divide_self_if
tff(fact_2147_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_2148_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),A) = one_one_real ) ) ).

% divide_self
tff(fact_2149_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_2150_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real = aa_real_real(divide_divide_real(A),B) )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2151_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_2152_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_2153_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(A),A) = one_one_int ) ) ).

% div_self
tff(fact_2154_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),A) = one_one_real ) ) ).

% div_self
tff(fact_2155_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_2156_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),B) = one_one_real )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2157_power__inject__exp,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_2158_power__inject__exp,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_nat_real(power_power_real(A),Mb) = aa_nat_real(power_power_real(A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_2159_power__inject__exp,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( ( aa_nat_int(power_power_int(A),Mb) = aa_nat_int(power_power_int(A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_2160_max__0__1_I1_J,axiom,
    aa_real_real(ord_max_real(zero_zero_real),one_one_real) = one_one_real ).

% max_0_1(1)
tff(fact_2161_max__0__1_I1_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),one_one_nat) = one_one_nat ).

% max_0_1(1)
tff(fact_2162_max__0__1_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),one_one_int) = one_one_int ).

% max_0_1(1)
tff(fact_2163_max__0__1_I1_J,axiom,
    aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(1)
tff(fact_2164_max__0__1_I2_J,axiom,
    aa_real_real(ord_max_real(one_one_real),zero_zero_real) = one_one_real ).

% max_0_1(2)
tff(fact_2165_max__0__1_I2_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),zero_zero_nat) = one_one_nat ).

% max_0_1(2)
tff(fact_2166_max__0__1_I2_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),zero_zero_int) = one_one_int ).

% max_0_1(2)
tff(fact_2167_max__0__1_I2_J,axiom,
    aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(one_on7984719198319812577d_enat),zero_z5237406670263579293d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(2)
tff(fact_2168_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(5)
tff(fact_2169_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(Xb)) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(5)
tff(fact_2170_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),numeral_numeral_int(Xb)) = numeral_numeral_int(Xb) ).

% max_0_1(5)
tff(fact_2171_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_real_real(ord_max_real(one_one_real),numeral_numeral_real(Xb)) = numeral_numeral_real(Xb) ).

% max_0_1(5)
tff(fact_2172_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xb)),one_one_nat) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(6)
tff(fact_2173_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(Xb)),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(6)
tff(fact_2174_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,numeral_numeral_int(Xb)),one_one_int) = numeral_numeral_int(Xb) ).

% max_0_1(6)
tff(fact_2175_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_real_real(ord_max_real(numeral_numeral_real(Xb)),one_one_real) = numeral_numeral_real(Xb) ).

% max_0_1(6)
tff(fact_2176_less__one,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),one_one_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% less_one
tff(fact_2177_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_divide_1_iff
tff(fact_2178_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% divide_le_0_1_iff
tff(fact_2179_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(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_2180_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(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_2181_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(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_2182_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(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_2183_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(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_2184_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_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_2185_power__strict__increasing__iff,axiom,
    ! [B: nat,Xb: 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),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_2186_power__strict__increasing__iff,axiom,
    ! [B: real,Xb: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(B),Xb)),aa_nat_real(power_power_real(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_2187_power__strict__increasing__iff,axiom,
    ! [B: int,Xb: 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),Xb)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_2188_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_2189_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_2190_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_2191_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_2192_one__add__one,axiom,
    aa_complex_complex(plus_plus_complex(one_one_complex),one_one_complex) = numera6690914467698888265omplex(bit0(one)) ).

% one_add_one
tff(fact_2193_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,bit0(one)) ).

% one_add_one
tff(fact_2194_one__add__one,axiom,
    aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(bit0(one)) ).

% one_add_one
tff(fact_2195_one__add__one,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int) = numeral_numeral_int(bit0(one)) ).

% one_add_one
tff(fact_2196_one__add__one,axiom,
    aa_real_real(plus_plus_real(one_one_real),one_one_real) = numeral_numeral_real(bit0(one)) ).

% one_add_one
tff(fact_2197_power__strict__decreasing__iff,axiom,
    ! [B: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2198_power__strict__decreasing__iff,axiom,
    ! [B: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(B),Mb)),aa_nat_real(power_power_real(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2199_power__strict__decreasing__iff,axiom,
    ! [B: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),Mb)),aa_nat_int(power_power_int(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2200_power__increasing__iff,axiom,
    ! [B: real,Xb: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(B),Xb)),aa_nat_real(power_power_real(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_2201_power__increasing__iff,axiom,
    ! [B: nat,Xb: 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),Xb)),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,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_2202_power__increasing__iff,axiom,
    ! [B: int,Xb: 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),Xb)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_2203_Suc__1,axiom,
    aa_nat_nat(suc,one_one_nat) = aa_num_nat(numeral_numeral_nat,bit0(one)) ).

% Suc_1
tff(fact_2204_numeral__plus__one,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),one_one_complex) = numera6690914467698888265omplex(plus_plus_num(Nb,one)) ).

% numeral_plus_one
tff(fact_2205_numeral__plus__one,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) = aa_num_nat(numeral_numeral_nat,plus_plus_num(Nb,one)) ).

% numeral_plus_one
tff(fact_2206_numeral__plus__one,axiom,
    ! [Nb: num] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Nb)),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(plus_plus_num(Nb,one)) ).

% numeral_plus_one
tff(fact_2207_numeral__plus__one,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(Nb)),one_one_int) = numeral_numeral_int(plus_plus_num(Nb,one)) ).

% numeral_plus_one
tff(fact_2208_numeral__plus__one,axiom,
    ! [Nb: num] : aa_real_real(plus_plus_real(numeral_numeral_real(Nb)),one_one_real) = numeral_numeral_real(plus_plus_num(Nb,one)) ).

% numeral_plus_one
tff(fact_2209_one__plus__numeral,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(plus_plus_num(one,Nb)) ).

% one_plus_numeral
tff(fact_2210_one__plus__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,plus_plus_num(one,Nb)) ).

% one_plus_numeral
tff(fact_2211_one__plus__numeral,axiom,
    ! [Nb: num] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(Nb)) = numera1916890842035813515d_enat(plus_plus_num(one,Nb)) ).

% one_plus_numeral
tff(fact_2212_one__plus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),numeral_numeral_int(Nb)) = numeral_numeral_int(plus_plus_num(one,Nb)) ).

% one_plus_numeral
tff(fact_2213_one__plus__numeral,axiom,
    ! [Nb: num] : aa_real_real(plus_plus_real(one_one_real),numeral_numeral_real(Nb)) = numeral_numeral_real(plus_plus_num(one,Nb)) ).

% one_plus_numeral
tff(fact_2214_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(Nb)),one_on7984719198319812577d_enat)
    <=> aa_num_o(ord_less_eq_num(Nb),one) ) ).

% numeral_le_one_iff
tff(fact_2215_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Nb)),one_one_real)
    <=> aa_num_o(ord_less_eq_num(Nb),one) ) ).

% numeral_le_one_iff
tff(fact_2216_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat)
    <=> aa_num_o(ord_less_eq_num(Nb),one) ) ).

% numeral_le_one_iff
tff(fact_2217_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(Nb)),one_one_int)
    <=> aa_num_o(ord_less_eq_num(Nb),one) ) ).

% numeral_le_one_iff
tff(fact_2218_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(ord_less_num(one),Nb) ) ).

% one_less_numeral_iff
tff(fact_2219_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,one_on7984719198319812577d_enat),numera1916890842035813515d_enat(Nb))
    <=> aa_num_o(ord_less_num(one),Nb) ) ).

% one_less_numeral_iff
tff(fact_2220_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),numeral_numeral_int(Nb))
    <=> aa_num_o(ord_less_num(one),Nb) ) ).

% one_less_numeral_iff
tff(fact_2221_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),numeral_numeral_real(Nb))
    <=> aa_num_o(ord_less_num(one),Nb) ) ).

% one_less_numeral_iff
tff(fact_2222_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,bit0(one))) = zero_zero_nat ).

% one_div_two_eq_zero
tff(fact_2223_one__div__two__eq__zero,axiom,
    aa_int_int(divide_divide_int(one_one_int),numeral_numeral_int(bit0(one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_2224_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,bit0(one))) = zero_zero_nat ).

% bits_1_div_2
tff(fact_2225_bits__1__div__2,axiom,
    aa_int_int(divide_divide_int(one_one_int),numeral_numeral_int(bit0(one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_2226_power__decreasing__iff,axiom,
    ! [B: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(B),Mb)),aa_nat_real(power_power_real(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2227_power__decreasing__iff,axiom,
    ! [B: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2228_power__decreasing__iff,axiom,
    ! [B: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),Mb)),aa_nat_int(power_power_int(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2229_one__reorient,axiom,
    ! [Xb: nat] :
      ( ( one_one_nat = Xb )
    <=> ( Xb = one_one_nat ) ) ).

% one_reorient
tff(fact_2230_one__reorient,axiom,
    ! [Xb: int] :
      ( ( one_one_int = Xb )
    <=> ( Xb = one_one_int ) ) ).

% one_reorient
tff(fact_2231_one__reorient,axiom,
    ! [Xb: real] :
      ( ( one_one_real = Xb )
    <=> ( Xb = one_one_real ) ) ).

% one_reorient
tff(fact_2232_one__reorient,axiom,
    ! [Xb: complex] :
      ( ( one_one_complex = Xb )
    <=> ( Xb = one_one_complex ) ) ).

% one_reorient
tff(fact_2233_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_2234_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_2235_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_2236_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_2237_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_2238_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_2239_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_2240_zero__neq__one,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_neq_one
tff(fact_2241_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_2242_less__numeral__extra_I4_J,axiom,
    ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,one_on7984719198319812577d_enat),one_on7984719198319812577d_enat) ).

% less_numeral_extra(4)
tff(fact_2243_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_2244_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_2245_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_2246_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_2247_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_2248_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_2249_not__one__le__zero,axiom,
    ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,one_on7984719198319812577d_enat),zero_z5237406670263579293d_enat) ).

% not_one_le_zero
tff(fact_2250_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_2251_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_2252_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_2253_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2254_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_2255_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_2256_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_2257_zero__less__one__class_Ozero__le__one,axiom,
    aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat) ).

% zero_less_one_class.zero_le_one
tff(fact_2258_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_2259_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_2260_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_2261_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_2262_not__one__less__zero,axiom,
    ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,one_on7984719198319812577d_enat),zero_z5237406670263579293d_enat) ).

% not_one_less_zero
tff(fact_2263_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_2264_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_2265_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_2266_zero__less__one,axiom,
    aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat) ).

% zero_less_one
tff(fact_2267_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_2268_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_2269_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_2270_less__numeral__extra_I1_J,axiom,
    aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat) ).

% less_numeral_extra(1)
tff(fact_2271_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_2272_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_2273_one__le__numeral,axiom,
    ! [Nb: num] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,one_on7984719198319812577d_enat),numera1916890842035813515d_enat(Nb)) ).

% one_le_numeral
tff(fact_2274_one__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),numeral_numeral_real(Nb)) ).

% one_le_numeral
tff(fact_2275_one__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% one_le_numeral
tff(fact_2276_one__le__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),numeral_numeral_int(Nb)) ).

% one_le_numeral
tff(fact_2277_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) ).

% not_numeral_less_one
tff(fact_2278_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,numera1916890842035813515d_enat(Nb)),one_on7984719198319812577d_enat) ).

% not_numeral_less_one
tff(fact_2279_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,numeral_numeral_int(Nb)),one_one_int) ).

% not_numeral_less_one
tff(fact_2280_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Nb)),one_one_real) ).

% not_numeral_less_one
tff(fact_2281_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_2282_less__add__one,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(plus_plus_real(A),one_one_real)) ).

% less_add_one
tff(fact_2283_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_2284_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_2285_add__mono1,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),one_on7984719198319812577d_enat)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B),one_on7984719198319812577d_enat)) ) ).

% add_mono1
tff(fact_2286_add__mono1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),one_one_real)),aa_real_real(plus_plus_real(B),one_one_real)) ) ).

% add_mono1
tff(fact_2287_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_2288_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_2289_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(divide_divide_real(A),B) = one_one_real )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2290_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Xb)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Xb)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_2291_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Xb)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_2292_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(Xb)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Xb)),one_on7984719198319812577d_enat) ).

% one_plus_numeral_commute
tff(fact_2293_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),numeral_numeral_int(Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(Xb)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_2294_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_real_real(plus_plus_real(one_one_real),numeral_numeral_real(Xb)) = aa_real_real(plus_plus_real(numeral_numeral_real(Xb)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_2295_one__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(power_power_real(A),Nb)) ) ).

% one_le_power
tff(fact_2296_one__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% one_le_power
tff(fact_2297_one__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),Nb)) ) ).

% one_le_power
tff(fact_2298_div__eq__dividend__iff,axiom,
    ! [Mb: nat,Nb: 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),Nb) = Mb )
      <=> ( Nb = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_2299_div__less__dividend,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Mb) ) ) ).

% div_less_dividend
tff(fact_2300_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_2301_power__0,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_2302_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_2303_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_2304_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_2305_Suc__eq__plus1__left,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),Nb) ).

% Suc_eq_plus1_left
tff(fact_2306_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_2307_Suc__eq__plus1,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat) ).

% Suc_eq_plus1
tff(fact_2308_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_2309_zero__less__two,axiom,
    aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat),one_on7984719198319812577d_enat)) ).

% zero_less_two
tff(fact_2310_zero__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(one_one_real),one_one_real)) ).

% zero_less_two
tff(fact_2311_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_2312_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(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_2313_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_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_2314_power__le__one,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),Nb)),one_one_real) ) ) ).

% power_le_one
tff(fact_2315_power__le__one,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),one_one_nat) ) ) ).

% power_le_one
tff(fact_2316_power__le__one,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),one_one_int) ) ) ).

% power_le_one
tff(fact_2317_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B)),aa_real_real(plus_plus_real(one_one_real),one_one_real))),B) ) ).

% gt_half_sum
tff(fact_2318_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B)),aa_real_real(plus_plus_real(one_one_real),one_one_real))) ) ).

% less_half_sum
tff(fact_2319_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_Extended_enat(power_8040749407984259932d_enat(zero_z5237406670263579293d_enat),Nb) = $ite(Nb = zero_zero_nat,one_on7984719198319812577d_enat,zero_z5237406670263579293d_enat) ).

% power_0_left
tff(fact_2320_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% power_0_left
tff(fact_2321_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_real(power_power_real(zero_zero_real),Nb) = $ite(Nb = zero_zero_nat,one_one_real,zero_zero_real) ).

% power_0_left
tff(fact_2322_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_int(power_power_int(zero_zero_int),Nb) = $ite(Nb = zero_zero_nat,one_one_int,zero_zero_int) ).

% power_0_left
tff(fact_2323_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_complex(power_power_complex(zero_zero_complex),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% power_0_left
tff(fact_2324_power__gt1,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_2325_power__gt1,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(power_power_real(A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_2326_power__gt1,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_2327_power__strict__increasing,axiom,
    ! [Nb: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)) ) ) ).

% power_strict_increasing
tff(fact_2328_power__strict__increasing,axiom,
    ! [Nb: nat,N4: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),Nb)),aa_nat_real(power_power_real(A),N4)) ) ) ).

% power_strict_increasing
tff(fact_2329_power__strict__increasing,axiom,
    ! [Nb: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(A),N4)) ) ) ).

% power_strict_increasing
tff(fact_2330_power__less__imp__less__exp,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_2331_power__less__imp__less__exp,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),Mb)),aa_nat_real(power_power_real(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_2332_power__less__imp__less__exp,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_2333_power__increasing,axiom,
    ! [Nb: nat,N4: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),Nb)),aa_nat_real(power_power_real(A),N4)) ) ) ).

% power_increasing
tff(fact_2334_power__increasing,axiom,
    ! [Nb: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)) ) ) ).

% power_increasing
tff(fact_2335_power__increasing,axiom,
    ! [Nb: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(A),N4)) ) ) ).

% power_increasing
tff(fact_2336_nat__induct__non__zero,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_non_zero
tff(fact_2337_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_2338_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_2339_power__Suc__le__self,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_2340_power__Suc__le__self,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_2341_power__Suc__le__self,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_2342_power__Suc__less__one,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_2343_power__Suc__less__one,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,Nb))),one_one_real) ) ) ).

% power_Suc_less_one
tff(fact_2344_power__Suc__less__one,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_2345_power__strict__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2346_power__strict__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),N4)),aa_nat_real(power_power_real(A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2347_power__strict__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),N4)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2348_power__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),N4)),aa_nat_real(power_power_real(A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2349_power__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2350_power__decreasing,axiom,
    ! [Nb: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),N4)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2351_power__le__imp__le__exp,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),Mb)),aa_nat_real(power_power_real(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_2352_power__le__imp__le__exp,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_2353_power__le__imp__le__exp,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_2354_self__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(power_power_real(A),Nb)) ) ) ).

% self_le_power
tff(fact_2355_self__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% self_le_power
tff(fact_2356_self__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(A),Nb)) ) ) ).

% self_le_power
tff(fact_2357_one__less__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% one_less_power
tff(fact_2358_one__less__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(power_power_real(A),Nb)) ) ) ).

% one_less_power
tff(fact_2359_one__less__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),Nb)) ) ) ).

% one_less_power
tff(fact_2360_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,bit0(one)) ).

% nat_1_add_1
tff(fact_2361_div__le__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Mb) ).

% div_le_dividend
tff(fact_2362_div__le__mono,axiom,
    ! [Mb: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),K)) ) ).

% div_le_mono
tff(fact_2363_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,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_2364_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,bit0(one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,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_2365_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Nb = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_2366_Suc__div__le__mono,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Mb)),Nb)) ).

% Suc_div_le_mono
tff(fact_2367_div__greater__zero__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% div_greater_zero_iff
tff(fact_2368_div__le__mono2,axiom,
    ! [Mb: nat,Nb: 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),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Mb)) ) ) ).

% div_le_mono2
tff(fact_2369_exp__add__not__zero__imp__left,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2370_exp__add__not__zero__imp__left,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Mb) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2371_exp__add__not__zero__imp__right,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2372_exp__add__not__zero__imp__right,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2373_nat__induct2,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct2
tff(fact_2374_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,bit0(one))),D))),L) ).

% bit_concat_def
tff(fact_2375_low__inv,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))),Xb)),Nb) = Xb ) ) ).

% low_inv
tff(fact_2376_high__inv,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))),Xb)),Nb) = Y ) ) ).

% high_inv
tff(fact_2377_enat__ord__number_I1_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(Mb)),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(1)
tff(fact_2378_enat__ord__number_I2_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,numera1916890842035813515d_enat(Mb)),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(2)
tff(fact_2379_pos2,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% pos2
tff(fact_2380_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_2381_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_2382_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_2383_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_2384_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_2385_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_2386_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_2387_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_2388_mult__zero__left,axiom,
    ! [A: real] : aa_real_real(times_times_real(zero_zero_real),A) = zero_zero_real ).

% mult_zero_left
tff(fact_2389_mult__zero__left,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A) = zero_zero_complex ).

% mult_zero_left
tff(fact_2390_mult__zero__left,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(zero_z5237406670263579293d_enat),A) = zero_z5237406670263579293d_enat ).

% mult_zero_left
tff(fact_2391_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_2392_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_2393_mult__zero__right,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),zero_zero_real) = zero_zero_real ).

% mult_zero_right
tff(fact_2394_mult__zero__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_2395_mult__zero__right,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),zero_z5237406670263579293d_enat) = zero_z5237406670263579293d_enat ).

% mult_zero_right
tff(fact_2396_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_2397_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_2398_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(times_times_real(A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_2399_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_2400_mult__eq__0__iff,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B) = zero_z5237406670263579293d_enat )
    <=> ( ( A = zero_z5237406670263579293d_enat )
        | ( B = zero_z5237406670263579293d_enat ) ) ) ).

% mult_eq_0_iff
tff(fact_2401_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_2402_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_2403_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(times_times_real(C),A) = aa_real_real(times_times_real(C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2404_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_2405_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_2406_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_2407_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(times_times_real(A),C) = aa_real_real(times_times_real(B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2408_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_2409_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_2410_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_2411_mult__1,axiom,
    ! [A: real] : aa_real_real(times_times_real(one_one_real),A) = A ).

% mult_1
tff(fact_2412_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_2413_mult__1,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(one_on7984719198319812577d_enat),A) = A ).

% mult_1
tff(fact_2414_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_2415_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_2416_mult_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),one_one_real) = A ).

% mult.right_neutral
tff(fact_2417_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_2418_mult_Oright__neutral,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),one_on7984719198319812577d_enat) = A ).

% mult.right_neutral
tff(fact_2419_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_2420_times__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(divide_divide_real(B),C)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),B)),C) ).

% times_divide_eq_right
tff(fact_2421_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_2422_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(A),aa_real_real(divide_divide_real(B),C)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),C)),B) ).

% divide_divide_eq_right
tff(fact_2423_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_2424_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(A),B)),C) = aa_real_real(divide_divide_real(A),aa_real_real(times_times_real(B),C)) ).

% divide_divide_eq_left
tff(fact_2425_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_2426_times__divide__eq__left,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(times_times_real(aa_real_real(divide_divide_real(B),C)),A) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(B),A)),C) ).

% times_divide_eq_left
tff(fact_2427_mult__is__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = zero_zero_nat )
    <=> ( ( Mb = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_2428_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_2429_mult__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: 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),Nb) )
    <=> ( ( Mb = Nb )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_2430_mult__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: 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,Nb),K) )
    <=> ( ( Mb = Nb )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_2431_nat__mult__eq__1__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = one_one_nat )
    <=> ( ( Mb = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_2432_nat__1__eq__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
    <=> ( ( Mb = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_2433_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_2434_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(times_times_real(C),B) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_2435_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_2436_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_2437_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( aa_real_real(times_times_real(C),A) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_2438_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_2439_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_2440_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(times_times_real(B),C) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_2441_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_2442_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_2443_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( aa_real_real(times_times_real(A),C) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_2444_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_2445_sum__squares__eq__zero__iff,axiom,
    ! [Xb: 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,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2446_sum__squares__eq__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Y),Y)) = zero_zero_real )
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2447_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_2448_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) = $ite(C = zero_zero_real,zero_zero_real,aa_real_real(divide_divide_real(A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_2449_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_2450_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2451_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_2452_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_2453_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_2454_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2455_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_2456_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(B),C)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2457_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_2458_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2459_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_2460_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_2461_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_2462_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2463_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_2464_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(C),B)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2465_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_2466_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_2467_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_2468_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_2469_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_2470_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_2471_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_2472_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_2473_distrib__left__numeral,axiom,
    ! [V: num,B: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(V)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B),C)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(V)),B)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(V)),C)) ).

% distrib_left_numeral
tff(fact_2474_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_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,numeral_numeral_int(V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V)),C)) ).

% distrib_left_numeral
tff(fact_2475_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(times_times_real(numeral_numeral_real(V)),aa_real_real(plus_plus_real(B),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(numeral_numeral_real(V)),B)),aa_real_real(times_times_real(numeral_numeral_real(V)),C)) ).

% distrib_left_numeral
tff(fact_2476_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_2477_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_2478_distrib__right__numeral,axiom,
    ! [A: extended_enat,B: extended_enat,V: num] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)),numera1916890842035813515d_enat(V)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),numera1916890842035813515d_enat(V))),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B),numera1916890842035813515d_enat(V))) ).

% distrib_right_numeral
tff(fact_2479_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)),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),numeral_numeral_int(V))),aa_int_int(aa_int_fun_int_int(times_times_int,B),numeral_numeral_int(V))) ).

% distrib_right_numeral
tff(fact_2480_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),numeral_numeral_real(V)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),numeral_numeral_real(V))),aa_real_real(times_times_real(B),numeral_numeral_real(V))) ).

% distrib_right_numeral
tff(fact_2481_one__eq__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
    <=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_2482_mult__eq__1__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_2483_mult__less__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% mult_less_cancel2
tff(fact_2484_nat__0__less__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% nat_0_less_mult_iff
tff(fact_2485_mult__Suc__right,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ).

% mult_Suc_right
tff(fact_2486_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(divide_divide_real(B),numeral_numeral_real(W2)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),numeral_numeral_real(W2))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2487_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(divide_divide_real(B),numeral_numeral_real(W2))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),numeral_numeral_real(W2))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2488_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_2489_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A = aa_real_real(divide_divide_real(B),numeral_numeral_real(W2)) )
    <=> $ite(numeral_numeral_real(W2) != zero_zero_real,aa_real_real(times_times_real(A),numeral_numeral_real(W2)) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2490_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_2491_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( aa_real_real(divide_divide_real(B),numeral_numeral_real(W2)) = A )
    <=> $ite(numeral_numeral_real(W2) != zero_zero_real,B = aa_real_real(times_times_real(A),numeral_numeral_real(W2)),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2492_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(divide_divide_real(B),numeral_numeral_real(W2))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),numeral_numeral_real(W2))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_2493_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(divide_divide_real(B),numeral_numeral_real(W2)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),numeral_numeral_real(W2))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_2494_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_2495_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(divide_divide_real(B),aa_real_real(times_times_real(A),B)) = aa_real_real(divide_divide_real(one_one_real),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2496_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_2497_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),aa_real_real(times_times_real(A),B)) = aa_real_real(divide_divide_real(one_one_real),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2498_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_2499_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_2500_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_2501_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_2502_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_2503_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_2504_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_2505_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_2506_one__le__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) ) ) ).

% one_le_mult_iff
tff(fact_2507_mult__le__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% mult_le_cancel2
tff(fact_2508_div__mult__self__is__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Nb) = Mb ) ) ).

% div_mult_self_is_m
tff(fact_2509_div__mult__self1__is__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)),Nb) = Mb ) ) ).

% div_mult_self1_is_m
tff(fact_2510_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_2511_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_2512_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(times_times_real(B),aa_real_real(times_times_real(A),C)) = aa_real_real(times_times_real(A),aa_real_real(times_times_real(B),C)) ).

% mult.left_commute
tff(fact_2513_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_2514_mult_Oleft__commute,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),C)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B),C)) ).

% mult.left_commute
tff(fact_2515_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_2516_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_2517_mult_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(A),B) = aa_real_real(times_times_real(B),A) ).

% mult.commute
tff(fact_2518_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_2519_mult_Ocommute,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B),A) ).

% mult.commute
tff(fact_2520_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_2521_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_2522_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(times_times_real(A),B)),C) = aa_real_real(times_times_real(A),aa_real_real(times_times_real(B),C)) ).

% mult.assoc
tff(fact_2523_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_2524_mult_Oassoc,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B)),C) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B),C)) ).

% mult.assoc
tff(fact_2525_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_2526_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_2527_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(times_times_real(A),B)),C) = aa_real_real(times_times_real(A),aa_real_real(times_times_real(B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2528_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_2529_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B)),C) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2530_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_2531_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_2532_mult__not__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(times_times_real(A),B) != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_2533_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_2534_mult__not__zero,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B) != zero_z5237406670263579293d_enat )
     => ( ( A != zero_z5237406670263579293d_enat )
        & ( B != zero_z5237406670263579293d_enat ) ) ) ).

% mult_not_zero
tff(fact_2535_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_2536_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_2537_divisors__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(times_times_real(A),B) = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_2538_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_2539_divisors__zero,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B) = zero_z5237406670263579293d_enat )
     => ( ( A = zero_z5237406670263579293d_enat )
        | ( B = zero_z5237406670263579293d_enat ) ) ) ).

% divisors_zero
tff(fact_2540_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_2541_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_2542_no__zero__divisors,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(times_times_real(A),B) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_2543_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_2544_no__zero__divisors,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A != zero_z5237406670263579293d_enat )
     => ( ( B != zero_z5237406670263579293d_enat )
       => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B) != zero_z5237406670263579293d_enat ) ) ) ).

% no_zero_divisors
tff(fact_2545_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_2546_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_2547_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(C),A) = aa_real_real(times_times_real(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2548_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_2549_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_2550_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_2551_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(A),C) = aa_real_real(times_times_real(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2552_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_2553_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_2554_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_2555_mult_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),one_one_real) = A ).

% mult.comm_neutral
tff(fact_2556_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_2557_mult_Ocomm__neutral,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),one_on7984719198319812577d_enat) = A ).

% mult.comm_neutral
tff(fact_2558_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_2559_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_2560_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] : aa_real_real(times_times_real(one_one_real),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2561_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_2562_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(one_on7984719198319812577d_enat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2563_crossproduct__eq,axiom,
    ! [W2: nat,Y: nat,Xb: 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,Xb),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,Xb),Y)) )
    <=> ( ( W2 = Xb )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2564_crossproduct__eq,axiom,
    ! [W2: int,Y: int,Xb: 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,Xb),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,Xb),Y)) )
    <=> ( ( W2 = Xb )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2565_crossproduct__eq,axiom,
    ! [W2: real,Y: real,Xb: real,Z: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(W2),Y)),aa_real_real(times_times_real(Xb),Z)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(W2),Z)),aa_real_real(times_times_real(Xb),Y)) )
    <=> ( ( W2 = Xb )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2566_crossproduct__eq,axiom,
    ! [W2: complex,Y: complex,Xb: complex,Z: complex] :
      ( ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(W2),Y)),aa_complex_complex(times_times_complex(Xb),Z)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(W2),Z)),aa_complex_complex(times_times_complex(Xb),Y)) )
    <=> ( ( W2 = Xb )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2567_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_2568_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_2569_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) != aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),D)),aa_real_real(times_times_real(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2570_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_2571_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_2572_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2573_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_2574_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_2575_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(plus_plus_real(B),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),B)),aa_real_real(times_times_real(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2576_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_2577_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_2578_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_2579_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ).

% comm_semiring_class.distrib
tff(fact_2580_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_2581_comm__semiring__class_Odistrib,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)),C) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),C)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B),C)) ).

% comm_semiring_class.distrib
tff(fact_2582_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_2583_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_2584_distrib__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(plus_plus_real(B),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),B)),aa_real_real(times_times_real(A),C)) ).

% distrib_left
tff(fact_2585_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_2586_distrib__left,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B),C)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),C)) ).

% distrib_left
tff(fact_2587_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_2588_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_2589_distrib__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ).

% distrib_right
tff(fact_2590_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_2591_distrib__right,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)),C) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),C)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B),C)) ).

% distrib_right
tff(fact_2592_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_2593_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_2594_combine__common__factor,axiom,
    ! [A: real,E2: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E2)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E2)),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2595_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_2596_combine__common__factor,axiom,
    ! [A: extended_enat,E2: extended_enat,B: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),E2)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B),E2)),C)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2597_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_2598_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(A),B)),C) = aa_real_real(divide_divide_real(A),aa_real_real(times_times_real(C),B)) ).

% divide_divide_eq_left'
tff(fact_2599_divide__divide__times__eq,axiom,
    ! [Xb: complex,Y: complex,Z: complex,W2: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(Xb),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(Xb),W2)),aa_complex_complex(times_times_complex(Y),Z)) ).

% divide_divide_times_eq
tff(fact_2600_divide__divide__times__eq,axiom,
    ! [Xb: real,Y: real,Z: real,W2: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(Xb),Y)),aa_real_real(divide_divide_real(Z),W2)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(Xb),W2)),aa_real_real(times_times_real(Y),Z)) ).

% divide_divide_times_eq
tff(fact_2601_times__divide__times__eq,axiom,
    ! [Xb: complex,Y: complex,Z: complex,W2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(Xb),Z)),aa_complex_complex(times_times_complex(Y),W2)) ).

% times_divide_times_eq
tff(fact_2602_times__divide__times__eq,axiom,
    ! [Xb: real,Y: real,Z: real,W2: real] : aa_real_real(times_times_real(aa_real_real(divide_divide_real(Xb),Y)),aa_real_real(divide_divide_real(Z),W2)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(Y),W2)) ).

% times_divide_times_eq
tff(fact_2603_Suc__mult__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: 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)),Nb) )
    <=> ( Mb = Nb ) ) ).

% Suc_mult_cancel1
tff(fact_2604_mult__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% mult_0
tff(fact_2605_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_2606_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_2607_mult__le__mono,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),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,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L)) ) ) ).

% mult_le_mono
tff(fact_2608_mult__le__mono1,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ).

% mult_le_mono1
tff(fact_2609_mult__le__mono2,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),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),Ia)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ).

% mult_le_mono2
tff(fact_2610_add__mult__distrib,axiom,
    ! [Mb: nat,Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),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,Nb),K)) ).

% add_mult_distrib
tff(fact_2611_add__mult__distrib2,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = 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),Nb)) ).

% add_mult_distrib2
tff(fact_2612_nat__mult__1__right,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),one_one_nat) = Nb ).

% nat_mult_1_right
tff(fact_2613_nat__mult__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Nb) = Nb ).

% nat_mult_1
tff(fact_2614_nat__mult__max__left,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ).

% nat_mult_max_left
tff(fact_2615_nat__mult__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q2)) ).

% nat_mult_max_right
tff(fact_2616_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_2617_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_2618_mult__mono,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),D)
       => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),B)
         => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),C)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),C)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2619_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2620_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_2621_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_2622_mult__mono_H,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),D)
       => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),A)
         => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),C)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),C)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2623_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2624_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_2625_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_2626_zero__le__square,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),A)) ).

% zero_le_square
tff(fact_2627_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_2628_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),B)) ) ).

% split_mult_pos_le
tff(fact_2629_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_2630_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2631_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_2632_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2633_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_2634_mult__left__mono,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),C)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(C),A)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(C),B)) ) ) ).

% mult_left_mono
tff(fact_2635_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_left_mono
tff(fact_2636_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_2637_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_2638_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2639_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_2640_mult__right__mono,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),C)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),C)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B),C)) ) ) ).

% mult_right_mono
tff(fact_2641_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_right_mono
tff(fact_2642_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_2643_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_2644_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% mult_le_0_iff
tff(fact_2645_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_2646_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),zero_zero_real) ) ).

% split_mult_neg_le
tff(fact_2647_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_2648_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_2649_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2650_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_2651_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_2652_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos
tff(fact_2653_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_2654_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_2655_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),zero_zero_real) ) ) ).

% mult_nonpos_nonneg
tff(fact_2656_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_2657_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_2658_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(B),A)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2659_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_2660_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_2661_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_mult_iff
tff(fact_2662_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_2663_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),C)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(C),A)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2664_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2665_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_2666_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_2667_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(times_times_real(A),B)) ) ) ).

% mult_neg_neg
tff(fact_2668_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_2669_not__square__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),A)),zero_zero_real) ).

% not_square_less_zero
tff(fact_2670_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_2671_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_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_2672_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_2673_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_2674_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(times_times_real(A),B)),zero_zero_real) ) ) ).

% mult_neg_pos
tff(fact_2675_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_2676_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_2677_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(times_times_real(A),B)),zero_zero_real) ) ) ).

% mult_pos_neg
tff(fact_2678_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_2679_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_2680_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(times_times_real(A),B)) ) ) ).

% mult_pos_pos
tff(fact_2681_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_2682_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_2683_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(times_times_real(B),A)),zero_zero_real) ) ) ).

% mult_pos_neg2
tff(fact_2684_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_2685_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(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_2686_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_2687_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_2688_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(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_2689_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_2690_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_2691_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(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_2692_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_2693_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(times_times_real(C),A)),aa_real_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_2694_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_2695_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(times_times_real(C),A)),aa_real_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_2696_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_2697_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(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2698_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_2699_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_2700_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(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2701_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_2702_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(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
          & aa_real_o(aa_real_fun_real_o(ord_less_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_2703_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_2704_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(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2705_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_2706_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_2707_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(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2708_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_2709_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(times_times_real(A),C)),aa_real_real(times_times_real(B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
          & aa_real_o(aa_real_fun_real_o(ord_less_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_2710_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_2711_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_2712_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(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2713_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_2714_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_2715_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_2716_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_real_real(plus_plus_real(A),aa_real_real(times_times_real(R2),C)) != aa_real_real(plus_plus_real(B),aa_real_real(times_times_real(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2717_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_2718_less__1__mult,axiom,
    ! [Mb: nat,Nb: 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),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_2719_less__1__mult,axiom,
    ! [Mb: real,Nb: 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),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(times_times_real(Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_2720_less__1__mult,axiom,
    ! [Mb: int,Nb: 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),Nb)
       => 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),Nb)) ) ) ).

% less_1_mult
tff(fact_2721_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_2722_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( A = aa_real_real(divide_divide_real(B),C) )
      <=> ( aa_real_real(times_times_real(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2723_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_2724_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(divide_divide_real(B),C) = A )
      <=> ( B = aa_real_real(times_times_real(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2725_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_2726_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(A),C) = B )
       => ( A = aa_real_real(divide_divide_real(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2727_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_2728_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B = aa_real_real(times_times_real(A),C) )
       => ( aa_real_real(divide_divide_real(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2729_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_2730_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(divide_divide_real(B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(A),C) = B,A = zero_zero_real) ) ).

% eq_divide_eq
tff(fact_2731_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_2732_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(divide_divide_real(B),C) = A )
    <=> $ite(C != zero_zero_real,B = aa_real_real(times_times_real(A),C),A = zero_zero_real) ) ).

% divide_eq_eq
tff(fact_2733_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,Xb: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( aa_complex_complex(divide1717551699836669952omplex(Xb),Y) = aa_complex_complex(divide1717551699836669952omplex(W2),Z) )
        <=> ( aa_complex_complex(times_times_complex(Xb),Z) = aa_complex_complex(times_times_complex(W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2734_frac__eq__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(divide_divide_real(Xb),Y) = aa_real_real(divide_divide_real(W2),Z) )
        <=> ( aa_real_real(times_times_real(Xb),Z) = aa_real_real(times_times_real(W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2735_power__Suc2,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),A) ).

% power_Suc2
tff(fact_2736_power__Suc2,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Nb)),A) ).

% power_Suc2
tff(fact_2737_power__Suc2,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(aa_nat_real(power_power_real(A),Nb)),A) ).

% power_Suc2
tff(fact_2738_power__Suc2,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Nb)),A) ).

% power_Suc2
tff(fact_2739_power__Suc2,axiom,
    ! [A: extended_enat,Nb: nat] : aa_nat_Extended_enat(power_8040749407984259932d_enat(A),aa_nat_nat(suc,Nb)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(power_8040749407984259932d_enat(A),Nb)),A) ).

% power_Suc2
tff(fact_2740_power__Suc,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_Suc
tff(fact_2741_power__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb)) ).

% power_Suc
tff(fact_2742_power__Suc,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(A),aa_nat_real(power_power_real(A),Nb)) ).

% power_Suc
tff(fact_2743_power__Suc,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_Suc
tff(fact_2744_power__Suc,axiom,
    ! [A: extended_enat,Nb: nat] : aa_nat_Extended_enat(power_8040749407984259932d_enat(A),aa_nat_nat(suc,Nb)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),aa_nat_Extended_enat(power_8040749407984259932d_enat(A),Nb)) ).

% power_Suc
tff(fact_2745_Suc__mult__less__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_2746_power__add,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_add
tff(fact_2747_power__add,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ).

% power_add
tff(fact_2748_power__add,axiom,
    ! [A: real,Mb: nat,Nb: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_real_real(times_times_real(aa_nat_real(power_power_real(A),Mb)),aa_nat_real(power_power_real(A),Nb)) ).

% power_add
tff(fact_2749_power__add,axiom,
    ! [A: complex,Mb: nat,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_add
tff(fact_2750_power__add,axiom,
    ! [A: extended_enat,Mb: nat,Nb: nat] : aa_nat_Extended_enat(power_8040749407984259932d_enat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(power_8040749407984259932d_enat(A),Mb)),aa_nat_Extended_enat(power_8040749407984259932d_enat(A),Nb)) ).

% power_add
tff(fact_2751_mult__less__mono1,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),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,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ) ).

% mult_less_mono1
tff(fact_2752_mult__less__mono2,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),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),Ia)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ) ).

% mult_less_mono2
tff(fact_2753_Suc__mult__le__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_2754_mult__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ).

% mult_Suc
tff(fact_2755_mult__eq__self__implies__10,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
     => ( ( Nb = one_one_nat )
        | ( Mb = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_2756_less__mult__imp__div__less,axiom,
    ! [Mb: nat,Ia: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ia),Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Ia) ) ).

% less_mult_imp_div_less
tff(fact_2757_times__div__less__eq__dividend,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))),Mb) ).

% times_div_less_eq_dividend
tff(fact_2758_div__times__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb)),Mb) ).

% div_times_less_eq_dividend
tff(fact_2759_power__odd__eq,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% power_odd_eq
tff(fact_2760_power__odd__eq,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Nb)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% power_odd_eq
tff(fact_2761_power__odd__eq,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))) = aa_real_real(times_times_real(A),aa_nat_real(power_power_real(aa_nat_real(power_power_real(A),Nb)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% power_odd_eq
tff(fact_2762_power__odd__eq,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Nb)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% power_odd_eq
tff(fact_2763_power__odd__eq,axiom,
    ! [A: extended_enat,Nb: nat] : aa_nat_Extended_enat(power_8040749407984259932d_enat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),aa_nat_Extended_enat(power_8040749407984259932d_enat(aa_nat_Extended_enat(power_8040749407984259932d_enat(A),Nb)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% power_odd_eq
tff(fact_2764_double__not__eq__Suc__double,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) ).

% double_not_eq_Suc_double
tff(fact_2765_Suc__double__not__eq__double,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb) ).

% Suc_double_not_eq_double
tff(fact_2766_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2767_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_2768_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2769_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_2770_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2771_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_2772_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_2773_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2774_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_2775_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_2776_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2777_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_2778_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2779_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_2780_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_2781_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2782_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_2783_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_2784_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2785_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_2786_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2787_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_2788_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2789_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_2790_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2791_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_2792_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_2793_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2794_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_2795_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_2796_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2797_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_2798_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_2799_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2800_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_2801_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_2802_mult__left__le__one__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Y),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_2803_mult__left__le__one__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( 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),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_2804_mult__right__le__one__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Xb),Y)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_2805_mult__right__le__one__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( 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,Xb),Y)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_2806_mult__le__one,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),one_on7984719198319812577d_enat)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),B)
       => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),one_on7984719198319812577d_enat)
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B)),one_on7984719198319812577d_enat) ) ) ) ).

% mult_le_one
tff(fact_2807_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),one_one_real) ) ) ) ).

% mult_le_one
tff(fact_2808_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_2809_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_2810_mult__left__le,axiom,
    ! [C: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),one_on7984719198319812577d_enat)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),A)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),C)),A) ) ) ).

% mult_left_le
tff(fact_2811_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),A) ) ) ).

% mult_left_le
tff(fact_2812_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_2813_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_2814_sum__squares__le__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Y),Y))),zero_zero_real)
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2815_sum__squares__le__zero__iff,axiom,
    ! [Xb: 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,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2816_sum__squares__ge__zero,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2817_sum__squares__ge__zero,axiom,
    ! [Xb: 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,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2818_sum__squares__gt__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Y),Y)))
    <=> ( ( Xb != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2819_sum__squares__gt__zero__iff,axiom,
    ! [Xb: 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,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))
    <=> ( ( Xb != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2820_not__sum__squares__lt__zero,axiom,
    ! [Xb: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Y),Y))),zero_zero_real) ).

% not_sum_squares_lt_zero
tff(fact_2821_not__sum__squares__lt__zero,axiom,
    ! [Xb: 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,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_2822_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),C)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_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_2823_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(divide_divide_real(B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(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(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_2824_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(divide_divide_real(B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_2825_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(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_2826_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(divide_divide_real(B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_2827_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(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_2828_mult__imp__div__pos__less,axiom,
    ! [Y: real,Xb: 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,Xb),aa_real_real(times_times_real(Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xb),Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_2829_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,Xb: 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(times_times_real(Z),Y)),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(divide_divide_real(Xb),Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_2830_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(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_2831_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(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2832_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_2833_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( numeral_numeral_real(W2) = aa_real_real(divide_divide_real(B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(numeral_numeral_real(W2)),C) = B,numeral_numeral_real(W2) = zero_zero_real) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2834_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_2835_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( aa_real_real(divide_divide_real(B),C) = numeral_numeral_real(W2) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(times_times_real(numeral_numeral_real(W2)),C),numeral_numeral_real(W2) = zero_zero_real) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2836_divide__add__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2837_divide__add__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(Xb),Z)),Y) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2838_add__divide__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xb),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2839_add__divide__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(Y),Z)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2840_add__num__frac,axiom,
    ! [Y: complex,Z: complex,Xb: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2841_add__num__frac,axiom,
    ! [Y: real,Z: real,Xb: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(plus_plus_real(Z),aa_real_real(divide_divide_real(Xb),Y)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2842_add__frac__num,axiom,
    ! [Y: complex,Xb: complex,Z: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Y)),Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2843_add__frac__num,axiom,
    ! [Y: real,Xb: real,Z: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(Xb),Y)),Z) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2844_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,Xb: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Y)),aa_complex_complex(divide1717551699836669952omplex(W2),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xb),Z)),aa_complex_complex(times_times_complex(W2),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2845_add__frac__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(Xb),Y)),aa_real_real(divide_divide_real(W2),Z)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(W2),Y))),aa_real_real(times_times_real(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2846_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_2847_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(plus_plus_real(A),aa_real_real(divide_divide_real(B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2848_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_2849_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(A),Z)),B) = $ite(Z = zero_zero_real,B,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),aa_real_real(times_times_real(B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2850_power__less__power__Suc,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))) ) ).

% power_less_power_Suc
tff(fact_2851_power__less__power__Suc,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),Nb)),aa_real_real(times_times_real(A),aa_nat_real(power_power_real(A),Nb))) ) ).

% power_less_power_Suc
tff(fact_2852_power__less__power__Suc,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb))) ) ).

% power_less_power_Suc
tff(fact_2853_power__gt1__lemma,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))) ) ).

% power_gt1_lemma
tff(fact_2854_power__gt1__lemma,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(times_times_real(A),aa_nat_real(power_power_real(A),Nb))) ) ).

% power_gt1_lemma
tff(fact_2855_power__gt1__lemma,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb))) ) ).

% power_gt1_lemma
tff(fact_2856_one__less__mult,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).

% one_less_mult
tff(fact_2857_n__less__m__mult__n,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).

% n_less_m_mult_n
tff(fact_2858_n__less__n__mult__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)) ) ) ).

% n_less_n_mult_m
tff(fact_2859_div__less__iff__less__mult,axiom,
    ! [Q2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Q2)),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ) ) ).

% div_less_iff_less_mult
tff(fact_2860_realpow__pos__nth2,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ? [R3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
          & ( aa_nat_real(power_power_real(R3),aa_nat_nat(suc,Nb)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_2861_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(times_times_real(C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2862_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_2863_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2864_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_2865_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(times_times_real(B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2866_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_2867_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2868_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_2869_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(times_times_real(C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2870_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_2871_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2872_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_2873_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(times_times_real(B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2874_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_2875_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2876_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_2877_field__le__mult__one__interval,axiom,
    ! [Xb: real,Y: real] :
      ( ! [Z2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Z2),Xb)),Y) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% field_le_mult_one_interval
tff(fact_2878_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% divide_le_eq
tff(fact_2879_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_divide_eq
tff(fact_2880_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_left_mono
tff(fact_2881_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_2882_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_2883_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_2884_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_2885_mult__imp__div__pos__le,axiom,
    ! [Y: real,Xb: 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,Xb),aa_real_real(times_times_real(Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_2886_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Z),Y)),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),aa_real_real(divide_divide_real(Xb),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_2887_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_2888_convex__bound__le,axiom,
    ! [Xb: real,A: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),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(plus_plus_real(U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(U),Xb)),aa_real_real(times_times_real(V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2889_convex__bound__le,axiom,
    ! [Xb: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),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),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2890_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(divide_divide_real(B),C)),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(times_times_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(times_times_real(numeral_numeral_real(W2)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),numeral_numeral_real(W2))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2891_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(W2)),aa_real_real(divide_divide_real(B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(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(times_times_real(numeral_numeral_real(W2)),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(W2)),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2892_power__Suc__less,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2893_power__Suc__less,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),aa_nat_real(power_power_real(A),Nb))),aa_nat_real(power_power_real(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2894_power__Suc__less,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb))),aa_nat_int(power_power_int(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2895_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(bit0(one))),A)),B) ).

% left_add_twice
tff(fact_2896_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,bit0(one))),A)),B) ).

% left_add_twice
tff(fact_2897_left__add__twice,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(bit0(one))),A)),B) ).

% left_add_twice
tff(fact_2898_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,numeral_numeral_int(bit0(one))),A)),B) ).

% left_add_twice
tff(fact_2899_left__add__twice,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),A)),B) ).

% left_add_twice
tff(fact_2900_mult__2__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),numera6690914467698888265omplex(bit0(one))) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2_right
tff(fact_2901_mult__2__right,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2_right
tff(fact_2902_mult__2__right,axiom,
    ! [Z: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(Z),numera1916890842035813515d_enat(bit0(one))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Z),Z) ).

% mult_2_right
tff(fact_2903_mult__2__right,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z),numeral_numeral_int(bit0(one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).

% mult_2_right
tff(fact_2904_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(times_times_real(Z),numeral_numeral_real(bit0(one))) = aa_real_real(plus_plus_real(Z),Z) ).

% mult_2_right
tff(fact_2905_mult__2,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),Z) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2
tff(fact_2906_mult__2,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Z) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2
tff(fact_2907_mult__2,axiom,
    ! [Z: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(bit0(one))),Z) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Z),Z) ).

% mult_2
tff(fact_2908_mult__2,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),Z) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).

% mult_2
tff(fact_2909_mult__2,axiom,
    ! [Z: real] : aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),Z) = aa_real_real(plus_plus_real(Z),Z) ).

% mult_2
tff(fact_2910_div__nat__eqI,axiom,
    ! [Nb: nat,Q2: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q2)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_2911_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Q2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q2)),Nb) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_2912_split__div,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,zero_zero_nat) )
        & ( ( Nb != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I4)),J3) )
               => aa_nat_o(P,I4) ) ) ) ) ) ).

% split_div
tff(fact_2913_dividend__less__div__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_2914_dividend__less__times__div,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)))) ) ).

% dividend_less_times_div
tff(fact_2915_convex__bound__lt,axiom,
    ! [Xb: real,A: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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(plus_plus_real(U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(U),Xb)),aa_real_real(times_times_real(V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2916_convex__bound__lt,axiom,
    ! [Xb: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),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),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2917_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(divide_divide_real(B),C)),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(times_times_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(times_times_real(numeral_numeral_real(W2)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),numeral_numeral_real(W2))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2918_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,numeral_numeral_real(W2)),aa_real_real(divide_divide_real(B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(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(times_times_real(numeral_numeral_real(W2)),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(W2)),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2919_sum__squares__bound,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),Xb)),Y)),aa_real_real(plus_plus_real(aa_nat_real(power_power_real(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% sum_squares_bound
tff(fact_2920_split__div_H,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
          & aa_nat_o(P,zero_zero_nat) )
        | ? [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,Nb),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,Nb),aa_nat_nat(suc,Q5)))
            & aa_nat_o(P,Q5) ) ) ) ).

% split_div'
tff(fact_2921_power2__sum,axiom,
    ! [Xb: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Xb),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),Xb)),Y)) ).

% power2_sum
tff(fact_2922_power2__sum,axiom,
    ! [Xb: 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,Xb),Y)),aa_num_nat(numeral_numeral_nat,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,Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,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,bit0(one))),Xb)),Y)) ).

% power2_sum
tff(fact_2923_power2__sum,axiom,
    ! [Xb: extended_enat,Y: extended_enat] : aa_nat_Extended_enat(power_8040749407984259932d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Xb),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(power_8040749407984259932d_enat(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_Extended_enat(power_8040749407984259932d_enat(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(bit0(one))),Xb)),Y)) ).

% power2_sum
tff(fact_2924_power2__sum,axiom,
    ! [Xb: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),aa_num_nat(numeral_numeral_nat,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(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),Xb)),Y)) ).

% power2_sum
tff(fact_2925_power2__sum,axiom,
    ! [Xb: real,Y: real] : aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Xb),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(aa_nat_real(power_power_real(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),Xb)),Y)) ).

% power2_sum
tff(fact_2926_zero__le__even__power_H,axiom,
    ! [A: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))) ).

% zero_le_even_power'
tff(fact_2927_zero__le__even__power_H,axiom,
    ! [A: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))) ).

% zero_le_even_power'
tff(fact_2928_nat__bit__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one))),N))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_bit_induct
tff(fact_2929_arith__geo__mean,axiom,
    ! [U: real,Xb: real,Y: real] :
      ( ( aa_nat_real(power_power_real(U),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_real_real(times_times_real(Xb),Y) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Xb),Y)),numeral_numeral_real(bit0(one)))) ) ) ) ).

% arith_geo_mean
tff(fact_2930_triangle__def,axiom,
    ! [Nb: nat] : aa_nat_nat(nat_triangle,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% triangle_def
tff(fact_2931_realpow__pos__nth,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ( aa_nat_real(power_power_real(R3),Nb) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_2932_realpow__pos__nth__unique,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X4)
            & ( aa_nat_real(power_power_real(X4),Nb) = A )
            & ! [Y4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4)
                  & ( aa_nat_real(power_power_real(Y4),Nb) = A ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_2933_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2934_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2935_odd__power__less__zero,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))),zero_zero_real) ) ).

% odd_power_less_zero
tff(fact_2936_odd__power__less__zero,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))),zero_zero_int) ) ).

% odd_power_less_zero
tff(fact_2937_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_2938_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) = $ite(K = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) ).

% nat_mult_div_cancel_disj
tff(fact_2939_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_2940_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,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% set_bit_0
tff(fact_2941_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,numeral_numeral_int(bit0(one))),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one))))) ).

% set_bit_0
tff(fact_2942_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,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% unset_bit_0
tff(fact_2943_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,numeral_numeral_int(bit0(one))),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one)))) ).

% unset_bit_0
tff(fact_2944_nat__mult__div__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_2945_nat__mult__le__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_2946_mult__le__cancel__iff1,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_2947_mult__le__cancel__iff1,axiom,
    ! [Z: int,Xb: 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,Xb),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,Xb),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_2948_mult__le__cancel__iff2,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Z),Xb)),aa_real_real(times_times_real(Z),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_2949_mult__le__cancel__iff2,axiom,
    ! [Z: int,Xb: 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),Xb)),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,Xb),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_2950_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: 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),Nb) )
    <=> ( ( K = zero_zero_nat )
        | ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_2951_left__add__mult__distrib,axiom,
    ! [Ia: 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,Ia),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,Ia),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_2952_mult__less__iff1,axiom,
    ! [Z: real,Xb: 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(times_times_real(Xb),Z)),aa_real_real(times_times_real(Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).

% mult_less_iff1
tff(fact_2953_mult__less__iff1,axiom,
    ! [Z: int,Xb: 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,Xb),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,Xb),Y) ) ) ).

% mult_less_iff1
tff(fact_2954_nat__mult__less__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% nat_mult_less_cancel1
tff(fact_2955_nat__mult__eq__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_2956_divides__aux__eq,axiom,
    ! [Q2: nat,R2: nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Q2),R2))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_2957_divides__aux__eq,axiom,
    ! [Q2: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_2958_low__def,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) ).

% low_def
tff(fact_2959_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT2(X4,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X4),Nb) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summarya),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb) )
         => ( ( Mb = aa_nat_nat(suc,Nb) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summarya),X_12)
               => ( ! [X4: vEBT_VEBT] :
                      ( member_VEBT_VEBT2(X4,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X4),X_12) )
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_2960_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT2(X4,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X4),Nb) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summarya),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb) )
         => ( ( Mb = Nb )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summarya),X_12)
               => ( ! [X4: vEBT_VEBT] :
                      ( member_VEBT_VEBT2(X4,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X4),X_12) )
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_2961_even__succ__div__exp,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_2962_even__succ__div__exp,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( 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(numeral_numeral_int(bit0(one))),Nb)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_2963_set__decode__Suc,axiom,
    ! [Nb: nat,Xb: nat] :
      ( aa_set_nat_o(member_nat2(aa_nat_nat(suc,Nb)),aa_nat_set_nat(nat_set_decode,Xb))
    <=> aa_set_nat_o(member_nat2(Nb),aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% set_decode_Suc
tff(fact_2964_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] : size_s7466405169056248089T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_2965_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int] : size_s3661962791536183091BT_int(produc7292646706713671643BT_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_2966_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat] : size_s6152045936467909847BT_nat(produc7295137177222721919BT_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_2967_length__product,axiom,
    ! [Xs: list_int,Ys: list_VEBT_VEBT] : size_s6639371672096860321T_VEBT(produc662631939642741121T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_2968_length__product,axiom,
    ! [Xs: list_int,Ys: list_int] : size_s5157815400016825771nt_int(product_int_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_2969_length__product,axiom,
    ! [Xs: list_int,Ys: list_nat] : size_s7647898544948552527nt_nat(product_int_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_2970_length__product,axiom,
    ! [Xs: list_nat,Ys: list_VEBT_VEBT] : size_s4762443039079500285T_VEBT(produc7156399406898700509T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_2971_length__product,axiom,
    ! [Xs: list_nat,Ys: list_int] : size_s2970893825323803983at_int(product_nat_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_2972_length__product,axiom,
    ! [Xs: list_nat,Ys: list_nat] : size_s5460976970255530739at_nat(product_nat_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_2973_num_Osize__gen_I2_J,axiom,
    ! [X22: num] : aa_num_nat(size_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_2974_Leaf__0__not,axiom,
    ! [A: $o,B: $o] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_Leaf((A),(B))),zero_zero_nat) ).

% Leaf_0_not
tff(fact_2975_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),one_one_nat)
    <=> ? [A5: $o,B5: $o] : Ta = vEBT_Leaf((A5),(B5)) ) ).

% deg1Leaf
tff(fact_2976_deg__1__Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),one_one_nat)
     => ? [A3: $o,B3: $o] : Ta = vEBT_Leaf((A3),(B3)) ) ).

% deg_1_Leaf
tff(fact_2977_deg__1__Leafy,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( Nb = one_one_nat )
       => ? [A3: $o,B3: $o] : Ta = vEBT_Leaf((A3),(B3)) ) ) ).

% deg_1_Leafy
tff(fact_2978_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_2979_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_2980_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_2981_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_2982_dvd__0__right,axiom,
    ! [A: complex] : aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),zero_zero_complex) ).

% dvd_0_right
tff(fact_2983_dvd__0__right,axiom,
    ! [A: extended_enat] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,A),zero_z5237406670263579293d_enat) ).

% dvd_0_right
tff(fact_2984_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_2985_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_2986_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_2987_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_2988_dvd__0__left__iff,axiom,
    ! [A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,zero_z5237406670263579293d_enat),A)
    <=> ( A = zero_z5237406670263579293d_enat ) ) ).

% dvd_0_left_iff
tff(fact_2989_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_2990_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_2991_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_2992_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_2993_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_2994_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(B),A))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_2995_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_2996_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_2997_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_2998_mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_2999_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_3000_mod__by__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),zero_zero_int) = A ).

% mod_by_0
tff(fact_3001_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_3002_mod__self,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),A) = zero_zero_int ).

% mod_self
tff(fact_3003_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_3004_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_3005_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_3006_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_3007_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_3008_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_3009_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_3010_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_3011_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
    <=> ( ( K = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_3012_mod__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = Mb ) ) ).

% mod_less
tff(fact_3013_not__None__eq,axiom,
    ! [Xb: option4927543243414619207at_nat] :
      ( ( Xb != none_P5556105721700978146at_nat )
    <=> ? [Y2: product_prod_nat_nat] : Xb = some_P7363390416028606310at_nat(Y2) ) ).

% not_None_eq
tff(fact_3014_not__None__eq,axiom,
    ! [Xb: option_num] :
      ( ( Xb != none_num )
    <=> ? [Y2: num] : Xb = some_num(Y2) ) ).

% not_None_eq
tff(fact_3015_not__Some__eq,axiom,
    ! [Xb: option4927543243414619207at_nat] :
      ( ! [Y2: product_prod_nat_nat] : Xb != some_P7363390416028606310at_nat(Y2)
    <=> ( Xb = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_3016_not__Some__eq,axiom,
    ! [Xb: option_num] :
      ( ! [Y2: num] : Xb != some_num(Y2)
    <=> ( Xb = none_num ) ) ).

% not_Some_eq
tff(fact_3017_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_3018_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_3019_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_3020_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_3021_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_3022_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(times_times_real(A),C)),aa_real_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_3023_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_3024_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_3025_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(times_times_real(C),A)),aa_real_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_3026_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_3027_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_3028_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_3029_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_3030_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_3031_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(aa_real_real(times_times_real(C),A)),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3032_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_3033_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_3034_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_3035_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(B),aa_real_real(times_times_real(C),A)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3036_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_3037_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_3038_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_3039_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_3040_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_3041_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_3042_mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_3043_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_3044_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_3045_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_3046_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_3047_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_3048_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_3049_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_3050_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_3051_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_3052_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_3053_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_3054_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_3055_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_3056_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_3057_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_3058_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_3059_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_3060_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_3061_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_3062_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_3063_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_3064_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_3065_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_3066_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_3067_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_3068_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_3069_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_3070_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_3071_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_3072_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_3073_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_3074_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_3075_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_3076_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_3077_even__Suc__Suc__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb) ) ).

% even_Suc_Suc_iff
tff(fact_3078_even__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb) ) ).

% even_Suc
tff(fact_3079_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_3080_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_3081_Suc__mod__mult__self1,axiom,
    ! [Mb: nat,K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self1
tff(fact_3082_Suc__mod__mult__self2,axiom,
    ! [Mb: nat,Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self2
tff(fact_3083_Suc__mod__mult__self3,axiom,
    ! [K: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)),Mb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self3
tff(fact_3084_Suc__mod__mult__self4,axiom,
    ! [Nb: 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,Nb),K)),Mb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self4
tff(fact_3085_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one))),A)
        <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B) ) ) ).

% odd_add
tff(fact_3086_odd__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(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,numeral_numeral_int(bit0(one))),A)
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),B) ) ) ).

% odd_add
tff(fact_3087_even__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one))),A)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B) ) ) ).

% even_add
tff(fact_3088_even__add,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(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,numeral_numeral_int(bit0(one))),A)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),B) ) ) ).

% even_add
tff(fact_3089_even__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one))) ) ) ).

% even_Suc_div_two
tff(fact_3090_odd__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ) ).

% odd_Suc_div_two
tff(fact_3091_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,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% mod2_Suc_Suc
tff(fact_3092_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,Nb: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) != one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),Nb))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_3093_set__decode__0,axiom,
    ! [Xb: nat] :
      ( aa_set_nat_o(member_nat2(zero_zero_nat),aa_nat_set_nat(nat_set_decode,Xb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Xb) ) ).

% set_decode_0
tff(fact_3094_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(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,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,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_3095_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,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,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_3096_power__less__zero__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),Nb)),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq
tff(fact_3097_power__less__zero__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_3098_power__less__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,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_3099_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,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_3100_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one))),A) ) ).

% even_plus_one_iff
tff(fact_3101_even__plus__one__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(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,numeral_numeral_int(bit0(one))),A) ) ).

% even_plus_one_iff
tff(fact_3102_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,bit0(one))) != zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_3103_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one))) != zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_3104_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,bit0(one))) != one_one_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_3105_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one))) != one_one_int )
    <=> ( aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_3106_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_3107_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,bit0(one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_3108_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(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,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,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_3109_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,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,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_3110_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,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,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,bit0(one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_3111_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),numeral_numeral_int(bit0(one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_3112_even__succ__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) ) ) ).

% even_succ_div_two
tff(fact_3113_even__succ__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),numeral_numeral_int(bit0(one))) = aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one))) ) ) ).

% even_succ_div_two
tff(fact_3114_even__succ__div__2,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) ) ) ).

% even_succ_div_2
tff(fact_3115_even__succ__div__2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),numeral_numeral_int(bit0(one))) = aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one))) ) ) ).

% even_succ_div_2
tff(fact_3116_even__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_3117_even__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_nat_int(power_power_int(A),Nb))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_3118_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,bit0(one))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_3119_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,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,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))))),one_one_nat) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3120_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(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,numeral_numeral_int(bit0(one))),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one))))),one_one_int) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3121_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(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,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,bit0(one))),aa_num_nat(numeral_numeral_nat,W2))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3122_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,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,bit0(one))),aa_num_nat(numeral_numeral_nat,W2))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3123_even__succ__mod__exp,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3124_even__succ__mod__exp,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( 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(numeral_numeral_int(bit0(one))),Nb)) = 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(numeral_numeral_int(bit0(one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3125_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_3126_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_3127_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_3128_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_3129_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_3130_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_3131_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_3132_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_3133_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_3134_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_3135_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_3136_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_3137_dvd__refl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),A) ).

% dvd_refl
tff(fact_3138_dvd__refl,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),A) ).

% dvd_refl
tff(fact_3139_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_3140_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_3141_mod__greater__zero__iff__not__dvd,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_3142_VEBT__internal_OminNull_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ( Xb != vEBT_Leaf($false,$false) )
     => ( ! [Uv2: $o] : Xb != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_3143_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_3144_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( ( Xb != vEBT_Leaf($false,$false) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_3145_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_3146_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_3147_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_3148_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_3149_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_3150_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_3151_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_3152_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_3153_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_3154_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_3155_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_3156_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_3157_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_3158_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_3159_dvd__0__left,axiom,
    ! [A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,zero_z5237406670263579293d_enat),A)
     => ( A = zero_z5237406670263579293d_enat ) ) ).

% dvd_0_left
tff(fact_3160_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D4: nat] : Xb != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Leaf((Uu2),(Uv2))),D4)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg2: nat] : Xb != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(Mima,Deg,TreeList,Summary)),Deg2) ) ).

% VEBT_internal.valid'.cases
tff(fact_3161_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_3162_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_3163_mod__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% mod_Suc_eq
tff(fact_3164_mod__Suc__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),Nb) ).

% mod_Suc_Suc_eq
tff(fact_3165_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_3166_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_3167_dvd__triv__right,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(times_times_real(B),A)) ).

% dvd_triv_right
tff(fact_3168_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_3169_dvd__triv__right,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,A),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B),A)) ).

% dvd_triv_right
tff(fact_3170_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_3171_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_3172_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_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_3173_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_3174_dvd__mult__right,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B)),C)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,B),C) ) ).

% dvd_mult_right
tff(fact_3175_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_3176_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_3177_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(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3178_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_3179_mult__dvd__mono,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,C),D)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),C)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3180_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_3181_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_3182_dvd__triv__left,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(times_times_real(A),B)) ).

% dvd_triv_left
tff(fact_3183_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_3184_dvd__triv__left,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,A),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B)) ).

% dvd_triv_left
tff(fact_3185_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_3186_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_3187_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_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_3188_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_3189_dvd__mult__left,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B)),C)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,A),C) ) ).

% dvd_mult_left
tff(fact_3190_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_3191_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_3192_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(times_times_real(B),C)) ) ).

% dvd_mult2
tff(fact_3193_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_3194_dvd__mult2,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,A),B)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,A),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B),C)) ) ).

% dvd_mult2
tff(fact_3195_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_3196_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_3197_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(times_times_real(B),C)) ) ).

% dvd_mult
tff(fact_3198_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_3199_dvd__mult,axiom,
    ! [A: extended_enat,C: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,A),C)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,A),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B),C)) ) ).

% dvd_mult
tff(fact_3200_dvd__def,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) ) ).

% dvd_def
tff(fact_3201_dvd__def,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) ) ).

% dvd_def
tff(fact_3202_dvd__def,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
    <=> ? [K2: real] : A = aa_real_real(times_times_real(B),K2) ) ).

% dvd_def
tff(fact_3203_dvd__def,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) ) ).

% dvd_def
tff(fact_3204_dvd__def,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,B),A)
    <=> ? [K2: extended_enat] : A = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B),K2) ) ).

% dvd_def
tff(fact_3205_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_3206_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_3207_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A = aa_real_real(times_times_real(B),K) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A) ) ).

% dvdI
tff(fact_3208_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_3209_dvdI,axiom,
    ! [A: extended_enat,B: extended_enat,K: extended_enat] :
      ( ( A = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B),K) )
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,B),A) ) ).

% dvdI
tff(fact_3210_dvdE,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) ) ).

% dvdE
tff(fact_3211_dvdE,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) ) ).

% dvdE
tff(fact_3212_dvdE,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ~ ! [K3: real] : A != aa_real_real(times_times_real(B),K3) ) ).

% dvdE
tff(fact_3213_dvdE,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) ) ).

% dvdE
tff(fact_3214_dvdE,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,B),A)
     => ~ ! [K3: extended_enat] : A != aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B),K3) ) ).

% dvdE
tff(fact_3215_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_3216_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_3217_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_3218_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_3219_one__dvd,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,one_one_nat),A) ).

% one_dvd
tff(fact_3220_one__dvd,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,one_one_int),A) ).

% one_dvd
tff(fact_3221_one__dvd,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,one_one_real),A) ).

% one_dvd
tff(fact_3222_one__dvd,axiom,
    ! [A: complex] : aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,one_one_complex),A) ).

% one_dvd
tff(fact_3223_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_3224_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_3225_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(B),C)) ) ) ).

% dvd_add
tff(fact_3226_dvd__add,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,A),C)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,A),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B),C)) ) ) ).

% dvd_add
tff(fact_3227_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_3228_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_3229_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3230_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_3231_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_3232_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_3233_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_3234_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_3235_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_3236_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_3237_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),C) = aa_real_real(divide_divide_real(B),C) )
     => ( aa_real_o(aa_real_fun_real_o(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_3238_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_3239_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_3240_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(divide_divide_real(A),C) = aa_real_real(divide_divide_real(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3241_mod__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Mb) ).

% mod_less_eq_dividend
tff(fact_3242_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_3243_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_3244_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_3245_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_3246_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_3247_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X22) ).

% option.distinct(1)
tff(fact_3248_option_Odistinct_I1_J,axiom,
    ! [X22: num] : none_num != some_num(X22) ).

% option.distinct(1)
tff(fact_3249_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X22) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_3250_option_OdiscI,axiom,
    ! [Option: option_num,X22: num] :
      ( ( Option = some_num(X22) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_3251_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] : Y != some_P7363390416028606310at_nat(X23) ) ).

% option.exhaust
tff(fact_3252_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X23: num] : Y != some_num(X23) ) ).

% option.exhaust
tff(fact_3253_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)
        | ? [X3: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X3)) ) ) ).

% split_option_ex
tff(fact_3254_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)
        | ? [X3: num] : aa_option_num_o(P,some_num(X3)) ) ) ).

% split_option_ex
tff(fact_3255_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)
        & ! [X3: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X3)) ) ) ).

% split_option_all
tff(fact_3256_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)
        & ! [X3: num] : aa_option_num_o(P,some_num(X3)) ) ) ).

% split_option_all
tff(fact_3257_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P: fun_op1221328669798608686_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) )
       => ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( ( Xb = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_3258_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P: fun_op7653269905477231939_num_o,Y: option_num] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) )
       => ( ! [A3: product_prod_nat_nat,B3: num] :
              ( ( Xb = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_num(B3) )
               => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) ) )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_3259_combine__options__cases,axiom,
    ! [Xb: option_num,P: fun_op5373868528577402299_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( Xb = none_num )
       => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) )
       => ( ! [A3: num,B3: product_prod_nat_nat] :
              ( ( Xb = some_num(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_3260_combine__options__cases,axiom,
    ! [Xb: option_num,P: fun_op493924057549976310_num_o,Y: option_num] :
      ( ( ( Xb = none_num )
       => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) )
       => ( ! [A3: num,B3: num] :
              ( ( Xb = some_num(A3) )
             => ( ( Y = some_num(B3) )
               => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) ) )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_3261_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_3262_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_3263_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_3264_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
      <=> (Y) )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ~ (Y) )
       => ( ( ? [Uv2: $o] : Xb = vEBT_Leaf($true,(Uv2))
           => (Y) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => (Y) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
               => ~ (Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)
                 => (Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_3265_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,bit0(one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_3266_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
    <=> ( aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_3267_subset__decode__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(nat_set_decode,Mb)),aa_nat_set_nat(nat_set_decode,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% subset_decode_imp_le
tff(fact_3268_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_3269_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_3270_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_3271_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_3272_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_3273_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_3274_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_3275_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_3276_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_3277_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_3278_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_3279_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S)) ) ) ).

% pinf(9)
tff(fact_3280_pinf_I9_J,axiom,
    ! [D: extended_enat,S: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z2),X)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,D),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),S))
      <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,D),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),S)) ) ) ).

% pinf(9)
tff(fact_3281_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),S))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),S)) ) ) ).

% pinf(9)
tff(fact_3282_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S)) ) ) ).

% pinf(9)
tff(fact_3283_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S)) ) ) ).

% pinf(10)
tff(fact_3284_pinf_I10_J,axiom,
    ! [D: extended_enat,S: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z2),X)
     => ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,D),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),S))
      <=> ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,D),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),S)) ) ) ).

% pinf(10)
tff(fact_3285_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),S))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),S)) ) ) ).

% pinf(10)
tff(fact_3286_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S)) ) ) ).

% pinf(10)
tff(fact_3287_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S)) ) ) ).

% minf(9)
tff(fact_3288_minf_I9_J,axiom,
    ! [D: extended_enat,S: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Z2)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,D),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),S))
      <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,D),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),S)) ) ) ).

% minf(9)
tff(fact_3289_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),S))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),S)) ) ) ).

% minf(9)
tff(fact_3290_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S)) ) ) ).

% minf(9)
tff(fact_3291_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S)) ) ) ).

% minf(10)
tff(fact_3292_minf_I10_J,axiom,
    ! [D: extended_enat,S: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Z2)
     => ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,D),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),S))
      <=> ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,D),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),S)) ) ) ).

% minf(10)
tff(fact_3293_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),S))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),S)) ) ) ).

% minf(10)
tff(fact_3294_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S)) ) ) ).

% minf(10)
tff(fact_3295_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_3296_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_3297_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_3298_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(divide_divide_real(A),B) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3299_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_3300_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_3301_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_3302_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_3303_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_3304_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_3305_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_3306_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_3307_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_3308_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_3309_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_3310_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_3311_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_3312_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_3313_mod__Suc,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = Nb,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))) ).

% mod_Suc
tff(fact_3314_mod__induct,axiom,
    ! [P: fun_nat_o,Nb: nat,P2: nat,Mb: nat] :
      ( aa_nat_o(P,Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),P2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),P2)
         => ( ! [N: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),P2)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),P2)) ) )
           => aa_nat_o(P,Mb) ) ) ) ) ).

% mod_induct
tff(fact_3315_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_3316_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_3317_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_3318_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_3319_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_3320_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_3321_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_3322_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_3323_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_3324_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_3325_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_3326_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_3327_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Mb: nat,Nb: 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),Nb) ) ) ).

% gcd_nat_induct
tff(fact_3328_mod__less__divisor,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Nb) ) ).

% mod_less_divisor
tff(fact_3329_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_3330_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_3331_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_3332_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_3333_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_3334_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_3335_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_3336_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_3337_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_3338_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_3339_mod__Suc__le__divisor,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,Nb))),Nb) ).

% mod_Suc_le_divisor
tff(fact_3340_mod__eq__0D,axiom,
    ! [Mb: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),D) = zero_zero_nat )
     => ? [Q3: nat] : Mb = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q3) ) ).

% mod_eq_0D
tff(fact_3341_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3342_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(power_power_real(A),Mb)),aa_nat_real(power_power_real(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3343_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3344_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3345_power__le__dvd,axiom,
    ! [A: nat,Nb: nat,B: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3346_power__le__dvd,axiom,
    ! [A: real,Nb: nat,B: real,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(power_power_real(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(power_power_real(A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3347_power__le__dvd,axiom,
    ! [A: int,Nb: nat,B: int,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3348_power__le__dvd,axiom,
    ! [A: complex,Nb: nat,B: complex,Mb: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3349_dvd__power__le,axiom,
    ! [Xb: nat,Y: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_3350_dvd__power__le,axiom,
    ! [Xb: real,Y: real,Nb: nat,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(power_power_real(Xb),Nb)),aa_nat_real(power_power_real(Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_3351_dvd__power__le,axiom,
    ! [Xb: int,Y: int,Nb: nat,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),Nb)),aa_nat_int(power_power_int(Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_3352_dvd__power__le,axiom,
    ! [Xb: complex,Y: complex,Nb: nat,Mb: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_3353_nat__mod__eq__iff,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Nb) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_3354_dvd__pos__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb) ) ) ).

% dvd_pos_nat
tff(fact_3355_nat__dvd__not__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb) ) ) ).

% nat_dvd_not_less
tff(fact_3356_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,Xb: 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),Xb) = 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),Xb) = 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) ) )
         => ? [X4: 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),X4) = 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)),X4) = 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_3357_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D4: nat,X4: 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),X4) = 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),X4) = 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_3358_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_3359_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,vEBT_Leaf((Uu),(Uv))),D)
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_3360_mod2__eq__if,axiom,
    ! [A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A),zero_zero_nat,one_one_nat) ).

% mod2_eq_if
tff(fact_3361_mod2__eq__if,axiom,
    ! [A: int] :
      aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A),zero_zero_int,one_one_int) ).

% mod2_eq_if
tff(fact_3362_parity__cases,axiom,
    ! [A: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) != zero_zero_nat ) )
     => ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_3363_parity__cases,axiom,
    ! [A: int] :
      ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
       => ( aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one))) != zero_zero_int ) )
     => ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
         => ( aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_3364_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)),Xb) ).

% vebt_member.simps(2)
tff(fact_3365_vebt__insert_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X4: nat] : Xb != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Leaf((A3),(B3))),X4)
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] : Xb != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(Info2,zero_zero_nat,Ts2,S2)),X4)
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] : Xb != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S2)),X4)
         => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] : Xb != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary)),X4)
           => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] : Xb != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)),X4) ) ) ) ) ).

% vebt_insert.cases
tff(fact_3366_vebt__member_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X4: nat] : Xb != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Leaf((A3),(B3))),X4)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] : Xb != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)),X4)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] : Xb != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)),X4)
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] : Xb != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)),X4)
           => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] : Xb != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)),X4) ) ) ) ) ).

% vebt_member.cases
tff(fact_3367_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] : Xb != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Leaf((Uu2),(Uv2))),Uw2)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : Xb != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)),Uz2)
       => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] : Xb != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3,Vb2)),X4)
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] : Xb != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)),X4)
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X4: nat] : Xb != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd)),X4) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_3368_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_3369_find__None__iff,axiom,
    ! [P: fun_real_o,Xs: list_real] :
      ( ( find_real(P,Xs) = none_real )
    <=> ~ ? [X3: real] :
            ( aa_set_real_o(member_real2(X3),set_real2(Xs))
            & aa_real_o(P,X3) ) ) ).

% find_None_iff
tff(fact_3370_find__None__iff,axiom,
    ! [P: fun_o_o,Xs: list_o] :
      ( ( find_o(P,Xs) = none_o )
    <=> ~ ? [X3: $o] :
            ( aa_set_o_o(member_o2((X3)),set_o2(Xs))
            & aa_o_o(P,(X3)) ) ) ).

% find_None_iff
tff(fact_3371_find__None__iff,axiom,
    ! [P: fun_set_nat_o,Xs: list_set_nat] :
      ( ( find_set_nat(P,Xs) = none_set_nat )
    <=> ~ ? [X3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat2(X3),set_set_nat2(Xs))
            & aa_set_nat_o(P,X3) ) ) ).

% find_None_iff
tff(fact_3372_find__None__iff,axiom,
    ! [P: fun_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ( find_VEBT_VEBT(P,Xs) = none_VEBT_VEBT )
    <=> ~ ? [X3: vEBT_VEBT] :
            ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(Xs))
            & aa_VEBT_VEBT_o(P,X3) ) ) ).

% find_None_iff
tff(fact_3373_find__None__iff,axiom,
    ! [P: fun_int_o,Xs: list_int] :
      ( ( find_int(P,Xs) = none_int )
    <=> ~ ? [X3: int] :
            ( aa_set_int_o(member_int2(X3),set_int2(Xs))
            & aa_int_o(P,X3) ) ) ).

% find_None_iff
tff(fact_3374_find__None__iff,axiom,
    ! [P: fun_nat_o,Xs: list_nat] :
      ( ( find_nat(P,Xs) = none_nat )
    <=> ~ ? [X3: nat] :
            ( aa_set_nat_o(member_nat2(X3),set_nat2(Xs))
            & aa_nat_o(P,X3) ) ) ).

% find_None_iff
tff(fact_3375_find__None__iff,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Xs: list_P6011104703257516679at_nat] :
      ( ( find_P8199882355184865565at_nat(P,Xs) = none_P5556105721700978146at_nat )
    <=> ~ ? [X3: product_prod_nat_nat] :
            ( member8440522571783428010at_nat(X3,set_Pr5648618587558075414at_nat(Xs))
            & aa_Pro6150637604445964947_nat_o(P,X3) ) ) ).

% find_None_iff
tff(fact_3376_find__None__iff,axiom,
    ! [P: fun_num_o,Xs: list_num] :
      ( ( find_num(P,Xs) = none_num )
    <=> ~ ? [X3: num] :
            ( member_num(X3,set_num2(Xs))
            & aa_num_o(P,X3) ) ) ).

% find_None_iff
tff(fact_3377_find__None__iff2,axiom,
    ! [P: fun_real_o,Xs: list_real] :
      ( ( none_real = find_real(P,Xs) )
    <=> ~ ? [X3: real] :
            ( aa_set_real_o(member_real2(X3),set_real2(Xs))
            & aa_real_o(P,X3) ) ) ).

% find_None_iff2
tff(fact_3378_find__None__iff2,axiom,
    ! [P: fun_o_o,Xs: list_o] :
      ( ( none_o = find_o(P,Xs) )
    <=> ~ ? [X3: $o] :
            ( aa_set_o_o(member_o2((X3)),set_o2(Xs))
            & aa_o_o(P,(X3)) ) ) ).

% find_None_iff2
tff(fact_3379_find__None__iff2,axiom,
    ! [P: fun_set_nat_o,Xs: list_set_nat] :
      ( ( none_set_nat = find_set_nat(P,Xs) )
    <=> ~ ? [X3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat2(X3),set_set_nat2(Xs))
            & aa_set_nat_o(P,X3) ) ) ).

% find_None_iff2
tff(fact_3380_find__None__iff2,axiom,
    ! [P: fun_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ( none_VEBT_VEBT = find_VEBT_VEBT(P,Xs) )
    <=> ~ ? [X3: vEBT_VEBT] :
            ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(Xs))
            & aa_VEBT_VEBT_o(P,X3) ) ) ).

% find_None_iff2
tff(fact_3381_find__None__iff2,axiom,
    ! [P: fun_int_o,Xs: list_int] :
      ( ( none_int = find_int(P,Xs) )
    <=> ~ ? [X3: int] :
            ( aa_set_int_o(member_int2(X3),set_int2(Xs))
            & aa_int_o(P,X3) ) ) ).

% find_None_iff2
tff(fact_3382_find__None__iff2,axiom,
    ! [P: fun_nat_o,Xs: list_nat] :
      ( ( none_nat = find_nat(P,Xs) )
    <=> ~ ? [X3: nat] :
            ( aa_set_nat_o(member_nat2(X3),set_nat2(Xs))
            & aa_nat_o(P,X3) ) ) ).

% find_None_iff2
tff(fact_3383_find__None__iff2,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Xs: list_P6011104703257516679at_nat] :
      ( ( none_P5556105721700978146at_nat = find_P8199882355184865565at_nat(P,Xs) )
    <=> ~ ? [X3: product_prod_nat_nat] :
            ( member8440522571783428010at_nat(X3,set_Pr5648618587558075414at_nat(Xs))
            & aa_Pro6150637604445964947_nat_o(P,X3) ) ) ).

% find_None_iff2
tff(fact_3384_find__None__iff2,axiom,
    ! [P: fun_num_o,Xs: list_num] :
      ( ( none_num = find_num(P,Xs) )
    <=> ~ ? [X3: num] :
            ( member_num(X3,set_num2(Xs))
            & aa_num_o(P,X3) ) ) ).

% find_None_iff2
tff(fact_3385_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_3386_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_3387_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_3388_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_3389_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,bit0(Nb))),aa_num_nat(numeral_numeral_nat,bit0(Q2))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_3390_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q2: num] :
      ( ( aa_int_int(modulo_modulo_int(numeral_numeral_int(bit0(Nb))),numeral_numeral_int(bit0(Q2))) = zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(numeral_numeral_int(Nb)),numeral_numeral_int(Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_3391_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_3392_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_int_int(modulo_modulo_int(numeral_numeral_int(Nb)),numeral_numeral_int(one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_3393_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X4: nat] : Xb != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Leaf((A3),(B3))),X4)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : Xb != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] : Xb != aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2)),X4) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_3394_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_3395_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_3396_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_3397_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_3398_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_3399_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_3400_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_3401_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_3402_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_3403_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_3404_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_3405_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_3406_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_3407_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_3408_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_3409_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_3410_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B: $o] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat)) ).

% invar_vebt.intros(1)
tff(fact_3411_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 )
         => ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C3) ) ) ).

% unit_dvdE
tff(fact_3412_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 )
         => ! [C3: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C3) ) ) ).

% unit_dvdE
tff(fact_3413_unity__coeff__ex,axiom,
    ! [P: fun_nat_o,L: nat] :
      ( ? [X3: nat] : aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X3))
    <=> ? [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),zero_zero_nat))
          & aa_nat_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_3414_unity__coeff__ex,axiom,
    ! [P: fun_int_o,L: int] :
      ( ? [X3: int] : aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X3))
    <=> ? [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),zero_zero_int))
          & aa_int_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_3415_unity__coeff__ex,axiom,
    ! [P: fun_real_o,L: real] :
      ( ? [X3: real] : aa_real_o(P,aa_real_real(times_times_real(L),X3))
    <=> ? [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),aa_real_real(plus_plus_real(X3),zero_zero_real))
          & aa_real_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_3416_unity__coeff__ex,axiom,
    ! [P: fun_complex_o,L: complex] :
      ( ? [X3: complex] : aa_complex_o(P,aa_complex_complex(times_times_complex(L),X3))
    <=> ? [X3: complex] :
          ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,L),aa_complex_complex(plus_plus_complex(X3),zero_zero_complex))
          & aa_complex_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_3417_unity__coeff__ex,axiom,
    ! [P: fun_Extended_enat_o,L: extended_enat] :
      ( ? [X3: extended_enat] : aa_Extended_enat_o(P,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(L),X3))
    <=> ? [X3: extended_enat] :
          ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,L),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X3),zero_z5237406670263579293d_enat))
          & aa_Extended_enat_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_3418_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_3419_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_3420_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_3421_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_3422_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_3423_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_3424_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_3425_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_3426_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_3427_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_3428_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_3429_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_3430_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_3431_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_3432_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_3433_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_3434_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_3435_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_3436_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_3437_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_3438_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_3439_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_3440_mod__le__divisor,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Nb) ) ).

% mod_le_divisor
tff(fact_3441_is__unit__power__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3442_is__unit__power__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3443_vebt__insert_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] :
      vEBT_vebt_insert(vEBT_Leaf((A),(B)),Xb) = $ite(
        Xb = zero_zero_nat,
        vEBT_Leaf($true,(B)),
        $ite(Xb = one_one_nat,vEBT_Leaf((A),$true),vEBT_Leaf((A),(B))) ) ).

% vebt_insert.simps(1)
tff(fact_3444_div__less__mono,axiom,
    ! [A2: nat,B2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Nb) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),Nb) = zero_zero_nat )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_3445_nat__mod__eq__lemma,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
       => ? [Q3: nat] : Xb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_3446_mod__eq__nat2E,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ~ ! [S2: nat] : Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S2)) ) ) ).

% mod_eq_nat2E
tff(fact_3447_mod__eq__nat1E,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ~ ! [S2: nat] : Mb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S2)) ) ) ).

% mod_eq_nat1E
tff(fact_3448_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_3449_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,vEBT_Leaf((A),(B))),Xb)
    <=> $ite(
          Xb = zero_zero_nat,
          (A),
          $ite(Xb = one_one_nat,(B),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_3450_dvd__imp__le,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ) ).

% dvd_imp_le
tff(fact_3451_div__mod__decomp,axiom,
    ! [A2: nat,Nb: 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),Nb)),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Nb)) ).

% div_mod_decomp
tff(fact_3452_mod__mult2__eq,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Q2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ).

% mod_mult2_eq
tff(fact_3453_dvd__mult__cancel,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% dvd_mult_cancel
tff(fact_3454_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_3455_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,vEBT_Leaf((A),(B))),Xb)
    <=> $ite(
          Xb = zero_zero_nat,
          (A),
          $ite(Xb = one_one_nat,(B),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_3456_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D4: nat,X4: 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),X4) = 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_3457_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( ! [Uv2: $o] : Xb != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_3458_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy)),Uz) ).

% VEBT_internal.membermima.simps(2)
tff(fact_3459_option_Osize__gen_I1_J,axiom,
    ! [Xb: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(Xb,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_3460_option_Osize__gen_I1_J,axiom,
    ! [Xb: fun_num_nat] : size_option_num(Xb,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_3461_even__zero,axiom,
    aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),zero_zero_nat) ).

% even_zero
tff(fact_3462_even__zero,axiom,
    aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),zero_zero_int) ).

% even_zero
tff(fact_3463_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_3464_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_3465_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_3466_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_3467_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_3468_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_3469_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,bit0(one))),A)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% odd_even_add
tff(fact_3470_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% odd_even_add
tff(fact_3471_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_3472_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_3473_dvd__power__iff,axiom,
    ! [Xb: nat,Mb: nat,Nb: nat] :
      ( ( Xb != 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,Xb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_3474_dvd__power__iff,axiom,
    ! [Xb: int,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),Mb)),aa_nat_int(power_power_int(Xb),Nb))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_3475_dvd__power,axiom,
    ! [Nb: nat,Xb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)) ) ).

% dvd_power
tff(fact_3476_dvd__power,axiom,
    ! [Nb: nat,Xb: real] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_real ) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_nat_real(power_power_real(Xb),Nb)) ) ).

% dvd_power
tff(fact_3477_dvd__power,axiom,
    ! [Nb: nat,Xb: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_nat_int(power_power_int(Xb),Nb)) ) ).

% dvd_power
tff(fact_3478_dvd__power,axiom,
    ! [Nb: nat,Xb: complex] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_complex ) )
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),aa_nat_complex(power_power_complex(Xb),Nb)) ) ).

% dvd_power
tff(fact_3479_split__mod,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,Mb) )
        & ( ( Nb != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I4)),J3) )
               => aa_nat_o(P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_3480_dvd__mult__cancel1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Mb)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_3481_dvd__mult__cancel2,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)),Mb)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_3482_power__dvd__imp__le,axiom,
    ! [Ia: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ia),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ia),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Ia)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_3483_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_3484_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_3485_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_Extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s3941691890525107288d_enat(Ys)))
     => ( nth_Pr7509392720524132704d_enat(produc4894985897562121079d_enat(Xs,Ys),Nb) = aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s3941691890525107288d_enat(Ys)))),aa_nat_Extended_enat(nth_Extended_enat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s3941691890525107288d_enat(Ys)))) ) ) ).

% product_nth
tff(fact_3486_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys),Nb) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_3487_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys)))
     => ( nth_Pr6837108013167703752BT_int(produc7292646706713671643BT_int(Xs,Ys),Nb) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_3488_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys),Nb) = aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys)))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_3489_product__nth,axiom,
    ! [Nb: nat,Xs: list_int,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr3474266648193625910T_VEBT(produc662631939642741121T_VEBT(Xs,Ys),Nb) = produc3329399203697025711T_VEBT(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_3490_product__nth,axiom,
    ! [Nb: nat,Xs: list_int,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_int(Ys)))
     => ( nth_Pr4439495888332055232nt_int(product_int_int(Xs,Ys),Nb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_int(Ys)))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_3491_product__nth,axiom,
    ! [Nb: nat,Xs: list_int,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr8617346907841251940nt_nat(product_int_nat(Xs,Ys),Nb) = product_Pair_int_nat(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_3492_product__nth,axiom,
    ! [Nb: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr744662078594809490T_VEBT(produc7156399406898700509T_VEBT(Xs,Ys),Nb) = produc599794634098209291T_VEBT(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_3493_product__nth,axiom,
    ! [Nb: nat,Xs: list_nat,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_int(Ys)))
     => ( nth_Pr3440142176431000676at_int(product_nat_int(Xs,Ys),Nb) = product_Pair_nat_int(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_3494_product__nth,axiom,
    ! [Nb: nat,Xs: list_nat,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr7617993195940197384at_nat(product_nat_nat(Xs,Ys),Nb) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys)))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_3495_power__mono__odd,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),Nb)),aa_nat_real(power_power_real(B),Nb)) ) ) ).

% power_mono_odd
tff(fact_3496_power__mono__odd,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono_odd
tff(fact_3497_Suc__times__mod__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))),Mb) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_3498_odd__pos,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% odd_pos
tff(fact_3499_dvd__power__iff__le,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,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),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_3500_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,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,bit0(one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3501_even__unset__bit__iff,axiom,
    ! [Mb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se4203085406695923979it_int(Mb),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3502_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,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,bit0(one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3503_even__set__bit__iff,axiom,
    ! [Mb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se7879613467334960850it_int(Mb),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3504_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_3505_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xb)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista,Summarya) ).

% vebt_insert.simps(4)
tff(fact_3506_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,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,bit0(one))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_3507_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,numeral_numeral_int(bit0(one))),B))),B)
       => ( aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B)) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_3508_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,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,bit0(one)))) = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
tff(fact_3509_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one))) = A )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_3510_div__exp__mod__exp__eq,axiom,
    ! [A: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_3511_div__exp__mod__exp__eq,axiom,
    ! [A: int,Nb: nat,Mb: nat] : aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Mb)) = aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_3512_oddE,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one))),B3)),one_one_nat) ) ).

% oddE
tff(fact_3513_oddE,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(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,numeral_numeral_int(bit0(one))),B3)),one_one_int) ) ).

% oddE
tff(fact_3514_zero__le__even__power,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),Nb)) ) ).

% zero_le_even_power
tff(fact_3515_zero__le__even__power,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb)) ) ).

% zero_le_even_power
tff(fact_3516_zero__le__odd__power,axiom,
    ! [Nb: nat,A: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),Nb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ).

% zero_le_odd_power
tff(fact_3517_zero__le__odd__power,axiom,
    ! [Nb: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_odd_power
tff(fact_3518_zero__le__power__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq
tff(fact_3519_zero__le__power__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq
tff(fact_3520_verit__le__mono__div,axiom,
    ! [A2: nat,B2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Nb)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),Nb) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),Nb)) ) ) ).

% verit_le_mono_div
tff(fact_3521_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,bit0(one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one))),B))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3522_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,numeral_numeral_int(bit0(one))),B))),B)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B))) = aa_int_int(divide_divide_int(A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3523_zero__less__power__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq
tff(fact_3524_zero__less__power__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq
tff(fact_3525_mod__double__modulus,axiom,
    ! [Mb: nat,Xb: 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),Xb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Mb) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,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,Xb),Mb)),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_3526_mod__double__modulus,axiom,
    ! [Mb: int,Xb: 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),Xb)
       => ( ( aa_int_int(modulo_modulo_int(Xb),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),Mb)) = aa_int_int(modulo_modulo_int(Xb),Mb) )
          | ( aa_int_int(modulo_modulo_int(Xb),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(Xb),Mb)),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_3527_unset__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% unset_bit_Suc
tff(fact_3528_unset__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se4203085406695923979it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se4203085406695923979it_int(Nb),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one)))))) ).

% unset_bit_Suc
tff(fact_3529_set__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% set_bit_Suc
tff(fact_3530_set__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se7879613467334960850it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se7879613467334960850it_int(Nb),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one)))))) ).

% set_bit_Suc
tff(fact_3531_power__le__zero__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),Nb)),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3532_power__le__zero__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3533_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,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,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,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_3534_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,numeral_numeral_int(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,numeral_numeral_int(bit0(one))),aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B)))),one_one_int) = aa_int_int(divide_divide_int(A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3535_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,A1),A22)
    <=> ( ( ? [A5: $o,B5: $o] : A1 = vEBT_Leaf((A5),(B5))
          & ( 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) )
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),N3) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),N3)
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N3) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3) )
            & ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X_13)
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_13) ) )
        | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList2,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),N3) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),aa_nat_nat(suc,N3))
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,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(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X_13)
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_13) ) )
        | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),N3) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),N3)
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N3) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3) )
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N3))
               => ( ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),X_13)
                <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),I4) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X3: vEBT_VEBT] :
                  ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),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,bit0(one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N3))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N3) = I4 )
                     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N3)) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N3) = I4 )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N3)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X3)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ma2) ) ) ) ) ) )
        | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),N3) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),aa_nat_nat(suc,N3))
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,N3)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) )
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,N3)))
               => ( ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),X_13)
                <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),I4) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X3: vEBT_VEBT] :
                  ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),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,bit0(one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,N3)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N3) = I4 )
                     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N3)) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N3) = I4 )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N3)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X3)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ma2) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_3536_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,A1),A22)
     => ( ( ? [A3: $o,B3: $o] : A1 = 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 )
               => ( ! [X: vEBT_VEBT] :
                      ( member_VEBT_VEBT2(X,set_VEBT_VEBT2(TreeList))
                     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),N) )
                 => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M)
                   => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M) )
                     => ( ( M = N )
                       => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
                         => ( ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X_1)
                           => ~ ! [X: vEBT_VEBT] :
                                  ( member_VEBT_VEBT2(X,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_1) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
                ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
               => ( ( A22 = Deg )
                 => ( ! [X: vEBT_VEBT] :
                        ( member_VEBT_VEBT2(X,set_VEBT_VEBT2(TreeList))
                       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),N) )
                   => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M)
                     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M) )
                       => ( ( M = aa_nat_nat(suc,N) )
                         => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
                           => ( ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X_1)
                             => ~ ! [X: vEBT_VEBT] :
                                    ( member_VEBT_VEBT2(X,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_1) ) ) ) ) ) ) ) ) )
           => ( ! [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(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg,TreeList,Summary) )
                 => ( ( A22 = Deg )
                   => ( ! [X: vEBT_VEBT] :
                          ( member_VEBT_VEBT2(X,set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),N) )
                     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M)
                       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M) )
                         => ( ( M = N )
                           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
                             => ( ! [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,bit0(one))),M))
                                   => ( ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),X_13)
                                    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),I3) ) )
                               => ( ( ( Mi = Ma )
                                   => ! [X: vEBT_VEBT] :
                                        ( member_VEBT_VEBT2(X,set_VEBT_VEBT2(TreeList))
                                       => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_1) ) )
                                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
                                     => ~ ( ( Mi != Ma )
                                         => ! [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,bit0(one))),M))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I3 )
                                                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N)) )
                                                & ! [X: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = I3 )
                                                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)) )
                                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X)
                                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
                    ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg,TreeList,Summary) )
                   => ( ( A22 = Deg )
                     => ( ! [X: vEBT_VEBT] :
                            ( member_VEBT_VEBT2(X,set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),N) )
                       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M)
                         => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M) )
                           => ( ( M = aa_nat_nat(suc,N) )
                             => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
                               => ( ! [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,bit0(one))),M))
                                     => ( ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),X_13)
                                      <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),I3) ) )
                                 => ( ( ( Mi = Ma )
                                     => ! [X: vEBT_VEBT] :
                                          ( member_VEBT_VEBT2(X,set_VEBT_VEBT2(TreeList))
                                         => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_1) ) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
                                       => ~ ( ( Mi != Ma )
                                           => ! [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,bit0(one))),M))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I3 )
                                                   => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N)) )
                                                  & ! [X: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = I3 )
                                                        & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)) )
                                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X)
                                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_3537_flip__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% flip_bit_Suc
tff(fact_3538_flip__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se2159334234014336723it_int(Nb),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one)))))) ).

% flip_bit_Suc
tff(fact_3539_signed__take__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one)))))) ).

% signed_take_bit_Suc
tff(fact_3540_vebt__insert_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => ( Y != $ite(
                  Xaa = zero_zero_nat,
                  vEBT_Leaf($true,(B3)),
                  $ite(Xaa = one_one_nat,vEBT_Leaf((A3),$true),vEBT_Leaf((A3),(B3))) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info2,zero_zero_nat,Ts2,S2) )
             => ( Y != vEBT_Node(Info2,zero_zero_nat,Ts2,S2) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S2) )
               => ( Y != vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S2) ) )
           => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
                 => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = Ma ) ) ),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Xaa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),
                                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),vEBT_vebt_insert(Summary,h),Summary)),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_3541_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3542_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(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(numeral_numeral_int(bit0(one))),Mb))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3543_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,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,bit0(one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_3544_even__flip__bit__iff,axiom,
    ! [Mb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se2159334234014336723it_int(Mb),A))
    <=> ~ ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_3545_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_3546_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : aa_int_int(modulo_modulo_int(one_one_int),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_3547_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one))),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_3548_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(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(numeral_numeral_int(bit0(one))),Mb)),one_one_int)),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)))
    <=> ( ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_3549_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique5052692396658037445od_int(one,bit0(Nb)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),numeral_numeral_int(one)) ).

% divmod_algorithm_code(3)
tff(fact_3550_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique5055182867167087721od_nat(one,bit0(Nb)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_3551_vebt__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xb),Xaa)
      <=> (Y) )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => (Y) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
             => (Y) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = 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] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)
                   => ( (Y)
                    <=> ~ $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_3552_set__vebt_H__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_VEBT_set_vebt(Ta) = collect_nat(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta)) ).

% set_vebt'_def
tff(fact_3553_Diff__cancel,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,A2) = bot_bot_set_real ).

% Diff_cancel
tff(fact_3554_Diff__cancel,axiom,
    ! [A2: set_o] : minus_minus_set_o(A2,A2) = bot_bot_set_o ).

% Diff_cancel
tff(fact_3555_Diff__cancel,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,A2) = bot_bot_set_int ).

% Diff_cancel
tff(fact_3556_Diff__cancel,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(A2,A2) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_3557_empty__Diff,axiom,
    ! [A2: set_real] : minus_minus_set_real(bot_bot_set_real,A2) = bot_bot_set_real ).

% empty_Diff
tff(fact_3558_empty__Diff,axiom,
    ! [A2: set_o] : minus_minus_set_o(bot_bot_set_o,A2) = bot_bot_set_o ).

% empty_Diff
tff(fact_3559_empty__Diff,axiom,
    ! [A2: set_int] : minus_minus_set_int(bot_bot_set_int,A2) = bot_bot_set_int ).

% empty_Diff
tff(fact_3560_empty__Diff,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(bot_bot_set_nat,A2) = bot_bot_set_nat ).

% empty_Diff
tff(fact_3561_Diff__empty,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,bot_bot_set_real) = A2 ).

% Diff_empty
tff(fact_3562_Diff__empty,axiom,
    ! [A2: set_o] : minus_minus_set_o(A2,bot_bot_set_o) = A2 ).

% Diff_empty
tff(fact_3563_Diff__empty,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,bot_bot_set_int) = A2 ).

% Diff_empty
tff(fact_3564_Diff__empty,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(A2,bot_bot_set_nat) = A2 ).

% Diff_empty
tff(fact_3565_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_3566_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_3567_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_3568_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),A) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3569_diff__zero,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_zero
tff(fact_3570_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_3571_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_3572_diff__zero,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),zero_zero_real) = A ).

% diff_zero
tff(fact_3573_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_3574_diff__0__right,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_0_right
tff(fact_3575_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_3576_diff__0__right,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),zero_zero_real) = A ).

% diff_0_right
tff(fact_3577_diff__self,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% diff_self
tff(fact_3578_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_3579_diff__self,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),A) = zero_zero_real ).

% diff_self
tff(fact_3580_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_3581_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_3582_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3583_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_3584_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_3585_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C)) = aa_real_real(minus_minus_real(A),B) ).

% add_diff_cancel_right
tff(fact_3586_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_3587_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_3588_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3589_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_3590_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_3591_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B)) = aa_real_real(minus_minus_real(A),B) ).

% add_diff_cancel_left
tff(fact_3592_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_3593_diff__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B)),B) = A ).

% diff_add_cancel
tff(fact_3594_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_3595_add__diff__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),B) = A ).

% add_diff_cancel
tff(fact_3596_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_3597_Diff__eq__empty__iff,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( minus_minus_set_o(A2,B2) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(A2),B2) ) ).

% Diff_eq_empty_iff
tff(fact_3598_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_3599_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_3600_diff__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% diff_Suc_Suc
tff(fact_3601_Suc__diff__diff,axiom,
    ! [Mb: nat,Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),K) ).

% Suc_diff_diff
tff(fact_3602_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_3603_diff__0__eq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_3604_diff__diff__cancel,axiom,
    ! [Ia: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ia)) = Ia ) ) ).

% diff_diff_cancel
tff(fact_3605_diff__diff__left,axiom,
    ! [Ia: 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,Ia),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ia),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ).

% diff_diff_left
tff(fact_3606_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(zero_n3304061248610475627l_real,(P))),aa_o_real(zero_n3304061248610475627l_real,(Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_3607_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_3608_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_3609_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_real(zero_n3304061248610475627l_real,(P)) = zero_zero_real )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_3610_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_complex(zero_n1201886186963655149omplex,(P)) = zero_zero_complex )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_3611_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_Extended_enat(zero_n1046097342994218471d_enat,(P)) = zero_z5237406670263579293d_enat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_3612_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_3613_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_3614_of__bool__eq_I1_J,axiom,
    aa_o_real(zero_n3304061248610475627l_real,$false) = zero_zero_real ).

% of_bool_eq(1)
tff(fact_3615_of__bool__eq_I1_J,axiom,
    aa_o_complex(zero_n1201886186963655149omplex,$false) = zero_zero_complex ).

% of_bool_eq(1)
tff(fact_3616_of__bool__eq_I1_J,axiom,
    aa_o_Extended_enat(zero_n1046097342994218471d_enat,$false) = zero_z5237406670263579293d_enat ).

% of_bool_eq(1)
tff(fact_3617_of__bool__eq_I1_J,axiom,
    aa_o_nat(zero_n2687167440665602831ol_nat,$false) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_3618_of__bool__eq_I1_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$false) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_3619_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_o_Extended_enat(zero_n1046097342994218471d_enat,(P))),aa_o_Extended_enat(zero_n1046097342994218471d_enat,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_3620_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_o_real(zero_n3304061248610475627l_real,(P))),aa_o_real(zero_n3304061248610475627l_real,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_3621_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_3622_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_3623_of__bool__eq_I2_J,axiom,
    aa_o_real(zero_n3304061248610475627l_real,$true) = one_one_real ).

% of_bool_eq(2)
tff(fact_3624_of__bool__eq_I2_J,axiom,
    aa_o_complex(zero_n1201886186963655149omplex,$true) = one_one_complex ).

% of_bool_eq(2)
tff(fact_3625_of__bool__eq_I2_J,axiom,
    aa_o_nat(zero_n2687167440665602831ol_nat,$true) = one_one_nat ).

% of_bool_eq(2)
tff(fact_3626_of__bool__eq_I2_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$true) = one_one_int ).

% of_bool_eq(2)
tff(fact_3627_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_real(zero_n3304061248610475627l_real,(P)) = one_one_real )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_3628_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_complex(zero_n1201886186963655149omplex,(P)) = one_one_complex )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_3629_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_3630_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_3631_signed__take__bit__of__0,axiom,
    ! [Nb: nat] : aa_int_int(bit_ri631733984087533419it_int(Nb),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_3632_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,
        ( (P)
        | (Q) )) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q))) ).

% of_bool_or_iff
tff(fact_3633_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,
        ( (P)
        | (Q) )) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q))) ).

% of_bool_or_iff
tff(fact_3634_finite__Collect__subsets,axiom,
    ! [A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => finite6551019134538273531omplex(collect_set_complex(aTP_Lamm_af(A2))) ) ).

% finite_Collect_subsets
tff(fact_3635_finite__Collect__subsets,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => finite6197958912794628473et_int(collect_set_int(aTP_Lamm_ag(A2))) ) ).

% finite_Collect_subsets
tff(fact_3636_finite__Collect__subsets,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => finite5468666774076196335d_enat(collec2260605976452661553d_enat(aTP_Lamm_ah(A2))) ) ).

% finite_Collect_subsets
tff(fact_3637_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_ai,A2))) ) ).

% finite_Collect_subsets
tff(fact_3638_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aj,K))) ).

% finite_Collect_less_nat
tff(fact_3639_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ak,K))) ).

% finite_Collect_le_nat
tff(fact_3640_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_3641_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_3642_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(minus_minus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_3643_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_3644_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(plus_plus_real(B),aa_real_real(minus_minus_real(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3645_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_3646_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_3647_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3648_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_3649_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_3650_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_3651_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_3652_diff__numeral__special_I9_J,axiom,
    aa_real_real(minus_minus_real(one_one_real),one_one_real) = zero_zero_real ).

% diff_numeral_special(9)
tff(fact_3653_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_3654_signed__take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),numeral_numeral_int(bit0(K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),numeral_numeral_int(K))),numeral_numeral_int(bit0(one))) ).

% signed_take_bit_Suc_bit0
tff(fact_3655_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_3656_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_o_real(zero_n3304061248610475627l_real,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_3657_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_3658_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_3659_zero__less__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% zero_less_diff
tff(fact_3660_diff__is__0__eq_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_3661_diff__is__0__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% diff_is_0_eq
tff(fact_3662_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_o_real(zero_n3304061248610475627l_real,(P))),one_one_real)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_3663_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_3664_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_3665_of__bool__not__iff,axiom,
    ! [P: $o] : aa_o_complex(zero_n1201886186963655149omplex,~ (P)) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_o_complex(zero_n1201886186963655149omplex,(P))) ).

% of_bool_not_iff
tff(fact_3666_of__bool__not__iff,axiom,
    ! [P: $o] : aa_o_real(zero_n3304061248610475627l_real,~ (P)) = aa_real_real(minus_minus_real(one_one_real),aa_o_real(zero_n3304061248610475627l_real,(P))) ).

% of_bool_not_iff
tff(fact_3667_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_3668_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,Ia: 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,Ia),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,Ia),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_3669_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,Ia: 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)),Ia) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Ia)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_3670_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,Ia: 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,Ia),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,Ia),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_3671_diff__Suc__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),one_one_nat) = Nb ).

% diff_Suc_1
tff(fact_3672_Suc__0__mod__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb != aa_nat_nat(suc,zero_zero_nat)) ).

% Suc_0_mod_eq
tff(fact_3673_signed__take__bit__Suc__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_3674_Suc__pred,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).

% Suc_pred
tff(fact_3675_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,Ia: 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,Ia),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,Ia),K)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_3676_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,Ia: 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))),Ia) = 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),Ia)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_3677_Suc__diff__1,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) = Nb ) ) ).

% Suc_diff_1
tff(fact_3678_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique5052692396658037445od_int(Mb,one) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,numeral_numeral_int(Mb)),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_3679_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique5055182867167087721od_nat(Mb,one) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_num_nat(numeral_numeral_nat,Mb)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_3680_even__diff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(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,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ).

% even_diff
tff(fact_3681_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,bit0(one))) = zero_zero_nat ).

% of_bool_half_eq_0
tff(fact_3682_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_int_int(divide_divide_int(aa_o_int(zero_n2684676970156552555ol_int,(B))),numeral_numeral_int(bit0(one))) = zero_zero_int ).

% of_bool_half_eq_0
tff(fact_3683_odd__Suc__minus__one,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).

% odd_Suc_minus_one
tff(fact_3684_even__diff__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ) ) ).

% even_diff_nat
tff(fact_3685_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one))),Nb)),one_one_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3686_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)),one_one_int))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3687_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_3688_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_3689_bits__1__div__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_3690_bits__1__div__exp,axiom,
    ! [Nb: nat] : aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_3691_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,bit0(one))),A))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% flip_bit_0
tff(fact_3692_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,numeral_numeral_int(bit0(one))),A))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one))))) ).

% flip_bit_0
tff(fact_3693_dvd__diff__nat,axiom,
    ! [K: nat,Mb: nat,Nb: 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),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).

% dvd_diff_nat
tff(fact_3694_dvd__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb)
       => ( Mb = Nb ) ) ) ).

% dvd_antisym
tff(fact_3695_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_3696_pred__subset__eq,axiom,
    ! [R: set_o,S3: set_o] :
      ( ord_less_eq_fun_o_o(aTP_Lamm_a(R),aTP_Lamm_a(S3))
    <=> aa_set_o_o(ord_less_eq_set_o(R),S3) ) ).

% pred_subset_eq
tff(fact_3697_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_3698_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_3699_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_3700_prop__restrict,axiom,
    ! [Xb: $o,Z4: set_o,X5: set_o,P: fun_o_o] :
      ( aa_set_o_o(member_o2((Xb)),Z4)
     => ( aa_set_o_o(ord_less_eq_set_o(Z4),collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_al(X5),P)))
       => aa_o_o(P,(Xb)) ) ) ).

% prop_restrict
tff(fact_3701_prop__restrict,axiom,
    ! [Xb: real,Z4: set_real,X5: set_real,P: fun_real_o] :
      ( aa_set_real_o(member_real2(Xb),Z4)
     => ( aa_set_real_o(ord_less_eq_set_real(Z4),collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_am(X5),P)))
       => aa_real_o(P,Xb) ) ) ).

% prop_restrict
tff(fact_3702_prop__restrict,axiom,
    ! [Xb: list_nat,Z4: set_list_nat,X5: set_list_nat,P: fun_list_nat_o] :
      ( member_list_nat(Xb,Z4)
     => ( ord_le6045566169113846134st_nat(Z4,collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_an(X5),P)))
       => aa_list_nat_o(P,Xb) ) ) ).

% prop_restrict
tff(fact_3703_prop__restrict,axiom,
    ! [Xb: set_nat,Z4: set_set_nat,X5: set_set_nat,P: fun_set_nat_o] :
      ( aa_set_set_nat_o(member_set_nat2(Xb),Z4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(Z4),collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ao(X5),P)))
       => aa_set_nat_o(P,Xb) ) ) ).

% prop_restrict
tff(fact_3704_prop__restrict,axiom,
    ! [Xb: int,Z4: set_int,X5: set_int,P: fun_int_o] :
      ( aa_set_int_o(member_int2(Xb),Z4)
     => ( aa_set_int_o(ord_less_eq_set_int(Z4),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ap(X5),P)))
       => aa_int_o(P,Xb) ) ) ).

% prop_restrict
tff(fact_3705_prop__restrict,axiom,
    ! [Xb: nat,Z4: set_nat,X5: set_nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat2(Xb),Z4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z4),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aq(X5),P)))
       => aa_nat_o(P,Xb) ) ) ).

% prop_restrict
tff(fact_3706_Collect__restrict,axiom,
    ! [X5: set_o,P: fun_o_o] : aa_set_o_o(ord_less_eq_set_o(collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_al(X5),P))),X5) ).

% Collect_restrict
tff(fact_3707_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_am(X5),P))),X5) ).

% Collect_restrict
tff(fact_3708_Collect__restrict,axiom,
    ! [X5: set_list_nat,P: fun_list_nat_o] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_an(X5),P)),X5) ).

% Collect_restrict
tff(fact_3709_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_ao(X5),P))),X5) ).

% Collect_restrict
tff(fact_3710_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_ap(X5),P))),X5) ).

% Collect_restrict
tff(fact_3711_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_aq(X5),P))),X5) ).

% Collect_restrict
tff(fact_3712_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_3713_less__eq__set__def,axiom,
    ! [A2: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
    <=> ord_less_eq_fun_o_o(aTP_Lamm_a(A2),aTP_Lamm_a(B2)) ) ).

% less_eq_set_def
tff(fact_3714_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_3715_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_3716_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_3717_Collect__subset,axiom,
    ! [A2: set_o,P: fun_o_o] : aa_set_o_o(ord_less_eq_set_o(collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_al(A2),P))),A2) ).

% Collect_subset
tff(fact_3718_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_am(A2),P))),A2) ).

% Collect_subset
tff(fact_3719_Collect__subset,axiom,
    ! [A2: set_list_nat,P: fun_list_nat_o] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_an(A2),P)),A2) ).

% Collect_subset
tff(fact_3720_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_ao(A2),P))),A2) ).

% Collect_subset
tff(fact_3721_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_ap(A2),P))),A2) ).

% Collect_subset
tff(fact_3722_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_aq(A2),P))),A2) ).

% Collect_subset
tff(fact_3723_of__bool__eq__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(P2)) = aa_o_nat(zero_n2687167440665602831ol_nat,(Q2)) )
    <=> ( (P2)
      <=> (Q2) ) ) ).

% of_bool_eq_iff
tff(fact_3724_of__bool__eq__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P2)) = aa_o_int(zero_n2684676970156552555ol_int,(Q2)) )
    <=> ( (P2)
      <=> (Q2) ) ) ).

% of_bool_eq_iff
tff(fact_3725_diff__commute,axiom,
    ! [Ia: 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,Ia),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ia),K)),J) ).

% diff_commute
tff(fact_3726_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_3727_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_real(minus_minus_real(C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_3728_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_3729_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_3730_diff__right__commute,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),C)),B) = aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),B)),C) ).

% diff_right_commute
tff(fact_3731_empty__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(aTP_Lamm_ar) ).

% empty_def
tff(fact_3732_empty__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(aTP_Lamm_as) ).

% empty_def
tff(fact_3733_empty__def,axiom,
    bot_bot_set_real = collect_real(aTP_Lamm_at) ).

% empty_def
tff(fact_3734_empty__def,axiom,
    bot_bot_set_o = collect_o(aTP_Lamm_au) ).

% empty_def
tff(fact_3735_empty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_av) ).

% empty_def
tff(fact_3736_empty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_aw) ).

% empty_def
tff(fact_3737_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Xa2: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X3),Xa2),R)
        <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X3),Xa2),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_3738_pred__equals__eq2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Xa2: product_prod_nat_nat] :
          ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,X3),Xa2),R)
        <=> member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,X3),Xa2),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_3739_pred__equals__eq2,axiom,
    ! [R: set_Pr7556676689462069481BT_nat,S3: set_Pr7556676689462069481BT_nat] :
      ( ! [X3: vEBT_VEBT,Xa2: nat] :
          ( member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,X3),Xa2),R)
        <=> member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,X3),Xa2),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_3740_pred__equals__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Xa2: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Xa2),R)
        <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Xa2),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_3741_pred__equals__eq2,axiom,
    ! [R: set_Pr2457182780427864761d_enat,S3: set_Pr2457182780427864761d_enat] :
      ( ! [X3: vEBT_VEBT,Xa2: extended_enat] :
          ( member38198578724832770d_enat(aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,X3),Xa2),R)
        <=> member38198578724832770d_enat(aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,X3),Xa2),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_3742_lambda__zero,axiom,
    aTP_Lamm_ax = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_3743_lambda__zero,axiom,
    aTP_Lamm_ay = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_3744_lambda__zero,axiom,
    aTP_Lamm_az = times_times_real(zero_zero_real) ).

% lambda_zero
tff(fact_3745_lambda__zero,axiom,
    aTP_Lamm_ba = times_times_complex(zero_zero_complex) ).

% lambda_zero
tff(fact_3746_lambda__zero,axiom,
    aTP_Lamm_bb = times_7803423173614009249d_enat(zero_z5237406670263579293d_enat) ).

% lambda_zero
tff(fact_3747_lambda__one,axiom,
    aTP_Lamm_bc = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_3748_lambda__one,axiom,
    aTP_Lamm_bd = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_3749_lambda__one,axiom,
    aTP_Lamm_be = times_times_real(one_one_real) ).

% lambda_one
tff(fact_3750_lambda__one,axiom,
    aTP_Lamm_bf = times_times_complex(one_one_complex) ).

% lambda_one
tff(fact_3751_lambda__one,axiom,
    aTP_Lamm_bg = times_7803423173614009249d_enat(one_on7984719198319812577d_enat) ).

% lambda_one
tff(fact_3752_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_bh,A))),collect_real(aa_real_fun_real_o(aTP_Lamm_bh,B)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% subset_divisors_dvd
tff(fact_3753_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_bi,A))),collect_int(aa_int_fun_int_o(aTP_Lamm_bi,B)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% subset_divisors_dvd
tff(fact_3754_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_bj,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bj,B)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% subset_divisors_dvd
tff(fact_3755_max__def__raw,axiom,
    ! [X: extended_enat,Xa: extended_enat] :
      aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X),Xa) = $ite(aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_3756_max__def__raw,axiom,
    ! [X: filter_nat,Xa: filter_nat] :
      ord_max_filter_nat(X,Xa) = $ite(ord_le2510731241096832064er_nat(X,Xa),Xa,X) ).

% max_def_raw
tff(fact_3757_max__def__raw,axiom,
    ! [X: real,Xa: real] :
      aa_real_real(ord_max_real(X),Xa) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_3758_max__def__raw,axiom,
    ! [X: set_nat,Xa: set_nat] :
      ord_max_set_nat(X,Xa) = $ite(aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_3759_max__def__raw,axiom,
    ! [X: nat,Xa: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Xa) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_3760_max__def__raw,axiom,
    ! [X: int,Xa: int] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,X),Xa) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_3761_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ord_le768339454706049738_nat_o(aTP_Lamm_bk(R),aTP_Lamm_bk(S3))
    <=> ord_le3146513528884898305at_nat(R,S3) ) ).

% pred_subset_eq2
tff(fact_3762_pred__subset__eq2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat] :
      ( ord_le3650406611588909774_nat_o(aTP_Lamm_bl(R),aTP_Lamm_bl(S3))
    <=> ord_le3000389064537975527at_nat(R,S3) ) ).

% pred_subset_eq2
tff(fact_3763_pred__subset__eq2,axiom,
    ! [R: set_Pr7556676689462069481BT_nat,S3: set_Pr7556676689462069481BT_nat] :
      ( ord_le5001929208632972944_nat_o(aTP_Lamm_bm(R),aTP_Lamm_bm(S3))
    <=> ord_le3442269383143156041BT_nat(R,S3) ) ).

% pred_subset_eq2
tff(fact_3764_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ord_le77980682512674322_int_o(aTP_Lamm_bn(R),aTP_Lamm_bn(S3))
    <=> ord_le2843351958646193337nt_int(R,S3) ) ).

% pred_subset_eq2
tff(fact_3765_pred__subset__eq2,axiom,
    ! [R: set_Pr2457182780427864761d_enat,S3: set_Pr2457182780427864761d_enat] :
      ( ord_le7796109748605290332enat_o(aTP_Lamm_bo(R),aTP_Lamm_bo(S3))
    <=> ord_le8566326065971749465d_enat(R,S3) ) ).

% pred_subset_eq2
tff(fact_3766_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_bh,A))),collect_real(aa_real_fun_real_o(aTP_Lamm_bh,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_3767_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_bj,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bj,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_3768_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_bi,A))),collect_int(aa_int_fun_int_o(aTP_Lamm_bi,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_3769_bot__empty__eq2,axiom,
    ! [X: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bot_bo1118997454112379798_nat_o,X),Xa)
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Xa),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq2
tff(fact_3770_bot__empty__eq2,axiom,
    ! [X: product_prod_nat_nat,Xa: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(bot_bo5970965355748268442_nat_o,X),Xa)
    <=> member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,X),Xa),bot_bo5327735625951526323at_nat) ) ).

% bot_empty_eq2
tff(fact_3771_bot__empty__eq2,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(bot_bo4779999733149731012_nat_o,X),Xa)
    <=> member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,X),Xa),bot_bo1642239108664514429BT_nat) ) ).

% bot_empty_eq2
tff(fact_3772_bot__empty__eq2,axiom,
    ! [X: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(bot_bo428638681919004382_int_o,X),Xa)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa),bot_bo1796632182523588997nt_int) ) ).

% bot_empty_eq2
tff(fact_3773_bot__empty__eq2,axiom,
    ! [X: vEBT_VEBT,Xa: extended_enat] :
      ( aa_Extended_enat_o(aa_VEB2705759856535948141enat_o(bot_bo6128915908787945000enat_o,X),Xa)
    <=> member38198578724832770d_enat(aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,X),Xa),bot_bo4330027929424010533d_enat) ) ).

% bot_empty_eq2
tff(fact_3774_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_o,Ia: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bp(P),Ia))) ).

% finite_M_bounded_by_nat
tff(fact_3775_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_bq(F),U))) ) ).

% finite_less_ub
tff(fact_3776_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      aa_o_real(zero_n3304061248610475627l_real,
        ( (P)
        & (Q) )) = aa_real_real(times_times_real(aa_o_real(zero_n3304061248610475627l_real,(P))),aa_o_real(zero_n3304061248610475627l_real,(Q))) ).

% of_bool_conj
tff(fact_3777_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      aa_o_complex(zero_n1201886186963655149omplex,
        ( (P)
        & (Q) )) = aa_complex_complex(times_times_complex(aa_o_complex(zero_n1201886186963655149omplex,(P))),aa_o_complex(zero_n1201886186963655149omplex,(Q))) ).

% of_bool_conj
tff(fact_3778_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      aa_o_Extended_enat(zero_n1046097342994218471d_enat,
        ( (P)
        & (Q) )) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_o_Extended_enat(zero_n1046097342994218471d_enat,(P))),aa_o_Extended_enat(zero_n1046097342994218471d_enat,(Q))) ).

% of_bool_conj
tff(fact_3779_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_3780_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_3781_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_real(minus_minus_real(C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3782_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_3783_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),C)) ) ).

% diff_right_mono
tff(fact_3784_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_3785_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(C),A)),aa_real_real(minus_minus_real(C),B)) ) ).

% diff_left_mono
tff(fact_3786_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_3787_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),D)) ) ) ).

% diff_mono
tff(fact_3788_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_3789_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_3790_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_3791_eq__iff__diff__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
    <=> ( aa_real_real(minus_minus_real(A),B) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_3792_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),C)) ) ).

% diff_strict_right_mono
tff(fact_3793_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_3794_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(C),A)),aa_real_real(minus_minus_real(C),B)) ) ).

% diff_strict_left_mono
tff(fact_3795_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_3796_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_real(minus_minus_real(C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_3797_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_3798_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),D)) ) ) ).

% diff_strict_mono
tff(fact_3799_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_3800_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_3801_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ).

% left_diff_distrib
tff(fact_3802_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_3803_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_3804_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),B)),aa_real_real(times_times_real(A),C)) ).

% right_diff_distrib
tff(fact_3805_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_3806_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_3807_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_3808_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),C)),A) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(B),A)),aa_real_real(times_times_real(C),A)) ).

% left_diff_distrib'
tff(fact_3809_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_3810_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_3811_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_3812_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),B)),aa_real_real(times_times_real(A),C)) ).

% right_diff_distrib'
tff(fact_3813_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_3814_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_3815_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_3816_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% diff_diff_eq
tff(fact_3817_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_3818_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_3819_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( aa_real_real(plus_plus_real(C),B) = A )
     => ( C = aa_real_real(minus_minus_real(A),B) ) ) ).

% add_implies_diff
tff(fact_3820_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_3821_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(minus_minus_real(A),aa_real_real(plus_plus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_3822_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_3823_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C)),B) ).

% diff_add_eq
tff(fact_3824_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_3825_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(minus_minus_real(A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C)),B) ).

% diff_diff_eq2
tff(fact_3826_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_3827_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),C) ).

% add_diff_eq
tff(fact_3828_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_3829_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A = aa_real_real(minus_minus_real(C),B) )
    <=> ( aa_real_real(plus_plus_real(A),B) = C ) ) ).

% eq_diff_eq
tff(fact_3830_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_3831_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = C )
    <=> ( A = aa_real_real(plus_plus_real(C),B) ) ) ).

% diff_eq_eq
tff(fact_3832_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_3833_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(plus_plus_real(K),A) )
     => ( aa_real_real(minus_minus_real(A2),B) = aa_real_real(plus_plus_real(K),aa_real_real(minus_minus_real(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_3834_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_3835_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) = aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B)),aa_real_real(minus_minus_real(C),D)) ).

% add_diff_add
tff(fact_3836_diff__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ).

% diff_divide_distrib
tff(fact_3837_dvd__diff,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Z)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ) ) ).

% dvd_diff
tff(fact_3838_dvd__diff,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Z)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_real_real(minus_minus_real(Y),Z)) ) ) ).

% dvd_diff
tff(fact_3839_zero__induct__lemma,axiom,
    ! [P: fun_nat_o,K: nat,Ia: 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),Ia)) ) ) ).

% zero_induct_lemma
tff(fact_3840_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_3841_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_3842_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_3843_diffs0__imp__equal,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb) = zero_zero_nat )
       => ( Mb = Nb ) ) ) ).

% diffs0_imp_equal
tff(fact_3844_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_3845_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Nb: 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),Nb)),K) ) ).

% less_imp_diff_less
tff(fact_3846_diff__less__mono2,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Mb)) ) ) ).

% diff_less_mono2
tff(fact_3847_dvd__minus__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% dvd_minus_self
tff(fact_3848_diff__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Mb)) ) ).

% diff_le_mono2
tff(fact_3849_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_3850_diff__le__self,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Mb) ).

% diff_le_self
tff(fact_3851_diff__le__mono,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),L)) ) ).

% diff_le_mono
tff(fact_3852_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,Mb: nat,Nb: 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),Nb)
       => ( 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,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_3853_le__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: 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),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% le_diff_iff
tff(fact_3854_eq__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: 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),Nb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K) )
        <=> ( Mb = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_3855_less__eq__dvd__minus,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% less_eq_dvd_minus
tff(fact_3856_dvd__diffD1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
     => ( 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,Nb),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb) ) ) ) ).

% dvd_diffD1
tff(fact_3857_dvd__diffD,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb) ) ) ) ).

% dvd_diffD
tff(fact_3858_diff__add__inverse2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),Nb) = Mb ).

% diff_add_inverse2
tff(fact_3859_diff__add__inverse,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),Nb) = Mb ).

% diff_add_inverse
tff(fact_3860_diff__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% diff_cancel2
tff(fact_3861_Nat_Odiff__cancel,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% Nat.diff_cancel
tff(fact_3862_diff__mult__distrib,axiom,
    ! [Mb: nat,Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),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,Nb),K)) ).

% diff_mult_distrib
tff(fact_3863_diff__mult__distrib2,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = 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),Nb)) ).

% diff_mult_distrib2
tff(fact_3864_max__diff__distrib__left,axiom,
    ! [Xb: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ).

% max_diff_distrib_left
tff(fact_3865_max__diff__distrib__left,axiom,
    ! [Xb: real,Y: real,Z: real] : aa_real_real(minus_minus_real(aa_real_real(ord_max_real(Xb),Y)),Z) = aa_real_real(ord_max_real(aa_real_real(minus_minus_real(Xb),Z)),aa_real_real(minus_minus_real(Y),Z)) ).

% max_diff_distrib_left
tff(fact_3866_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,bit0(Nb)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2) ) ).

% numeral_code(2)
tff(fact_3867_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numera1916890842035813515d_enat(bit0(Nb)) = $let(
        m2: extended_enat,
        m2:= numera1916890842035813515d_enat(Nb),
        aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(m2),m2) ) ).

% numeral_code(2)
tff(fact_3868_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numeral_numeral_int(bit0(Nb)) = $let(
        m2: int,
        m2:= numeral_numeral_int(Nb),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2) ) ).

% numeral_code(2)
tff(fact_3869_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numeral_numeral_real(bit0(Nb)) = $let(
        m2: real,
        m2:= numeral_numeral_real(Nb),
        aa_real_real(plus_plus_real(m2),m2) ) ).

% numeral_code(2)
tff(fact_3870_set__vebt__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_set_vebt(Ta) = collect_nat(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta)) ).

% set_vebt_def
tff(fact_3871_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_bj,Mb))) ) ).

% finite_divisors_nat
tff(fact_3872_signed__take__bit__int__less__eq,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)),K)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(suc,Nb)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_3873_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(zero_n3304061248610475627l_real,(P))) ).

% zero_less_eq_of_bool
tff(fact_3874_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_3875_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_3876_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(zero_n3304061248610475627l_real,(P))),one_one_real) ).

% of_bool_less_eq_one
tff(fact_3877_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_3878_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_3879_of__bool__def,axiom,
    ! [P2: $o] :
      aa_o_real(zero_n3304061248610475627l_real,(P2)) = $ite((P2),one_one_real,zero_zero_real) ).

% of_bool_def
tff(fact_3880_of__bool__def,axiom,
    ! [P2: $o] :
      aa_o_complex(zero_n1201886186963655149omplex,(P2)) = $ite((P2),one_one_complex,zero_zero_complex) ).

% of_bool_def
tff(fact_3881_of__bool__def,axiom,
    ! [P2: $o] :
      aa_o_Extended_enat(zero_n1046097342994218471d_enat,(P2)) = $ite((P2),one_on7984719198319812577d_enat,zero_z5237406670263579293d_enat) ).

% of_bool_def
tff(fact_3882_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_3883_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_3884_split__of__bool,axiom,
    ! [P: fun_real_o,P2: $o] :
      ( aa_real_o(P,aa_o_real(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_3885_split__of__bool,axiom,
    ! [P: fun_complex_o,P2: $o] :
      ( aa_complex_o(P,aa_o_complex(zero_n1201886186963655149omplex,(P2)))
    <=> ( ( (P2)
         => aa_complex_o(P,one_one_complex) )
        & ( ~ (P2)
         => aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool
tff(fact_3886_split__of__bool,axiom,
    ! [P: fun_Extended_enat_o,P2: $o] :
      ( aa_Extended_enat_o(P,aa_o_Extended_enat(zero_n1046097342994218471d_enat,(P2)))
    <=> ( ( (P2)
         => aa_Extended_enat_o(P,one_on7984719198319812577d_enat) )
        & ( ~ (P2)
         => aa_Extended_enat_o(P,zero_z5237406670263579293d_enat) ) ) ) ).

% split_of_bool
tff(fact_3887_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_3888_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_3889_split__of__bool__asm,axiom,
    ! [P: fun_real_o,P2: $o] :
      ( aa_real_o(P,aa_o_real(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_3890_split__of__bool__asm,axiom,
    ! [P: fun_complex_o,P2: $o] :
      ( aa_complex_o(P,aa_o_complex(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_3891_split__of__bool__asm,axiom,
    ! [P: fun_Extended_enat_o,P2: $o] :
      ( aa_Extended_enat_o(P,aa_o_Extended_enat(zero_n1046097342994218471d_enat,(P2)))
    <=> ~ ( ( (P2)
            & ~ aa_Extended_enat_o(P,one_on7984719198319812577d_enat) )
          | ( ~ (P2)
            & ~ aa_Extended_enat_o(P,zero_z5237406670263579293d_enat) ) ) ) ).

% split_of_bool_asm
tff(fact_3892_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_3893_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_3894_le__iff__diff__le__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),B)),zero_zero_real) ) ).

% le_iff_diff_le_0
tff(fact_3895_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_3896_less__iff__diff__less__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),B)),zero_zero_real) ) ).

% less_iff_diff_less_0
tff(fact_3897_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_3898_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(plus_plus_real(C),B)) ) ).

% diff_le_eq
tff(fact_3899_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_3900_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(minus_minus_real(C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),B)),C) ) ).

% le_diff_eq
tff(fact_3901_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_3902_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_3903_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_3904_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_3905_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_3906_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_3907_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_3908_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_3909_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_3910_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_3911_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_3912_add__le__imp__le__diff,axiom,
    ! [Ia: real,K: real,Nb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(Ia),K)),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ia),aa_real_real(minus_minus_real(Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_3913_add__le__imp__le__diff,axiom,
    ! [Ia: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_3914_add__le__imp__le__diff,axiom,
    ! [Ia: int,K: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Ia),K)),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_3915_add__le__add__imp__diff__le,axiom,
    ! [Ia: real,K: real,Nb: real,J: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(Ia),K)),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(plus_plus_real(J),K))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(Ia),K)),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(plus_plus_real(J),K))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3916_add__le__add__imp__diff__le,axiom,
    ! [Ia: nat,K: nat,Nb: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,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,Ia),K)),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,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,Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3917_add__le__add__imp__diff__le,axiom,
    ! [Ia: int,K: int,Nb: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Ia),K)),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Ia),K)),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3918_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(plus_plus_real(C),B)) ) ).

% diff_less_eq
tff(fact_3919_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_3920_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(minus_minus_real(C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),B)),C) ) ).

% less_diff_eq
tff(fact_3921_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_3922_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_3923_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(plus_plus_real(B),aa_real_real(minus_minus_real(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3924_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_3925_square__diff__square__factored,axiom,
    ! [Xb: 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,Xb),Xb)),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,Xb),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)) ).

% square_diff_square_factored
tff(fact_3926_square__diff__square__factored,axiom,
    ! [Xb: real,Y: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Y),Y)) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(Xb),Y)),aa_real_real(minus_minus_real(Xb),Y)) ).

% square_diff_square_factored
tff(fact_3927_square__diff__square__factored,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xb),Xb)),aa_complex_complex(times_times_complex(Y),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Xb),Y)),aa_complex_complex(minus_minus_complex(Xb),Y)) ).

% square_diff_square_factored
tff(fact_3928_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_3929_eq__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E2)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E2)),D) )
    <=> ( C = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_3930_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_3931_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_3932_eq__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E2)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E2)),D) )
    <=> ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_3933_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_3934_mult__diff__mult,axiom,
    ! [Xb: 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,Xb),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,Xb),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,Xb),A)),B)) ).

% mult_diff_mult
tff(fact_3935_mult__diff__mult,axiom,
    ! [Xb: real,Y: real,A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Y)),aa_real_real(times_times_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),aa_real_real(minus_minus_real(Y),B))),aa_real_real(times_times_real(aa_real_real(minus_minus_real(Xb),A)),B)) ).

% mult_diff_mult
tff(fact_3936_mult__diff__mult,axiom,
    ! [Xb: complex,Y: complex,A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xb),Y)),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xb),aa_complex_complex(minus_minus_complex(Y),B))),aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),A)),B)) ).

% mult_diff_mult
tff(fact_3937_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_3938_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_3939_Suc__diff__Suc,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ) ) ).

% Suc_diff_Suc
tff(fact_3940_diff__less__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(suc,Mb)) ).

% diff_less_Suc
tff(fact_3941_diff__less,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Mb) ) ) ).

% diff_less
tff(fact_3942_Suc__diff__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).

% Suc_diff_le
tff(fact_3943_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_3944_less__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: 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),Nb)
       => ( 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,Nb),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ) ).

% less_diff_iff
tff(fact_3945_diff__add__0,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = zero_zero_nat ).

% diff_add_0
tff(fact_3946_less__diff__conv,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),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,Ia),K)),J) ) ).

% less_diff_conv
tff(fact_3947_add__diff__inverse__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = Mb ) ) ).

% add_diff_inverse_nat
tff(fact_3948_le__diff__conv,axiom,
    ! [J: nat,K: nat,Ia: 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)),Ia)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)) ) ).

% le_diff_conv
tff(fact_3949_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,Ia: 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,Ia),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,Ia),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_3950_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,Ia: 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,Ia),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_3951_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,Ia: 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),Ia)),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)),Ia) ) ) ).

% Nat.diff_add_assoc2
tff(fact_3952_Nat_Ole__imp__diff__is__add,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Ia) = K )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Ia) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_3953_diff__Suc__eq__diff__pred,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb) ).

% diff_Suc_eq_diff_pred
tff(fact_3954_mod__geq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) ) ) ).

% mod_geq
tff(fact_3955_mod__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb),Mb,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ).

% mod_if
tff(fact_3956_le__mod__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) ) ) ).

% le_mod_geq
tff(fact_3957_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,Mb: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q2) )
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_3958_nat__minus__add__max,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Mb) ).

% nat_minus_add_max
tff(fact_3959_finite__lists__length__eq,axiom,
    ! [A2: set_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_br(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_3960_finite__lists__length__eq,axiom,
    ! [A2: set_Extended_enat,Nb: nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => finite1862508098717546133d_enat(collec8433460942617342167d_enat(aa_nat4163650157883470461enat_o(aTP_Lamm_bs(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_3961_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,Nb: nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_bt(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_3962_finite__lists__length__eq,axiom,
    ! [A2: set_int,Nb: nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_bu(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_3963_finite__lists__length__eq,axiom,
    ! [A2: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_bv(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_3964_finite__lists__length__le,axiom,
    ! [A2: set_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_bw(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_3965_finite__lists__length__le,axiom,
    ! [A2: set_Extended_enat,Nb: nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => finite1862508098717546133d_enat(collec8433460942617342167d_enat(aa_nat4163650157883470461enat_o(aTP_Lamm_bx(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_3966_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,Nb: nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_by(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_3967_finite__lists__length__le,axiom,
    ! [A2: set_int,Nb: nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_bz(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_3968_finite__lists__length__le,axiom,
    ! [A2: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_ca(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_3969_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(plus_plus_real(aa_real_real(times_times_real(A),E2)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),E2)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3970_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_3971_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(plus_plus_real(aa_real_real(times_times_real(A),E2)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3972_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_3973_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(plus_plus_real(aa_real_real(times_times_real(A),E2)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),E2)),C)),D) ) ).

% less_add_iff1
tff(fact_3974_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_3975_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(plus_plus_real(aa_real_real(times_times_real(A),E2)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),E2)),D)) ) ).

% less_add_iff2
tff(fact_3976_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_3977_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_3978_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(minus_minus_real(A),aa_real_real(divide_divide_real(B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_3979_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,Xb: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Y)),aa_complex_complex(divide1717551699836669952omplex(W2),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xb),Z)),aa_complex_complex(times_times_complex(W2),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3980_diff__frac__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(Xb),Y)),aa_real_real(divide_divide_real(W2),Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(W2),Y))),aa_real_real(times_times_real(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3981_diff__divide__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(Xb),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xb),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3982_diff__divide__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(minus_minus_real(Xb),aa_real_real(divide_divide_real(Y),Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3983_divide__diff__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Xb),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3984_divide__diff__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(Xb),Z)),Y) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Xb),aa_real_real(times_times_real(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3985_square__diff__one__factored,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),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,Xb),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),one_one_int)) ).

% square_diff_one_factored
tff(fact_3986_square__diff__one__factored,axiom,
    ! [Xb: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Xb)),one_one_real) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(Xb),one_one_real)),aa_real_real(minus_minus_real(Xb),one_one_real)) ).

% square_diff_one_factored
tff(fact_3987_square__diff__one__factored,axiom,
    ! [Xb: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xb),Xb)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Xb),one_one_complex)),aa_complex_complex(minus_minus_complex(Xb),one_one_complex)) ).

% square_diff_one_factored
tff(fact_3988_inf__period_I3_J,axiom,
    ! [D: int,D5: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D5)
     => ! [X: int,K4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5))),Ta)) ) ) ).

% inf_period(3)
tff(fact_3989_inf__period_I3_J,axiom,
    ! [D: real,D5: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D5)
     => ! [X: real,K4: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),Ta))
        <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(X),aa_real_real(times_times_real(K4),D5))),Ta)) ) ) ).

% inf_period(3)
tff(fact_3990_inf__period_I3_J,axiom,
    ! [D: complex,D5: complex,Ta: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,D),D5)
     => ! [X: complex,K4: complex] :
          ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,D),aa_complex_complex(plus_plus_complex(X),Ta))
        <=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,D),aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(X),aa_complex_complex(times_times_complex(K4),D5))),Ta)) ) ) ).

% inf_period(3)
tff(fact_3991_inf__period_I4_J,axiom,
    ! [D: int,D5: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D5)
     => ! [X: int,K4: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5))),Ta)) ) ) ).

% inf_period(4)
tff(fact_3992_inf__period_I4_J,axiom,
    ! [D: real,D5: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D5)
     => ! [X: real,K4: real] :
          ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),Ta))
        <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(X),aa_real_real(times_times_real(K4),D5))),Ta)) ) ) ).

% inf_period(4)
tff(fact_3993_inf__period_I4_J,axiom,
    ! [D: complex,D5: complex,Ta: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,D),D5)
     => ! [X: complex,K4: complex] :
          ( ~ aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,D),aa_complex_complex(plus_plus_complex(X),Ta))
        <=> ~ aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,D),aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(X),aa_complex_complex(times_times_complex(K4),D5))),Ta)) ) ) ).

% inf_period(4)
tff(fact_3994_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_3995_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_3996_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_3997_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_3998_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_3999_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_4000_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_4001_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_4002_diff__Suc__less,axiom,
    ! [Nb: nat,Ia: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,Ia))),Nb) ) ).

% diff_Suc_less
tff(fact_4003_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_4004_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_4005_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_4006_less__diff__conv2,axiom,
    ! [K: nat,J: nat,Ia: 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)),Ia)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)) ) ) ).

% less_diff_conv2
tff(fact_4007_nat__eq__add__iff1,axiom,
    ! [J: nat,Ia: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),Ia)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ia),U)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ia),J)),U)),Mb) = Nb ) ) ) ).

% nat_eq_add_iff1
tff(fact_4008_nat__eq__add__iff2,axiom,
    ! [Ia: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ia),U)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Ia)),U)),Nb) ) ) ) ).

% nat_eq_add_iff2
tff(fact_4009_nat__le__add__iff1,axiom,
    ! [J: nat,Ia: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),Ia)
     => ( 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,Ia),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ia),J)),U)),Mb)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_4010_nat__le__add__iff2,axiom,
    ! [Ia: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),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,Ia),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Ia)),U)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_4011_nat__diff__add__eq1,axiom,
    ! [J: nat,Ia: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),Ia)
     => ( 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,Ia),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ia),J)),U)),Mb)),Nb) ) ) ).

% nat_diff_add_eq1
tff(fact_4012_nat__diff__add__eq2,axiom,
    ! [Ia: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),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,Ia),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Ia)),U)),Nb)) ) ) ).

% nat_diff_add_eq2
tff(fact_4013_dvd__minus__add,axiom,
    ! [Q2: nat,Nb: nat,R2: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q2),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),Mb))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Q2))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),Mb)),Q2))) ) ) ) ).

% dvd_minus_add
tff(fact_4014_mod__nat__eqI,axiom,
    ! [R2: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R2),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Mb)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),R2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_4015_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb) != zero_zero_nat )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_4016_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_int_int(divide_divide_int(aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Mb)),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Mb) != zero_zero_int )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_4017_frac__le__eq,axiom,
    ! [Y: real,Z: real,Xb: 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(divide_divide_real(Xb),Y)),aa_real_real(divide_divide_real(W2),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(W2),Y))),aa_real_real(times_times_real(Y),Z))),zero_zero_real) ) ) ) ).

% frac_le_eq
tff(fact_4018_frac__less__eq,axiom,
    ! [Y: real,Z: real,Xb: 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(divide_divide_real(Xb),Y)),aa_real_real(divide_divide_real(W2),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(W2),Y))),aa_real_real(times_times_real(Y),Z))),zero_zero_real) ) ) ) ).

% frac_less_eq
tff(fact_4019_power__diff,axiom,
    ! [A: complex,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ) ) ) ).

% power_diff
tff(fact_4020_power__diff,axiom,
    ! [A: nat,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_diff
tff(fact_4021_power__diff,axiom,
    ! [A: int,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).

% power_diff
tff(fact_4022_power__diff,axiom,
    ! [A: real,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_real_real(divide_divide_real(aa_nat_real(power_power_real(A),Mb)),aa_nat_real(power_power_real(A),Nb)) ) ) ) ).

% power_diff
tff(fact_4023_Suc__diff__eq__diff__pred,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_4024_Suc__pred_H,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_4025_div__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ).

% div_if
tff(fact_4026_div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ) ) ).

% div_geq
tff(fact_4027_add__eq__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = $ite(Mb = zero_zero_nat,Nb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb))) ).

% add_eq_if
tff(fact_4028_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,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,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,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,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,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_4029_nat__less__add__iff2,axiom,
    ! [Ia: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),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,Ia),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Ia)),U)),Nb)) ) ) ).

% nat_less_add_iff2
tff(fact_4030_nat__less__add__iff1,axiom,
    ! [J: nat,Ia: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),Ia)
     => ( 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,Ia),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ia),J)),U)),Mb)),Nb) ) ) ).

% nat_less_add_iff1
tff(fact_4031_mult__eq__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = $ite(Mb = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb))) ).

% mult_eq_if
tff(fact_4032_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(plus_plus_real(U),aa_real_real(divide_divide_real(aa_real_real(times_times_real(R2),aa_real_real(minus_minus_real(V),U))),S))),V) ) ) ) ).

% scaling_mono
tff(fact_4033_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4034_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb) != zero_zero_int )
     => ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4035_power__diff__power__eq,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% power_diff_power_eq
tff(fact_4036_power__diff__power__eq,axiom,
    ! [A: int,Mb: nat,Nb: 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),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% power_diff_power_eq
tff(fact_4037_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_4038_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_4039_power__eq__if,axiom,
    ! [P2: real,Mb: nat] :
      aa_nat_real(power_power_real(P2),Mb) = $ite(Mb = zero_zero_nat,one_one_real,aa_real_real(times_times_real(P2),aa_nat_real(power_power_real(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_4040_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_4041_power__eq__if,axiom,
    ! [P2: extended_enat,Mb: nat] :
      aa_nat_Extended_enat(power_8040749407984259932d_enat(P2),Mb) = $ite(Mb = zero_zero_nat,one_on7984719198319812577d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(P2),aa_nat_Extended_enat(power_8040749407984259932d_enat(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_4042_vebt__buildup_Oelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = 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,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,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,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,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,bit0(one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_4043_power__minus__mult,axiom,
    ! [Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) ) ) ).

% power_minus_mult
tff(fact_4044_power__minus__mult,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_int(power_power_int(A),Nb) ) ) ).

% power_minus_mult
tff(fact_4045_power__minus__mult,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(times_times_real(aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_real(power_power_real(A),Nb) ) ) ).

% power_minus_mult
tff(fact_4046_power__minus__mult,axiom,
    ! [Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),Nb) ) ) ).

% power_minus_mult
tff(fact_4047_power__minus__mult,axiom,
    ! [Nb: nat,A: extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(power_8040749407984259932d_enat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_Extended_enat(power_8040749407984259932d_enat(A),Nb) ) ) ).

% power_minus_mult
tff(fact_4048_diff__le__diff__pow,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,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),Nb)),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),Nb))) ) ).

% diff_le_diff_pow
tff(fact_4049_le__div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ) ) ).

% le_div_geq
tff(fact_4050_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(Mb,Nb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(divide_divide_int(numeral_numeral_int(Mb)),numeral_numeral_int(Nb))),aa_int_int(modulo_modulo_int(numeral_numeral_int(Mb)),numeral_numeral_int(Nb))) ).

% divmod_def
tff(fact_4051_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod_def
tff(fact_4052_divmod_H__nat__def,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod'_nat_def
tff(fact_4053_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeLista: list_VEBT_VEBT,S: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,vEBT_Node(Uy,aa_nat_nat(suc,V),TreeLista,S)),Xb)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_4054_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,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,bit0(one))),A3))),aa_num_nat(numeral_numeral_nat,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,bit0(one))),A3))) ) )
       => aa_nat_o(P,A) ) ) ).

% bits_induct
tff(fact_4055_bits__induct,axiom,
    ! [P: fun_int_o,A: int] :
      ( ! [A3: int] :
          ( ( aa_int_int(divide_divide_int(A3),numeral_numeral_int(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,numeral_numeral_int(bit0(one))),A3))),numeral_numeral_int(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,numeral_numeral_int(bit0(one))),A3))) ) )
       => aa_int_o(P,A) ) ) ).

% bits_induct
tff(fact_4056_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Vd2: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeLista,Vd2)),Xb)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_4057_vebt__member_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya)),Xb)
    <=> $ite(
          Xb = Mia,
          $true,
          $ite(
            Xb = Maa,
            $true,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
              $false,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
                $false,
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),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,bit0(one)))),
                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_4058_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mia: nat,Maa: nat,V: nat,TreeLista: list_VEBT_VEBT,Vc: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,V),TreeLista,Vc)),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa )
        | $let(
            pos: nat,
            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_4059_exp__mod__exp,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)) ).

% exp_mod_exp
tff(fact_4060_exp__mod__exp,axiom,
    ! [Mb: nat,Nb: nat] : aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Mb)),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Mb)) ).

% exp_mod_exp
tff(fact_4061_power2__diff,axiom,
    ! [Xb: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Xb),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),Xb)),Y)) ).

% power2_diff
tff(fact_4062_power2__diff,axiom,
    ! [Xb: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),aa_num_nat(numeral_numeral_nat,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(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),Xb)),Y)) ).

% power2_diff
tff(fact_4063_power2__diff,axiom,
    ! [Xb: real,Y: real] : aa_nat_real(power_power_real(aa_real_real(minus_minus_real(Xb),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(aa_nat_real(power_power_real(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),Xb)),Y)) ).

% power2_diff
tff(fact_4064_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4065_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Mb))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4066_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xb),Xaa)
      <=> (Y) )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
           => (Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : Xb = 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,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_4067_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xb),Xaa)
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A3),
                  $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [S2: vEBT_VEBT] : Xb = 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,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_4068_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xb),Xaa)
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A3),
                $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb != vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : Xb = 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,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_4069_even__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(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,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ).

% even_mod_4_div_2
tff(fact_4070_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xb),Xaa)
     => ( ! [Mi: nat,Ma: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3,Vb2)
           => ~ ( ( Xaa = Mi )
                | ( Xaa = Ma ) ) )
       => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
             => ~ ( ( Xaa = Mi )
                  | ( Xaa = Ma )
                  | $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) )
         => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [Vd: vEBT_VEBT] : Xb = 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,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_4071_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,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,bit0(one))),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4072_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,numeral_numeral_int(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,numeral_numeral_int(bit0(one))),B))),B) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4073_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one))),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_4074_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(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(numeral_numeral_int(bit0(one))),Mb)),one_one_int)),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_4075_vebt__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xb),Xaa)
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A3),
                  $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)
             => ~ $ite(
                    Xaa = Mi,
                    $true,
                    $ite(
                      Xaa = Ma,
                      $true,
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                        $false,
                        $ite(
                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_4076_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xb),Xaa)
     => ( ! [Uu2: $o,Uv2: $o] : Xb != vEBT_Leaf((Uu2),(Uv2))
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3,Vb2)
               => ( ( Xaa = Mi )
                  | ( Xaa = Ma ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
                 => ( ( Xaa = Mi )
                    | ( Xaa = Ma )
                    | $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : Xb = 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,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_4077_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xb),Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => (Y) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
           => (Y) )
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3,Vb2)
               => ( (Y)
                <=> ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
                 => ( (Y)
                  <=> ~ ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : Xb = 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,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_4078_vebt__insert_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Mia,Xb),
        $let(
          h: nat,
          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista))
            & ~ ( ( Xb = Mia )
                | ( Xb = Maa ) ) ),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Xb,Mia)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Maa))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeLista,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),
              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,h)),vEBT_vebt_insert(Summarya,h),Summarya)),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_4079_vebt__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xb),Xaa)
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A3),
                $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != 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] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)
                   => $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_4080_divmod__step__eq,axiom,
    ! [L: num,Q2: int,R2: int] :
      unique5024387138958732305ep_int(L,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_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,numeral_numeral_int(bit0(one))),Q2)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),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,numeral_numeral_int(bit0(one))),Q2)),R2)) ).

% divmod_step_eq
tff(fact_4081_divmod__step__eq,axiom,
    ! [L: num,Q2: nat,R2: nat] :
      unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Q2),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(aa_nat8803788312118491388at_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,bit0(one))),Q2)),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(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Q2)),R2)) ).

% divmod_step_eq
tff(fact_4082_inrange,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,vEBT_VEBT_set_vebt(Ta)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)),one_one_nat))) ) ).

% inrange
tff(fact_4083_finite__nth__roots,axiom,
    ! [Nb: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_cb(Nb),C))) ) ).

% finite_nth_roots
tff(fact_4084_finite__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => aa_set_real_o(finite_finite_real,collect_real(aa_nat_fun_real_o(aTP_Lamm_cc,Nb))) ) ).

% finite_roots_unity
tff(fact_4085_finite__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_cd(Nb))) ) ).

% finite_roots_unity
tff(fact_4086_vebt__insert_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = $ite(
                      Xaa = zero_zero_nat,
                      vEBT_Leaf($true,(B3)),
                      $ite(Xaa = one_one_nat,vEBT_Leaf((A3),$true),vEBT_Leaf((A3),(B3))) ) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Leaf((A3),(B3))),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,zero_zero_nat,Ts2,S2) )
               => ( ( Y = vEBT_Node(Info2,zero_zero_nat,Ts2,S2) )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(Info2,zero_zero_nat,Ts2,S2)),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S2) )
                 => ( ( Y = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S2) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S2)),Xaa)) ) )
             => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
                   => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = Ma ) ) ),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,
                                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Xaa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),
                                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),vEBT_vebt_insert(Summary,h),Summary)),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) ) ) ) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_4087_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(Mb,Nb) = $ite(aa_num_o(ord_less_num(Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),numeral_numeral_int(Mb)),unique5024387138958732305ep_int(Nb,unique5052692396658037445od_int(Mb,bit0(Nb)))) ).

% divmod_divmod_step
tff(fact_4088_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(Mb,Nb) = $ite(aa_num_o(ord_less_num(Mb),Nb),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Mb)),unique5026877609467782581ep_nat(Nb,unique5055182867167087721od_nat(Mb,bit0(Nb)))) ).

% divmod_divmod_step
tff(fact_4089_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,Xb: fun_o_nat,Y: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_ce(I5),Xb)))
     => ( aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_ce(I5),Y)))
       => aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_nat_fun_o_o(aa_fun5833554292436378194un_o_o(aTP_Lamm_cf(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4090_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xb: fun_real_nat,Y: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun7775777430665565006real_o(aTP_Lamm_cg(I5),Xb)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun7775777430665565006real_o(aTP_Lamm_cg(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun7775777430665565006real_o(aa_fun1158036852828689646real_o(aTP_Lamm_ch(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4091_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xb: fun_nat_nat,Y: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_ci(I5),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_ci(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aa_fun6348600740168882050_nat_o(aTP_Lamm_cj(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4092_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,Xb: fun_complex_nat,Y: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun8723867720292580170plex_o(aTP_Lamm_ck(I5),Xb)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun8723867720292580170plex_o(aTP_Lamm_ck(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun8723867720292580170plex_o(aa_fun3914901917261573480plex_o(aTP_Lamm_cl(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4093_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xb: fun_int_nat,Y: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_cm(I5),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_cm(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aa_fun8703205213358249326_int_o(aTP_Lamm_cn(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4094_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Extended_enat,Xb: fun_Ex7660298398204468809at_nat,Y: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun3064223910257113474enat_o(aTP_Lamm_co(I5),Xb)))
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun3064223910257113474enat_o(aTP_Lamm_co(I5),Y)))
       => aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun3064223910257113474enat_o(aa_fun5506575416149999492enat_o(aTP_Lamm_cp(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4095_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,Xb: fun_o_real,Y: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun1386971751305049950un_o_o(aTP_Lamm_cq(I5),Xb)))
     => ( aa_set_o_o(finite_finite_o,collect_o(aa_fun1386971751305049950un_o_o(aTP_Lamm_cq(I5),Y)))
       => aa_set_o_o(finite_finite_o,collect_o(aa_fun1386971751305049950un_o_o(aa_fun3246320876514483410un_o_o(aTP_Lamm_cr(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4096_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xb: fun_real_real,Y: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_cs(I5),Xb)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_cs(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_ct(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4097_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xb: fun_nat_real,Y: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_cu(I5),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_cu(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_cv(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4098_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,Xb: fun_complex_real,Y: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_cw(I5),Xb)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_cw(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_cx(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4099_finite__interval__int1,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_cy(A),B))) ).

% finite_interval_int1
tff(fact_4100_finite__interval__int4,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_cz(A),B))) ).

% finite_interval_int4
tff(fact_4101_Icc__eq__Icc,axiom,
    ! [L: filter_nat,H: filter_nat,L2: filter_nat,H2: filter_nat] :
      ( ( set_or1955772592623580779er_nat(L,H) = set_or1955772592623580779er_nat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ ord_le2510731241096832064er_nat(L,H)
          & ~ ord_le2510731241096832064er_nat(L2,H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4102_Icc__eq__Icc,axiom,
    ! [L: set_nat,H: set_nat,L2: set_nat,H2: set_nat] :
      ( ( set_or4548717258645045905et_nat(L,H) = set_or4548717258645045905et_nat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,L),H)
          & ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4103_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L2: nat,H2: nat] :
      ( ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L),H) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L2),H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),H)
          & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4104_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L2: int,H2: int] :
      ( ( set_or1266510415728281911st_int(L,H) = set_or1266510415728281911st_int(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),H)
          & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4105_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_4106_atLeastAtMost__iff,axiom,
    ! [Ia: $o,L: $o,U: $o] :
      ( aa_set_o_o(member_o2((Ia)),set_or8904488021354931149Most_o((L),(U)))
    <=> ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(L)),(Ia))
        & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Ia)),(U)) ) ) ).

% atLeastAtMost_iff
tff(fact_4107_atLeastAtMost__iff,axiom,
    ! [Ia: filter_nat,L: filter_nat,U: filter_nat] :
      ( member_filter_nat(Ia,set_or1955772592623580779er_nat(L,U))
    <=> ( ord_le2510731241096832064er_nat(L,Ia)
        & ord_le2510731241096832064er_nat(Ia,U) ) ) ).

% atLeastAtMost_iff
tff(fact_4108_atLeastAtMost__iff,axiom,
    ! [Ia: set_nat,L: set_nat,U: set_nat] :
      ( aa_set_set_nat_o(member_set_nat2(Ia),set_or4548717258645045905et_nat(L,U))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,L),Ia)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Ia),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4109_atLeastAtMost__iff,axiom,
    ! [Ia: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat2(Ia),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L),U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),Ia)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4110_atLeastAtMost__iff,axiom,
    ! [Ia: int,L: int,U: int] :
      ( aa_set_int_o(member_int2(Ia),set_or1266510415728281911st_int(L,U))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),Ia)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4111_atLeastAtMost__iff,axiom,
    ! [Ia: real,L: real,U: real] :
      ( aa_set_real_o(member_real2(Ia),set_or1222579329274155063t_real(L,U))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),Ia)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ia),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4112_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_4113_finite__interval__int2,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_da(A),B))) ).

% finite_interval_int2
tff(fact_4114_finite__interval__int3,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_db(A),B))) ).

% finite_interval_int3
tff(fact_4115_atLeastatMost__empty__iff2,axiom,
    ! [A: $o,B: $o] :
      ( ( bot_bot_set_o = set_or8904488021354931149Most_o((A),(B)) )
    <=> ~ aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A)),(B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4116_atLeastatMost__empty__iff2,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( bot_bo498966703094740906er_nat = set_or1955772592623580779er_nat(A,B) )
    <=> ~ ord_le2510731241096832064er_nat(A,B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4117_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_set_nat = set_or4548717258645045905et_nat(A,B) )
    <=> ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4118_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(aa_nat1818925681583257441et_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_4119_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = set_or1266510415728281911st_int(A,B) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4120_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_4121_atLeastatMost__empty__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( set_or8904488021354931149Most_o((A),(B)) = bot_bot_set_o )
    <=> ~ aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A)),(B)) ) ).

% atLeastatMost_empty_iff
tff(fact_4122_atLeastatMost__empty__iff,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( set_or1955772592623580779er_nat(A,B) = bot_bo498966703094740906er_nat )
    <=> ~ ord_le2510731241096832064er_nat(A,B) ) ).

% atLeastatMost_empty_iff
tff(fact_4123_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( set_or4548717258645045905et_nat(A,B) = bot_bot_set_set_nat )
    <=> ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4124_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_set_nat(aa_nat1818925681583257441et_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_4125_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4126_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_4127_atLeastatMost__subset__iff,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat,D: filter_nat] :
      ( ord_le2426478655948331894er_nat(set_or1955772592623580779er_nat(A,B),set_or1955772592623580779er_nat(C,D))
    <=> ( ~ ord_le2510731241096832064er_nat(A,B)
        | ( ord_le2510731241096832064er_nat(C,A)
          & ord_le2510731241096832064er_nat(B,D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4128_atLeastatMost__subset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(A,B)),set_or4548717258645045905et_nat(C,D))
    <=> ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
        | ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A)
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4129_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,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,C),D))
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4130_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C,D))
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4131_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_4132_atLeastatMost__empty,axiom,
    ! [B: $o,A: $o] :
      ( aa_o_o(aa_o_fun_o_o(ord_less_o,(B)),(A))
     => ( set_or8904488021354931149Most_o((A),(B)) = bot_bot_set_o ) ) ).

% atLeastatMost_empty
tff(fact_4133_atLeastatMost__empty,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
     => ( set_or5403411693681687835d_enat(A,B) = bot_bo7653980558646680370d_enat ) ) ).

% atLeastatMost_empty
tff(fact_4134_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_4135_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_4136_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_4137_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_4138_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_4139_zdvd__mult__cancel,axiom,
    ! [K: int,Mb: int,Nb: 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),Nb))
     => ( ( K != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Nb) ) ) ).

% zdvd_mult_cancel
tff(fact_4140_zdvd__antisym__nonneg,axiom,
    ! [Mb: int,Nb: 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),Nb)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Nb)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Nb),Mb)
           => ( Mb = Nb ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_4141_zdvd__imp__le,axiom,
    ! [Z: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Nb) ) ) ).

% zdvd_imp_le
tff(fact_4142_zdvd__period,axiom,
    ! [A: int,D: int,Xb: int,Ta: 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,Xb),Ta))
      <=> 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,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))),Ta)) ) ) ).

% zdvd_period
tff(fact_4143_zdvd__reduce,axiom,
    ! [K: int,Nb: 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,Nb),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),Nb) ) ).

% zdvd_reduce
tff(fact_4144_zdvd__not__zless,axiom,
    ! [Mb: int,Nb: 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),Nb)
       => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Nb),Mb) ) ) ).

% zdvd_not_zless
tff(fact_4145_zdvd__zdiffD,axiom,
    ! [K: int,Mb: int,Nb: 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),Nb))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Mb) ) ) ).

% zdvd_zdiffD
tff(fact_4146_int__induct,axiom,
    ! [P: fun_int_o,K: int,Ia: int] :
      ( aa_int_o(P,K)
     => ( ! [I2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
           => ( aa_int_o(P,I2)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,Ia) ) ) ) ).

% int_induct
tff(fact_4147_int__ge__induct,axiom,
    ! [K: int,Ia: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),Ia)
     => ( aa_int_o(P,K)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
         => aa_int_o(P,Ia) ) ) ) ).

% int_ge_induct
tff(fact_4148_int__le__induct,axiom,
    ! [Ia: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),K)
     => ( aa_int_o(P,K)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,Ia) ) ) ) ).

% int_le_induct
tff(fact_4149_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_4150_less__eq__int__code_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% less_eq_int_code(1)
tff(fact_4151_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_4152_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_4153_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_4154_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_4155_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_4156_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_4157_int__less__induct,axiom,
    ! [Ia: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ia),K)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,Ia) ) ) ) ).

% int_less_induct
tff(fact_4158_pos__zmult__eq__1__iff,axiom,
    ! [Mb: int,Nb: 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),Nb) = one_one_int )
      <=> ( ( Mb = one_one_int )
          & ( Nb = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_4159_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_4160_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_4161_int__gr__induct,axiom,
    ! [K: int,Ia: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),Ia)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I2)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
         => aa_int_o(P,Ia) ) ) ) ).

% int_gr_induct
tff(fact_4162_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_4163_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_4164_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_4165_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_4166_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_4167_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_4168_zmult__zless__mono2,axiom,
    ! [Ia: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ia),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),Ia)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_4169_plus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),L) = L ).

% plus_int_code(2)
tff(fact_4170_plus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,K),zero_zero_int) = K ).

% plus_int_code(1)
tff(fact_4171_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_4172_all__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),Nb)
         => aa_nat_o(P,M5) )
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))
         => aa_nat_o(P,X3) ) ) ).

% all_nat_less
tff(fact_4173_ex__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),Nb)
          & aa_nat_o(P,M5) )
    <=> ? [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))
          & aa_nat_o(P,X3) ) ) ).

% ex_nat_less
tff(fact_4174_atLeastatMost__psubset__iff,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le2529575680413868914d_enat(set_or5403411693681687835d_enat(A,B)),set_or5403411693681687835d_enat(C,D))
    <=> ( ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
          | ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),A)
            & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),D)
            & ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),A)
              | aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),D) ) ) )
        & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4175_atLeastatMost__psubset__iff,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat,D: filter_nat] :
      ( ord_le6505334834405097322er_nat(set_or1955772592623580779er_nat(A,B),set_or1955772592623580779er_nat(C,D))
    <=> ( ( ~ ord_le2510731241096832064er_nat(A,B)
          | ( ord_le2510731241096832064er_nat(C,A)
            & ord_le2510731241096832064er_nat(B,D)
            & ( ord_less_filter_nat(C,A)
              | ord_less_filter_nat(B,D) ) ) )
        & ord_le2510731241096832064er_nat(C,D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4176_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(set_or4548717258645045905et_nat(A,B)),set_or4548717258645045905et_nat(C,D))
    <=> ( ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
          | ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),D)
            & ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),A)
              | aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),D) ) ) )
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4177_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,C),D))
    <=> ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D)
            & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),D) ) ) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4178_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_set_int_o(ord_less_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C,D))
    <=> ( ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
          | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D)
            & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
              | aa_int_o(aa_int_fun_int_o(ord_less_int,B),D) ) ) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4179_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_4180_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N4: set_nat,Nb: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N4),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))
     => aa_set_nat_o(finite_finite_nat,N4) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_4181_vebt__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xb),Xaa)
      <=> (Y) )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B3),$false) ) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Leaf((A3),(B3))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)),Xaa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ( ~ (Y)
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ~ (Y)
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                     => ( ( (Y)
                        <=> $ite(
                              Xaa = Mi,
                              $true,
                              $ite(
                                Xaa = Ma,
                                $true,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                  $false,
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) ) ) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_4182_vebt__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Leaf((A3),(B3))),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)),Xaa)) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)),Xaa)) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)),Xaa)) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)),Xaa))
                       => $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_4183_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Leaf((A3),(B3))),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)),Xaa)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2) )
                 => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2)),Xaa))
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_4184_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Leaf((A3),(B3))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2)),Xaa))
                 => ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_4185_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xb),Xaa)
      <=> (Y) )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B3),$false) ) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Leaf((A3),(B3))),Xaa)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)),Xaa)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xb = 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,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2)),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_4186_vebt__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Leaf((A3),(B3))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)),Xaa))
                 => ~ $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_4187_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xb),Xaa)
      <=> (Y) )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Y)
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Leaf((Uu2),(Uv2))),Xaa)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ( ~ (Y)
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)),Xaa)) ) )
           => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3,Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi )
                        | ( Xaa = Ma ) ) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3,Vb2)),Xaa)) ) )
             => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2) )
                   => ( ( (Y)
                      <=> ( ( Xaa = Mi )
                          | ( Xaa = Ma )
                          | $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)),Xaa)) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( Xb = 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,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd)),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_4188_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Xaa))
       => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3,Vb2) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3,Vb2)),Xaa))
               => ~ ( ( Xaa = Mi )
                    | ( Xaa = Ma ) ) ) )
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)),Xaa))
                 => ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma )
                      | $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) )
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd) )
                 => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd)),Xaa))
                   => ~ $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_4189_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Leaf((Uu2),(Uv2))),Xaa)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)),Xaa)) )
           => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3,Vb2) )
                 => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3,Vb2)),Xaa))
                   => ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
             => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2) )
                   => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)),Xaa))
                     => ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd) )
                     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd)),Xaa))
                       => $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_4190_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_4191_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_4192_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_4193_signed__take__bit__rec,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(Nb),A) = $ite(Nb = zero_zero_nat,aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one))))))) ).

% signed_take_bit_rec
tff(fact_4194_vebt__buildup_Opelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat) ) )
         => ( ( ( Xb = 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] :
                  ( ( Xb = 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,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,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,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,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,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_4195_diff__shunt__var,axiom,
    ! [Xb: set_real,Y: set_real] :
      ( ( minus_minus_set_real(Xb,Y) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(Xb),Y) ) ).

% diff_shunt_var
tff(fact_4196_diff__shunt__var,axiom,
    ! [Xb: set_o,Y: set_o] :
      ( ( minus_minus_set_o(Xb,Y) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(Xb),Y) ) ).

% diff_shunt_var
tff(fact_4197_diff__shunt__var,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ( minus_minus_set_int(Xb,Y) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(Xb),Y) ) ).

% diff_shunt_var
tff(fact_4198_diff__shunt__var,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ( minus_minus_set_nat(Xb,Y) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Y) ) ).

% diff_shunt_var
tff(fact_4199_artanh__def,axiom,
    ! [Xb: real] : aa_real_real(artanh_real,Xb) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(one_one_real),Xb)),aa_real_real(minus_minus_real(one_one_real),Xb)))),numeral_numeral_real(bit0(one))) ).

% artanh_def
tff(fact_4200_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_4201_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_4202_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_4203_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_4204_Compl__anti__mono,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(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,uminus5710092332889474511et_nat(B2)),uminus5710092332889474511et_nat(A2)) ) ).

% Compl_anti_mono
tff(fact_4205_Compl__subset__Compl__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,uminus5710092332889474511et_nat(A2)),uminus5710092332889474511et_nat(B2))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2) ) ).

% Compl_subset_Compl_iff
tff(fact_4206_compl__le__compl__iff,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,uminus5710092332889474511et_nat(Xb)),uminus5710092332889474511et_nat(Y))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Xb) ) ).

% compl_le_compl_iff
tff(fact_4207_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_4208_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_4209_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_4210_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_4211_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_4212_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_4213_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_4214_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_4215_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_4216_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_4217_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_4218_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_4219_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_4220_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_4221_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_4222_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_4223_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_4224_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_4225_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_4226_mult__minus__right,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(A),B)) ).

% mult_minus_right
tff(fact_4227_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_4228_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_4229_minus__mult__minus,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(times_times_real(A),B) ).

% minus_mult_minus
tff(fact_4230_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_4231_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_4232_mult__minus__left,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(A),B)) ).

% mult_minus_left
tff(fact_4233_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_4234_add__minus__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),B)) = B ).

% add_minus_cancel
tff(fact_4235_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_4236_minus__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(plus_plus_real(A),B)) = B ).

% minus_add_cancel
tff(fact_4237_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_4238_minus__add__distrib,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) ).

% minus_add_distrib
tff(fact_4239_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_4240_minus__diff__eq,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(minus_minus_real(A),B)) = aa_real_real(minus_minus_real(B),A) ).

% minus_diff_eq
tff(fact_4241_minus__dvd__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(uminus_uminus_int,Xb)),Y)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_4242_minus__dvd__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(uminus_uminus_real,Xb)),Y)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_4243_dvd__minus__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_int_int(uminus_uminus_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_4244_dvd__minus__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_real_real(uminus_uminus_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_4245_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_4246_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_4247_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_4248_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_4249_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_4250_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_4251_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_4252_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_4253_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_4254_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_4255_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_4256_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_4257_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_4258_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_4259_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_4260_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_4261_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_4262_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_4263_ab__left__minus,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_left_minus
tff(fact_4264_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_4265_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_4266_add_Oright__inverse,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),aa_real_real(uminus_uminus_real,A)) = zero_zero_real ).

% add.right_inverse
tff(fact_4267_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_4268_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_4269_verit__minus__simplify_I3_J,axiom,
    ! [B: real] : aa_real_real(minus_minus_real(zero_zero_real),B) = aa_real_real(uminus_uminus_real,B) ).

% verit_minus_simplify(3)
tff(fact_4270_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_4271_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_4272_diff__0,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(zero_zero_real),A) = aa_real_real(uminus_uminus_real,A) ).

% diff_0
tff(fact_4273_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(Mb))),aa_int_int(uminus_uminus_int,numeral_numeral_int(Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(Mb)),numeral_numeral_int(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4274_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) = aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(numeral_numeral_real(Mb)),numeral_numeral_real(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4275_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_4276_diff__minus__eq__add,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(plus_plus_real(A),B) ).

% diff_minus_eq_add
tff(fact_4277_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_4278_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(minus_minus_real(B),A) ).

% uminus_add_conv_diff
tff(fact_4279_divide__minus1,axiom,
    ! [Xb: complex] : aa_complex_complex(divide1717551699836669952omplex(Xb),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Xb) ).

% divide_minus1
tff(fact_4280_divide__minus1,axiom,
    ! [Xb: real] : aa_real_real(divide_divide_real(Xb),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Xb) ).

% divide_minus1
tff(fact_4281_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_4282_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_4283_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_4284_add__neg__numeral__special_I7_J,axiom,
    aa_real_real(plus_plus_real(one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% add_neg_numeral_special(7)
tff(fact_4285_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_4286_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_4287_add__neg__numeral__special_I8_J,axiom,
    aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = zero_zero_real ).

% add_neg_numeral_special(8)
tff(fact_4288_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_4289_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_4290_diff__numeral__special_I12_J,axiom,
    aa_real_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_4291_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_4292_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(ord_max_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V)),aa_real_real(uminus_uminus_real,numeral_numeral_real(U))) ).

% max_number_of(4)
tff(fact_4293_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(U))),aa_int_int(uminus_uminus_int,numeral_numeral_int(V))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(U))),aa_int_int(uminus_uminus_int,numeral_numeral_int(V))),aa_int_int(uminus_uminus_int,numeral_numeral_int(V)),aa_int_int(uminus_uminus_int,numeral_numeral_int(U))) ).

% max_number_of(4)
tff(fact_4294_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(ord_max_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)),numeral_numeral_real(V),aa_real_real(uminus_uminus_real,numeral_numeral_real(U))) ).

% max_number_of(3)
tff(fact_4295_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(U))),numeral_numeral_int(V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(U))),numeral_numeral_int(V)),numeral_numeral_int(V),aa_int_int(uminus_uminus_int,numeral_numeral_int(U))) ).

% max_number_of(3)
tff(fact_4296_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(ord_max_real(numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V)),numeral_numeral_real(U)) ).

% max_number_of(2)
tff(fact_4297_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,numeral_numeral_int(U)),aa_int_int(uminus_uminus_int,numeral_numeral_int(V))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(U)),aa_int_int(uminus_uminus_int,numeral_numeral_int(V))),aa_int_int(uminus_uminus_int,numeral_numeral_int(V)),numeral_numeral_int(U)) ).

% max_number_of(2)
tff(fact_4298_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,numeral_numeral_int(V))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(W2))),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(plus_plus_num(V,W2)))),Y) ).

% semiring_norm(168)
tff(fact_4299_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),Y)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(plus_plus_num(V,W2)))),Y) ).

% semiring_norm(168)
tff(fact_4300_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb)))
    <=> aa_num_o(ord_less_eq_num(Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_4301_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(Mb))),aa_int_int(uminus_uminus_int,numeral_numeral_int(Nb)))
    <=> aa_num_o(ord_less_eq_num(Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_4302_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(Mb))),aa_int_int(uminus_uminus_int,numeral_numeral_int(Nb)))
    <=> aa_num_o(ord_less_num(Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_4303_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb)))
    <=> aa_num_o(ord_less_num(Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_4304_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,numeral_numeral_real(Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4305_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,numeral_numeral_int(Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4306_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(divide_divide_real(B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4307_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(divide_divide_real(B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4308_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_4309_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A = aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))) )
    <=> $ite(aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)) != zero_zero_real,aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4310_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_4311_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))) = A )
    <=> $ite(aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)) != zero_zero_real,B = aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4312_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,numeral_numeral_int(Mb))),aa_int_int(uminus_uminus_int,one_one_int))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4313_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,numeral_numeral_real(Mb))),aa_real_real(uminus_uminus_real,one_one_real))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4314_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(divide_divide_real(B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_4315_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(divide_divide_real(B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_4316_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(bit0(one))) ).

% add_neg_numeral_special(9)
tff(fact_4317_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,numeral_numeral_int(bit0(one))) ).

% add_neg_numeral_special(9)
tff(fact_4318_add__neg__numeral__special_I9_J,axiom,
    aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(bit0(one))) ).

% add_neg_numeral_special(9)
tff(fact_4319_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,numeral_numeral_int(bit0(K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,numeral_numeral_int(K)))),numeral_numeral_int(bit0(one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_4320_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),numeral_numeral_int(bit0(one)))) ).

% signed_take_bit_0
tff(fact_4321_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_4322_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_4323_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_4324_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_4325_compl__le__swap2,axiom,
    ! [Y: set_nat,Xb: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,uminus5710092332889474511et_nat(Y)),Xb)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,uminus5710092332889474511et_nat(Xb)),Y) ) ).

% compl_le_swap2
tff(fact_4326_compl__le__swap1,axiom,
    ! [Y: set_nat,Xb: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),uminus5710092332889474511et_nat(Xb))
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),uminus5710092332889474511et_nat(Y)) ) ).

% compl_le_swap1
tff(fact_4327_compl__mono,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Y)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,uminus5710092332889474511et_nat(Y)),uminus5710092332889474511et_nat(Xb)) ) ).

% compl_mono
tff(fact_4328_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_4329_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_4330_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_4331_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_4332_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_4333_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_4334_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_4335_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_4336_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_4337_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_4338_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_4339_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_4340_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_4341_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_4342_minus__mult__commute,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,B)) ).

% minus_mult_commute
tff(fact_4343_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_4344_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_4345_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(times_times_real(A),A) = aa_real_real(times_times_real(B),B) )
    <=> ( ( A = B )
        | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% square_eq_iff
tff(fact_4346_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_4347_is__num__normalize_I8_J,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% is_num_normalize(8)
tff(fact_4348_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_4349_group__cancel_Oneg1,axiom,
    ! [A2: real,K: real,A: real] :
      ( ( A2 = aa_real_real(plus_plus_real(K),A) )
     => ( aa_real_real(uminus_uminus_real,A2) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_4350_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_4351_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% add.inverse_distrib_swap
tff(fact_4352_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_4353_minus__diff__commute,axiom,
    ! [B: real,A: real] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,B)),A) = aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,A)),B) ).

% minus_diff_commute
tff(fact_4354_minus__divide__left,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),B) ).

% minus_divide_left
tff(fact_4355_minus__divide__divide,axiom,
    ! [A: real,B: real] : aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(divide_divide_real(A),B) ).

% minus_divide_divide
tff(fact_4356_minus__divide__right,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(A),aa_real_real(uminus_uminus_real,B)) ).

% minus_divide_right
tff(fact_4357_uminus__int__code_I1_J,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% uminus_int_code(1)
tff(fact_4358_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Mb)),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_4359_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(Mb)),aa_int_int(uminus_uminus_int,numeral_numeral_int(Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_4360_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))),numeral_numeral_real(Nb)) ).

% neg_numeral_le_numeral
tff(fact_4361_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(Mb))),numeral_numeral_int(Nb)) ).

% neg_numeral_le_numeral
tff(fact_4362_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% zero_neq_neg_numeral
tff(fact_4363_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_int_int(uminus_uminus_int,numeral_numeral_int(Nb)) ).

% zero_neq_neg_numeral
tff(fact_4364_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_real != aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb)) ).

% zero_neq_neg_numeral
tff(fact_4365_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(Mb))),numeral_numeral_int(Nb)) ).

% neg_numeral_less_numeral
tff(fact_4366_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))),numeral_numeral_real(Nb)) ).

% neg_numeral_less_numeral
tff(fact_4367_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,numeral_numeral_int(Mb)),aa_int_int(uminus_uminus_int,numeral_numeral_int(Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_4368_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Mb)),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_4369_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_4370_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_4371_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_4372_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_4373_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_4374_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_4375_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_4376_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_4377_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_4378_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(plus_plus_real(A),B) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_4379_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_4380_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_4381_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( aa_real_real(plus_plus_real(A),B) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_4382_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_4383_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_4384_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,A) = B ) ) ).

% add.inverse_unique
tff(fact_4385_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_4386_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_4387_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_group_add_class.ab_left_minus
tff(fact_4388_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_4389_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_4390_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = zero_zero_real )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% add_eq_0_iff
tff(fact_4391_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_4392_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_4393_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_4394_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_4395_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_4396_nonzero__minus__divide__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_4397_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_4398_nonzero__minus__divide__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(A),aa_real_real(uminus_uminus_real,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_4399_square__eq__1__iff,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(times_times_complex(Xb),Xb) = one_one_complex )
    <=> ( ( Xb = one_one_complex )
        | ( Xb = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_4400_square__eq__1__iff,axiom,
    ! [Xb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb) = one_one_int )
    <=> ( ( Xb = one_one_int )
        | ( Xb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_4401_square__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(times_times_real(Xb),Xb) = one_one_real )
    <=> ( ( Xb = one_one_real )
        | ( Xb = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_4402_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_4403_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(A),B) = aa_real_real(plus_plus_real(A),aa_real_real(uminus_uminus_real,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_4404_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_4405_diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(A),B) = aa_real_real(plus_plus_real(A),aa_real_real(uminus_uminus_real,B)) ).

% diff_conv_add_uminus
tff(fact_4406_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_4407_group__cancel_Osub2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2 = aa_real_real(plus_plus_real(K),B) )
     => ( aa_real_real(minus_minus_real(A),B2) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,K)),aa_real_real(minus_minus_real(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_4408_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_4409_dvd__neg__div,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) ) ) ).

% dvd_neg_div
tff(fact_4410_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_4411_dvd__div__neg,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( aa_real_real(divide_divide_real(A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) ) ) ).

% dvd_div_neg
tff(fact_4412_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_4413_subset__Compl__self__eq,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),uminus_uminus_set_o(A2))
    <=> ( A2 = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
tff(fact_4414_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_4415_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_4416_zmult__eq__1__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
    <=> ( ( ( Mb = one_one_int )
          & ( Nb = one_one_int ) )
        | ( ( Mb = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( Nb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_4417_pos__zmult__eq__1__iff__lemma,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = 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_4418_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_4419_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))),zero_zero_real) ).

% neg_numeral_le_zero
tff(fact_4420_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(Nb))),zero_zero_int) ).

% neg_numeral_le_zero
tff(fact_4421_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) ).

% not_zero_le_neg_numeral
tff(fact_4422_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,numeral_numeral_int(Nb))) ).

% not_zero_le_neg_numeral
tff(fact_4423_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(Nb))),zero_zero_int) ).

% neg_numeral_less_zero
tff(fact_4424_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))),zero_zero_real) ).

% neg_numeral_less_zero
tff(fact_4425_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,numeral_numeral_int(Nb))) ).

% not_zero_less_neg_numeral
tff(fact_4426_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) ).

% not_zero_less_neg_numeral
tff(fact_4427_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_4428_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_4429_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_4430_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_4431_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_4432_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_4433_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_4434_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_4435_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,numeral_numeral_real(Mb))) ).

% not_one_le_neg_numeral
tff(fact_4436_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,numeral_numeral_int(Mb))) ).

% not_one_le_neg_numeral
tff(fact_4437_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Mb)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_le_neg_one
tff(fact_4438_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(Mb)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_le_neg_one
tff(fact_4439_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,numeral_numeral_real(Mb))),aa_real_real(uminus_uminus_real,one_one_real)) ).

% neg_numeral_le_neg_one
tff(fact_4440_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,numeral_numeral_int(Mb))),aa_int_int(uminus_uminus_int,one_one_int)) ).

% neg_numeral_le_neg_one
tff(fact_4441_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)),numeral_numeral_real(Mb)) ).

% neg_one_le_numeral
tff(fact_4442_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)),numeral_numeral_int(Mb)) ).

% neg_one_le_numeral
tff(fact_4443_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,numeral_numeral_real(Mb))),one_one_real) ).

% neg_numeral_le_one
tff(fact_4444_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,numeral_numeral_int(Mb))),one_one_int) ).

% neg_numeral_le_one
tff(fact_4445_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(Mb))),one_one_int) ).

% neg_numeral_less_one
tff(fact_4446_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))),one_one_real) ).

% neg_numeral_less_one
tff(fact_4447_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)),numeral_numeral_int(Mb)) ).

% neg_one_less_numeral
tff(fact_4448_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)),numeral_numeral_real(Mb)) ).

% neg_one_less_numeral
tff(fact_4449_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,numeral_numeral_int(Mb)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_less_neg_one
tff(fact_4450_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Mb)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_less_neg_one
tff(fact_4451_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,numeral_numeral_int(Mb))) ).

% not_one_less_neg_numeral
tff(fact_4452_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,numeral_numeral_real(Mb))) ).

% not_one_less_neg_numeral
tff(fact_4453_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,numeral_numeral_int(Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_4454_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,numeral_numeral_real(Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_4455_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_4456_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(divide_divide_real(A),B)) )
      <=> ( aa_real_real(times_times_real(C),B) = aa_real_real(uminus_uminus_real,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_4457_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_4458_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(divide_divide_real(A),B)) = C )
      <=> ( aa_real_real(uminus_uminus_real,A) = aa_real_real(times_times_real(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_4459_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_4460_minus__divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)) = A )
    <=> $ite(C != zero_zero_real,aa_real_real(uminus_uminus_real,B) = aa_real_real(times_times_real(A),C),A = zero_zero_real) ) ).

% minus_divide_eq_eq
tff(fact_4461_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_4462_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(A),C) = aa_real_real(uminus_uminus_real,B),A = zero_zero_real) ) ).

% eq_minus_divide_eq
tff(fact_4463_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_4464_divide__eq__minus__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_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_4465_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(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(times_times_real(A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_4466_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(divide_divide_real(B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_4467_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(divide_divide_real(B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_4468_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(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(times_times_real(A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_4469_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(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(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(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_4470_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(divide_divide_real(B),C)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(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(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_4471_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_4472_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)) = aa_real_real(divide_divide_real(B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),C) = B,aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)) = zero_zero_real) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4473_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_4474_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( aa_real_real(divide_divide_real(B),C) = aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),C),aa_real_real(uminus_uminus_real,numeral_numeral_real(W2)) = zero_zero_real) ) ).

% divide_eq_eq_numeral(2)
tff(fact_4475_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_4476_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),Z))),B) = $ite(Z = zero_zero_real,B,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(times_times_real(B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_4477_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(Xb),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4478_minus__divide__add__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(Xb),Z))),Y) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,Xb)),aa_real_real(times_times_real(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4479_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(Xb),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4480_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(Xb),Z))),Y) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,Xb)),aa_real_real(times_times_real(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4481_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_4482_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(A),Z)),B) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(A),aa_real_real(times_times_real(B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_4483_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_4484_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),Z))),B) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(times_times_real(B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_4485_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_minus_divide_eq
tff(fact_4486_minus__divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% minus_divide_le_eq
tff(fact_4487_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_4488_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_4489_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_4490_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_4491_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,numeral_numeral_real(W2))),aa_real_real(divide_divide_real(B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,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(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_4492_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(divide_divide_real(B),C)),aa_real_real(uminus_uminus_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(times_times_real(aa_real_real(uminus_uminus_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(times_times_real(aa_real_real(uminus_uminus_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,numeral_numeral_real(W2)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_4493_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4494_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4495_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4496_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,numeral_numeral_real(W2))),aa_real_real(divide_divide_real(B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,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(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W2))),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_4497_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(divide_divide_real(B),C)),aa_real_real(uminus_uminus_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(times_times_real(aa_real_real(uminus_uminus_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(times_times_real(aa_real_real(uminus_uminus_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,numeral_numeral_real(W2)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_4498_square__le__1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real) ) ) ).

% square_le_1
tff(fact_4499_square__le__1,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_int) ) ) ).

% square_le_1
tff(fact_4500_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_4501_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_4502_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_4503_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(suc,Nb)))),aa_int_int(bit_ri631733984087533419it_int(Nb),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_4504_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,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(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(K))),one_one_int))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_4505_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_dc(L),Qr) ).

% divmod_step_def
tff(fact_4506_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_dd(L)),Qr) ).

% divmod_step_def
tff(fact_4507_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,bit0(one))),aa_set_nat_nat(nat_set_encode,A2))
      <=> ~ aa_set_nat_o(member_nat2(zero_zero_nat),A2) ) ) ).

% even_set_encode_iff
tff(fact_4508_signed__take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),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(Nb),numeral_numeral_int(K))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_4509_take__bit__rec,axiom,
    ! [Nb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% take_bit_rec
tff(fact_4510_take__bit__rec,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),A) = $ite(Nb = zero_zero_nat,zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one))))),numeral_numeral_int(bit0(one)))),aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one))))) ).

% take_bit_rec
tff(fact_4511_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(bit0(Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(ord_less_eq_num(Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),numeral_numeral_int(bit0(Mb))),unique5024387138958732305ep_int(aa_num_num(bit1,Nb),unique5052692396658037445od_int(bit0(Mb),bit0(aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_4512_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(bit0(Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(ord_less_eq_num(Mb),Nb),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,bit0(Mb))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(bit0(Mb),bit0(aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_4513_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(ord_less_num(Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),numeral_numeral_int(aa_num_num(bit1,Mb))),unique5024387138958732305ep_int(aa_num_num(bit1,Nb),unique5052692396658037445od_int(aa_num_num(bit1,Mb),bit0(aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_4514_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(ord_less_num(Mb),Nb),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit1,Mb),bit0(aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_4515_take__bit__of__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),zero_zero_nat) = zero_zero_nat ).

% take_bit_of_0
tff(fact_4516_take__bit__of__0,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),zero_zero_int) = zero_zero_int ).

% take_bit_of_0
tff(fact_4517_case__prod__conv,axiom,
    ! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(F,A),B) ) ).

% case_prod_conv
tff(fact_4518_case__prod__conv,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat] : aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(F,A),B) ).

% case_prod_conv
tff(fact_4519_case__prod__conv,axiom,
    ! [F: fun_na1668100129766719005at_nat,A: nat,B: nat] : aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,A),B) ).

% case_prod_conv
tff(fact_4520_case__prod__conv,axiom,
    ! [F: fun_na5914473681949881652_nat_o,A: nat,B: nat] : aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,A),B) ).

% case_prod_conv
tff(fact_4521_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_4522_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_4523_take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(zero_zero_nat),A) = zero_zero_int ).

% take_bit_0
tff(fact_4524_take__bit__Suc__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),one_one_nat) = one_one_nat ).

% take_bit_Suc_1
tff(fact_4525_take__bit__Suc__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),one_one_int) = one_one_int ).

% take_bit_Suc_1
tff(fact_4526_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_4527_take__bit__of__1__eq__0__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),one_one_nat) = zero_zero_nat )
    <=> ( Nb = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_4528_take__bit__of__1__eq__0__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),one_one_int) = zero_zero_int )
    <=> ( Nb = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_4529_take__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_Suc_0
tff(fact_4530_take__bit__of__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),one_one_nat) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_1
tff(fact_4531_take__bit__of__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),one_one_int) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_1
tff(fact_4532_even__take__bit__eq,axiom,
    ! [Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))
    <=> ( ( Nb = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A) ) ) ).

% even_take_bit_eq
tff(fact_4533_even__take__bit__eq,axiom,
    ! [Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se2923211474154528505it_int(Nb),A))
    <=> ( ( Nb = zero_zero_nat )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A) ) ) ).

% even_take_bit_eq
tff(fact_4534_div__Suc__eq__div__add3,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% div_Suc_eq_div_add3
tff(fact_4535_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_4536_mod__Suc__eq__mod__add3,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% mod_Suc_eq_mod_add3
tff(fact_4537_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_4538_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,Nb)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),numeral_numeral_int(one)) ).

% divmod_algorithm_code(4)
tff(fact_4539_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,Nb)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_4540_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,bit0(one))) ).

% take_bit_Suc_0
tff(fact_4541_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),numeral_numeral_int(bit0(one))) ).

% take_bit_Suc_0
tff(fact_4542_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(bit0(Mb),bit0(Nb)) = produc4245557441103728435nt_int(aTP_Lamm_de,unique5052692396658037445od_int(Mb,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_4543_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(bit0(Mb),bit0(Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_df),unique5055182867167087721od_nat(Mb,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_4544_take__bit__of__exp,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) ).

% take_bit_of_exp
tff(fact_4545_take__bit__of__exp,axiom,
    ! [Mb: nat,Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Mb),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)) ).

% take_bit_of_exp
tff(fact_4546_take__bit__of__2,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,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,bit0(one))),Nb))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% take_bit_of_2
tff(fact_4547_take__bit__of__2,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),numeral_numeral_int(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,bit0(one))),Nb))),numeral_numeral_int(bit0(one))) ).

% take_bit_of_2
tff(fact_4548_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit1,Mb),bit0(Nb)) = produc4245557441103728435nt_int(aTP_Lamm_dg,unique5052692396658037445od_int(Mb,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_4549_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit1,Mb),bit0(Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_dh),unique5055182867167087721od_nat(Mb,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_4550_take__bit__add,axiom,
    ! [Nb: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B))) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% take_bit_add
tff(fact_4551_take__bit__add,axiom,
    ! [Nb: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),A)),aa_int_int(bit_se2923211474154528505it_int(Nb),B))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ).

% take_bit_add
tff(fact_4552_take__bit__tightened,axiom,
    ! [Nb: nat,A: nat,B: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),B) ) ) ) ).

% take_bit_tightened
tff(fact_4553_take__bit__tightened,axiom,
    ! [Nb: nat,A: int,B: int,Mb: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),A) = aa_int_int(bit_se2923211474154528505it_int(Nb),B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_int_int(bit_se2923211474154528505it_int(Mb),A) = aa_int_int(bit_se2923211474154528505it_int(Mb),B) ) ) ) ).

% take_bit_tightened
tff(fact_4554_take__bit__tightened__less__eq__nat,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Q2)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_4555_take__bit__nat__less__eq__self,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),Mb) ).

% take_bit_nat_less_eq_self
tff(fact_4556_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: fun_nat_fun_nat_o,G: fun_nat_fun_nat_o,P2: product_prod_nat_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3) = Q2 )
         => ( aa_nat_o(aa_nat_fun_nat_o(F,X4),Y3)
          <=> aa_nat_o(aa_nat_fun_nat_o(G,X4),Y3) ) )
     => ( ( P2 = Q2 )
       => ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),P2)
        <=> aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(G),Q2) ) ) ) ).

% split_cong
tff(fact_4557_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: fun_nat_fun_nat_nat,G: fun_nat_fun_nat_nat,P2: product_prod_nat_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3) = Q2 )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(F,X4),Y3) = aa_nat_nat(aa_nat_fun_nat_nat(G,X4),Y3) ) )
     => ( ( P2 = Q2 )
       => ( aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),P2) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(G),Q2) ) ) ) ).

% split_cong
tff(fact_4558_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: fun_na1668100129766719005at_nat,G: fun_na1668100129766719005at_nat,P2: product_prod_nat_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3) = Q2 )
         => ( aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X4),Y3) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(G,X4),Y3) ) )
     => ( ( P2 = Q2 )
       => ( aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),P2) = aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(G),Q2) ) ) ) ).

% split_cong
tff(fact_4559_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: fun_na5914473681949881652_nat_o,G: fun_na5914473681949881652_nat_o,P2: product_prod_nat_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3) = Q2 )
         => ( aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X4),Y3) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(G,X4),Y3) ) )
     => ( ( P2 = Q2 )
       => ( aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),P2) = aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(G),Q2) ) ) ) ).

% split_cong
tff(fact_4560_split__cong,axiom,
    ! [Q2: product_prod_int_int,F: fun_int_fun_int_o,G: fun_int_fun_int_o,P2: product_prod_int_int] :
      ( ! [X4: int,Y3: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3) = Q2 )
         => ( aa_int_o(aa_int_fun_int_o(F,X4),Y3)
          <=> aa_int_o(aa_int_fun_int_o(G,X4),Y3) ) )
     => ( ( P2 = Q2 )
       => ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),P2)
        <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(G),Q2) ) ) ) ).

% split_cong
tff(fact_4561_old_Oprod_Ocase,axiom,
    ! [F: fun_nat_fun_nat_o,X1: nat,X22: nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X22))
    <=> aa_nat_o(aa_nat_fun_nat_o(F,X1),X22) ) ).

% old.prod.case
tff(fact_4562_old_Oprod_Ocase,axiom,
    ! [F: fun_nat_fun_nat_nat,X1: nat,X22: nat] : aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X22)) = aa_nat_nat(aa_nat_fun_nat_nat(F,X1),X22) ).

% old.prod.case
tff(fact_4563_old_Oprod_Ocase,axiom,
    ! [F: fun_na1668100129766719005at_nat,X1: nat,X22: nat] : aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X22)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X1),X22) ).

% old.prod.case
tff(fact_4564_old_Oprod_Ocase,axiom,
    ! [F: fun_na5914473681949881652_nat_o,X1: nat,X22: nat] : aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X22)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X1),X22) ).

% old.prod.case
tff(fact_4565_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_4566_case__prodE2,axiom,
    ! [Q: fun_o_o,P: fun_nat_fun_nat_o,Z: product_prod_nat_nat] :
      ( aa_o_o(Q,aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(P),Z))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3) )
           => ~ aa_o_o(Q,aa_nat_o(aa_nat_fun_nat_o(P,X4),Y3)) ) ) ).

% case_prodE2
tff(fact_4567_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))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3) )
           => ~ aa_nat_o(Q,aa_nat_nat(aa_nat_fun_nat_nat(P,X4),Y3)) ) ) ).

% case_prodE2
tff(fact_4568_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))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3) )
           => ~ aa_fun3538214411680374423_nat_o(Q,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(P,X4),Y3)) ) ) ).

% case_prodE2
tff(fact_4569_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))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3) )
           => ~ aa_fun8408151826566693824at_o_o(Q,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(P,X4),Y3)) ) ) ).

% case_prodE2
tff(fact_4570_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))
     => ~ ! [X4: int,Y3: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3) )
           => ~ aa_o_o(Q,aa_int_o(aa_int_fun_int_o(P,X4),Y3)) ) ) ).

% case_prodE2
tff(fact_4571_case__prod__eta,axiom,
    ! [F: fun_Pr4796943543331071514_nat_o] : produc6081775807080527818_nat_o(aTP_Lamm_di(F)) = F ).

% case_prod_eta
tff(fact_4572_case__prod__eta,axiom,
    ! [F: fun_Pr2602117396471556196at_nat] : produc6842872674320459806at_nat(aTP_Lamm_dj(F)) = F ).

% case_prod_eta
tff(fact_4573_case__prod__eta,axiom,
    ! [F: fun_Pr2477239776880934807at_nat] : produc4528391901023682385at_nat(aTP_Lamm_dk(F)) = F ).

% case_prod_eta
tff(fact_4574_case__prod__eta,axiom,
    ! [F: fun_Pr2028404973800837934_nat_o] : produc6395135316500328296_nat_o(aTP_Lamm_dl(F)) = F ).

% case_prod_eta
tff(fact_4575_case__prod__eta,axiom,
    ! [F: fun_Pr3238374608701176162_int_o] : produc4947309494688390418_int_o(aTP_Lamm_dm(F)) = F ).

% case_prod_eta
tff(fact_4576_cond__case__prod__eta,axiom,
    ! [F: fun_nat_fun_nat_o,G: fun_Pr4796943543331071514_nat_o] :
      ( ! [X4: nat,Y3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(F,X4),Y3)
        <=> aa_Pro6150637604445964947_nat_o(G,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3)) )
     => ( produc6081775807080527818_nat_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_4577_cond__case__prod__eta,axiom,
    ! [F: fun_nat_fun_nat_nat,G: fun_Pr2602117396471556196at_nat] :
      ( ! [X4: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(F,X4),Y3) = aa_Pro4243173770044397973at_nat(G,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3))
     => ( produc6842872674320459806at_nat(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_4578_cond__case__prod__eta,axiom,
    ! [F: fun_na1668100129766719005at_nat,G: fun_Pr2477239776880934807at_nat] :
      ( ! [X4: nat,Y3: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X4),Y3) = aa_Pro8023792847356852296at_nat(G,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3))
     => ( produc4528391901023682385at_nat(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_4579_cond__case__prod__eta,axiom,
    ! [F: fun_na5914473681949881652_nat_o,G: fun_Pr2028404973800837934_nat_o] :
      ( ! [X4: nat,Y3: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X4),Y3) = aa_Pro989595756709598047_nat_o(G,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3))
     => ( produc6395135316500328296_nat_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_4580_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_o,G: fun_Pr3238374608701176162_int_o] :
      ( ! [X4: int,Y3: int] :
          ( aa_int_o(aa_int_fun_int_o(F,X4),Y3)
        <=> aa_Pro4592068669816069595_int_o(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3)) )
     => ( produc4947309494688390418_int_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_4581_take__bit__tightened__less__eq__int,axiom,
    ! [Mb: nat,Nb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Mb),K)),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_4582_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(Nb),A) = aa_int_int(bit_ri631733984087533419it_int(Nb),B) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),B) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_4583_signed__take__bit__take__bit,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A)) = aa_int_int($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),bit_se2923211474154528505it_int(Nb),bit_ri631733984087533419it_int(Mb)),A) ).

% signed_take_bit_take_bit
tff(fact_4584_take__bit__unset__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_unset_bit_eq
tff(fact_4585_take__bit__unset__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se4203085406695923979it_int(Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(bit_se4203085406695923979it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_unset_bit_eq
tff(fact_4586_take__bit__set__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_set_bit_eq
tff(fact_4587_take__bit__set__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se7879613467334960850it_int(Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(bit_se7879613467334960850it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_set_bit_eq
tff(fact_4588_take__bit__flip__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_flip_bit_eq
tff(fact_4589_take__bit__flip__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se2159334234014336723it_int(Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(bit_se2159334234014336723it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_flip_bit_eq
tff(fact_4590_take__bit__signed__take__bit,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_int_int(bit_se2923211474154528505it_int(Mb),aa_int_int(bit_ri631733984087533419it_int(Nb),A)) = aa_int_int(bit_se2923211474154528505it_int(Mb),A) ) ) ).

% take_bit_signed_take_bit
tff(fact_4591_numeral__Bit1,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb))),one_one_complex) ).

% numeral_Bit1
tff(fact_4592_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Nb))),one_one_nat) ).

% numeral_Bit1
tff(fact_4593_numeral__Bit1,axiom,
    ! [Nb: num] : numera1916890842035813515d_enat(aa_num_num(bit1,Nb)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Nb)),numera1916890842035813515d_enat(Nb))),one_on7984719198319812577d_enat) ).

% numeral_Bit1
tff(fact_4594_numeral__Bit1,axiom,
    ! [Nb: num] : numeral_numeral_int(aa_num_num(bit1,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(Nb)),numeral_numeral_int(Nb))),one_one_int) ).

% numeral_Bit1
tff(fact_4595_numeral__Bit1,axiom,
    ! [Nb: num] : numeral_numeral_real(aa_num_num(bit1,Nb)) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(numeral_numeral_real(Nb)),numeral_numeral_real(Nb))),one_one_real) ).

% numeral_Bit1
tff(fact_4596_eval__nat__numeral_I3_J,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,bit0(Nb))) ).

% eval_nat_numeral(3)
tff(fact_4597_take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,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,Nb),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_nat) ).

% take_bit_Suc_bit1
tff(fact_4598_take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),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(Nb),numeral_numeral_int(K))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% take_bit_Suc_bit1
tff(fact_4599_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = $let(
        m2: complex,
        m2:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(m2),m2)),one_one_complex) ) ).

% numeral_code(3)
tff(fact_4600_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2)),one_one_nat) ) ).

% numeral_code(3)
tff(fact_4601_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numera1916890842035813515d_enat(aa_num_num(bit1,Nb)) = $let(
        m2: extended_enat,
        m2:= numera1916890842035813515d_enat(Nb),
        aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(m2),m2)),one_on7984719198319812577d_enat) ) ).

% numeral_code(3)
tff(fact_4602_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numeral_numeral_int(aa_num_num(bit1,Nb)) = $let(
        m2: int,
        m2:= numeral_numeral_int(Nb),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2)),one_one_int) ) ).

% numeral_code(3)
tff(fact_4603_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numeral_numeral_real(aa_num_num(bit1,Nb)) = $let(
        m2: real,
        m2:= numeral_numeral_real(Nb),
        aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(m2),m2)),one_one_real) ) ).

% numeral_code(3)
tff(fact_4604_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_4605_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,bit0(Q2))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_4606_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q2: num] : aa_int_int(modulo_modulo_int(numeral_numeral_int(aa_num_num(bit1,Nb))),numeral_numeral_int(bit0(Q2))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_4607_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_4608_Suc3__eq__add__3,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb) ).

% Suc3_eq_add_3
tff(fact_4609_take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,bit0(K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% take_bit_Suc_bit0
tff(fact_4610_take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),numeral_numeral_int(bit0(K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),numeral_numeral_int(K))),numeral_numeral_int(bit0(one))) ).

% take_bit_Suc_bit0
tff(fact_4611_take__bit__nat__eq__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = Mb ) ) ).

% take_bit_nat_eq_self
tff(fact_4612_take__bit__nat__less__exp,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) ).

% take_bit_nat_less_exp
tff(fact_4613_take__bit__nat__eq__self__iff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = Mb )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_4614_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_4615_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_4616_take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,numeral_numeral_int(bit0(K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,numeral_numeral_int(K)))),numeral_numeral_int(bit0(one))) ).

% take_bit_Suc_minus_bit0
tff(fact_4617_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,bit0(Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,bit0(Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_4618_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Nb: num] :
      ( ( aa_int_int(modulo_modulo_int(numeral_numeral_int(one)),numeral_numeral_int(bit0(Q2))) = aa_int_int(modulo_modulo_int(numeral_numeral_int(aa_num_num(bit1,Nb))),numeral_numeral_int(bit0(Q2))) )
    <=> ( aa_int_int(modulo_modulo_int(numeral_numeral_int(Nb)),numeral_numeral_int(Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_4619_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q2: 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,bit0(Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,bit0(Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_4620_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q2: num] :
      ( ( aa_int_int(modulo_modulo_int(numeral_numeral_int(aa_num_num(bit1,Mb))),numeral_numeral_int(bit0(Q2))) = aa_int_int(modulo_modulo_int(numeral_numeral_int(one)),numeral_numeral_int(bit0(Q2))) )
    <=> ( aa_int_int(modulo_modulo_int(numeral_numeral_int(Mb)),numeral_numeral_int(Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_4621_Suc__div__eq__add3__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),Nb) ).

% Suc_div_eq_add3_div
tff(fact_4622_Suc__mod__eq__add3__mod,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),Nb) ).

% Suc_mod_eq_add3_mod
tff(fact_4623_take__bit__eq__0__iff,axiom,
    ! [Nb: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)),A) ) ).

% take_bit_eq_0_iff
tff(fact_4624_take__bit__eq__0__iff,axiom,
    ! [Nb: nat,A: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),A) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)),A) ) ).

% take_bit_eq_0_iff
tff(fact_4625_take__bit__nat__less__self__iff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),Mb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)),Mb) ) ).

% take_bit_nat_less_self_iff
tff(fact_4626_take__bit__Suc__minus__1__eq,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(suc,Nb))),one_one_int) ).

% take_bit_Suc_minus_1_eq
tff(fact_4627_take__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% take_bit_Suc
tff(fact_4628_take__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one))))),numeral_numeral_int(bit0(one)))),aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one)))) ).

% take_bit_Suc
tff(fact_4629_take__bit__int__less__eq,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb))) ) ) ).

% take_bit_int_less_eq
tff(fact_4630_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_4631_stable__imp__take__bit__eq,axiom,
    ! [A: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = A )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one))),Nb)),one_one_nat)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_4632_stable__imp__take__bit__eq,axiom,
    ! [A: int,Nb: nat] :
      ( ( aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one))) = A )
     => ( aa_int_int(bit_se2923211474154528505it_int(Nb),A) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)),one_one_int)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_4633_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_dd(L)),Qr) ).

% divmod_step_nat_def
tff(fact_4634_odd__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(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,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ).

% odd_mod_4_div_2
tff(fact_4635_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,bit0(bit0(one)))) = zero_zero_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = one_one_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = aa_num_nat(numeral_numeral_nat,bit0(one)) )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_4636_take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,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(Nb),aa_int_int(uminus_uminus_int,numeral_numeral_int(inc(K))))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_4637_divmod__nat__if,axiom,
    ! [Mb: nat,Nb: nat] :
      divmod_nat(Mb,Nb) = $ite(
        ( ( Nb = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ),
        aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),Mb),
        aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_dn),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb),Nb)) ) ).

% divmod_nat_if
tff(fact_4638_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(bit0(one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(divide_divide_int(K),numeral_numeral_int(bit0(one))))),
            $ite(aa_int_int(modulo_modulo_int(K),numeral_numeral_int(bit0(one))) = zero_zero_int,l,aa_complex_complex(plus_plus_complex(l),one_one_complex)) ) ) ) ).

% of_int_code_if
tff(fact_4639_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,numeral_numeral_int(bit0(one))),aa_int_int(ring_1_of_int_int,aa_int_int(divide_divide_int(K),numeral_numeral_int(bit0(one))))),
            $ite(aa_int_int(modulo_modulo_int(K),numeral_numeral_int(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_4640_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(times_times_real(numeral_numeral_real(bit0(one))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(K),numeral_numeral_int(bit0(one))))),
            $ite(aa_int_int(modulo_modulo_int(K),numeral_numeral_int(bit0(one))) = zero_zero_int,l,aa_real_real(plus_plus_real(l),one_one_real)) ) ) ) ).

% of_int_code_if
tff(fact_4641_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,bit0(one)))),one_one_nat) ).

% take_bit_numeral_bit1
tff(fact_4642_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),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)),numeral_numeral_int(K))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% take_bit_numeral_bit1
tff(fact_4643_concat__bit__Suc,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_concat_bit(aa_nat_nat(suc,Nb),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(K),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_concat_bit(Nb,aa_int_int(divide_divide_int(K),numeral_numeral_int(bit0(one)))),L))) ).

% concat_bit_Suc
tff(fact_4644_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_4645_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_4646_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_4647_case__prodI2,axiom,
    ! [P2: produc859450856879609959at_nat,C: fun_Pr2028404973800837934_nat_o] :
      ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
          ( ( P2 = aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,A3),B3) )
         => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(C,A3),B3) )
     => aa_Pro602569247023830573_nat_o(produc6590410687421337004_nat_o(C),P2) ) ).

% case_prodI2
tff(fact_4648_case__prodI2,axiom,
    ! [P2: produc9072475918466114483BT_nat,C: fun_VE6480751511521678896_nat_o] :
      ( ! [A3: vEBT_VEBT,B3: nat] :
          ( ( P2 = aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,A3),B3) )
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(C,A3),B3) )
     => aa_Pro632845113475883111_nat_o(produc7574032145190910526_nat_o(C),P2) ) ).

% case_prodI2
tff(fact_4649_case__prodI2,axiom,
    ! [P2: produc7272778201969148633d_enat,C: fun_VE3387248342596292540enat_o] :
      ( ! [A3: vEBT_VEBT,B3: extended_enat] :
          ( ( P2 = aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,A3),B3) )
         => aa_Extended_enat_o(aa_VEB2705759856535948141enat_o(C,A3),B3) )
     => aa_Pro1441479085436938555enat_o(produc1206667027950169146enat_o(C),P2) ) ).

% case_prodI2
tff(fact_4650_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C: fun_nat_fun_nat_o] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A3),B3) )
         => aa_nat_o(aa_nat_fun_nat_o(C,A3),B3) )
     => aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(C),P2) ) ).

% case_prodI2
tff(fact_4651_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_4652_case__prodI,axiom,
    ! [F: fun_Pr2028404973800837934_nat_o,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(F,A),B)
     => aa_Pro602569247023830573_nat_o(produc6590410687421337004_nat_o(F),aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,A),B)) ) ).

% case_prodI
tff(fact_4653_case__prodI,axiom,
    ! [F: fun_VE6480751511521678896_nat_o,A: vEBT_VEBT,B: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(F,A),B)
     => aa_Pro632845113475883111_nat_o(produc7574032145190910526_nat_o(F),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,A),B)) ) ).

% case_prodI
tff(fact_4654_case__prodI,axiom,
    ! [F: fun_VE3387248342596292540enat_o,A: vEBT_VEBT,B: extended_enat] :
      ( aa_Extended_enat_o(aa_VEB2705759856535948141enat_o(F,A),B)
     => aa_Pro1441479085436938555enat_o(produc1206667027950169146enat_o(F),aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,A),B)) ) ).

% case_prodI
tff(fact_4655_case__prodI,axiom,
    ! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(F,A),B)
     => aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)) ) ).

% case_prodI
tff(fact_4656_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_4657_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z: real,C: fun_na312559527389416956t_real] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A3),B3) )
         => aa_set_real_o(member_real2(Z),aa_nat_set_real(aa_nat3669977791431069885t_real(C,A3),B3)) )
     => aa_set_real_o(member_real2(Z),produc3668448655016342576t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_4658_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z: $o,C: fun_na3100059926620910090_set_o] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A3),B3) )
         => aa_set_o_o(member_o2((Z)),aa_nat_set_o(aa_nat_fun_nat_set_o(C,A3),B3)) )
     => aa_set_o_o(member_o2((Z)),produc59986286002894506_set_o(C,P2)) ) ).

% mem_case_prodI2
tff(fact_4659_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z: nat,C: fun_na6128946301132549280et_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A3),B3) )
         => aa_set_nat_o(member_nat2(Z),aa_nat_set_nat(aa_nat1818925681583257441et_nat(C,A3),B3)) )
     => aa_set_nat_o(member_nat2(Z),produc6189476227299908564et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_4660_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z: int,C: fun_na1315397882344630524et_int] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A3),B3) )
         => aa_set_int_o(member_int2(Z),aa_nat_set_int(aa_nat8552214573506306493et_int(C,A3),B3)) )
     => aa_set_int_o(member_int2(Z),produc2011625207790711856et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_4661_mem__case__prodI2,axiom,
    ! [P2: produc9072475918466114483BT_nat,Z: real,C: fun_VE3839197795564616566t_real] :
      ( ! [A3: vEBT_VEBT,B3: nat] :
          ( ( P2 = aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,A3),B3) )
         => aa_set_real_o(member_real2(Z),aa_nat_set_real(aa_VEB7442336959815234087t_real(C,A3),B3)) )
     => aa_set_real_o(member_real2(Z),produc4370036051912753340t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_4662_mem__case__prodI2,axiom,
    ! [P2: produc9072475918466114483BT_nat,Z: $o,C: fun_VE7913661350430487824_set_o] :
      ( ! [A3: vEBT_VEBT,B3: nat] :
          ( ( P2 = aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,A3),B3) )
         => aa_set_o_o(member_o2((Z)),aa_nat_set_o(aa_VEB5857738340502637065_set_o(C,A3),B3)) )
     => aa_set_o_o(member_o2((Z)),produc162892877563308318_set_o(C,P2)) ) ).

% mem_case_prodI2
tff(fact_4663_mem__case__prodI2,axiom,
    ! [P2: produc9072475918466114483BT_nat,Z: nat,C: fun_VE6544184183685019418et_nat] :
      ( ! [A3: vEBT_VEBT,B3: nat] :
          ( ( P2 = aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,A3),B3) )
         => aa_set_nat_o(member_nat2(Z),aa_nat_set_nat(aa_VEB7217921854499810763et_nat(C,A3),B3)) )
     => aa_set_nat_o(member_nat2(Z),produc8779078400790687328et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_4664_mem__case__prodI2,axiom,
    ! [P2: produc9072475918466114483BT_nat,Z: int,C: fun_VE1730635764897100662et_int] :
      ( ! [A3: vEBT_VEBT,B3: nat] :
          ( ( P2 = aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,A3),B3) )
         => aa_set_int_o(member_int2(Z),aa_nat_set_int(aa_VEB4727838709568084007et_int(C,A3),B3)) )
     => aa_set_int_o(member_int2(Z),produc4601227381281490620et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_4665_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_real2(Z),aa_int_set_real(aa_int4847878404602763637t_real(C,A3),B3)) )
     => aa_set_real_o(member_real2(Z),produc6452406959799940328t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_4666_mem__case__prodI2,axiom,
    ! [P2: product_prod_int_int,Z: $o,C: fun_in4068701459849370962_set_o] :
      ( ! [A3: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_set_o_o(member_o2((Z)),aa_int_set_o(aa_int_fun_int_set_o(C,A3),B3)) )
     => aa_set_o_o(member_o2((Z)),produc4257766111578684402_set_o(C,P2)) ) ).

% mem_case_prodI2
tff(fact_4667_mem__case__prodI,axiom,
    ! [Z: real,C: fun_na312559527389416956t_real,A: nat,B: nat] :
      ( aa_set_real_o(member_real2(Z),aa_nat_set_real(aa_nat3669977791431069885t_real(C,A),B))
     => aa_set_real_o(member_real2(Z),produc3668448655016342576t_real(C,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_4668_mem__case__prodI,axiom,
    ! [Z: $o,C: fun_na3100059926620910090_set_o,A: nat,B: nat] :
      ( aa_set_o_o(member_o2((Z)),aa_nat_set_o(aa_nat_fun_nat_set_o(C,A),B))
     => aa_set_o_o(member_o2((Z)),produc59986286002894506_set_o(C,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_4669_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_na6128946301132549280et_nat,A: nat,B: nat] :
      ( aa_set_nat_o(member_nat2(Z),aa_nat_set_nat(aa_nat1818925681583257441et_nat(C,A),B))
     => aa_set_nat_o(member_nat2(Z),produc6189476227299908564et_nat(C,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_4670_mem__case__prodI,axiom,
    ! [Z: int,C: fun_na1315397882344630524et_int,A: nat,B: nat] :
      ( aa_set_int_o(member_int2(Z),aa_nat_set_int(aa_nat8552214573506306493et_int(C,A),B))
     => aa_set_int_o(member_int2(Z),produc2011625207790711856et_int(C,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_4671_mem__case__prodI,axiom,
    ! [Z: real,C: fun_VE3839197795564616566t_real,A: vEBT_VEBT,B: nat] :
      ( aa_set_real_o(member_real2(Z),aa_nat_set_real(aa_VEB7442336959815234087t_real(C,A),B))
     => aa_set_real_o(member_real2(Z),produc4370036051912753340t_real(C,aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_4672_mem__case__prodI,axiom,
    ! [Z: $o,C: fun_VE7913661350430487824_set_o,A: vEBT_VEBT,B: nat] :
      ( aa_set_o_o(member_o2((Z)),aa_nat_set_o(aa_VEB5857738340502637065_set_o(C,A),B))
     => aa_set_o_o(member_o2((Z)),produc162892877563308318_set_o(C,aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_4673_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_VE6544184183685019418et_nat,A: vEBT_VEBT,B: nat] :
      ( aa_set_nat_o(member_nat2(Z),aa_nat_set_nat(aa_VEB7217921854499810763et_nat(C,A),B))
     => aa_set_nat_o(member_nat2(Z),produc8779078400790687328et_nat(C,aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_4674_mem__case__prodI,axiom,
    ! [Z: int,C: fun_VE1730635764897100662et_int,A: vEBT_VEBT,B: nat] :
      ( aa_set_int_o(member_int2(Z),aa_nat_set_int(aa_VEB4727838709568084007et_int(C,A),B))
     => aa_set_int_o(member_int2(Z),produc4601227381281490620et_int(C,aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_4675_mem__case__prodI,axiom,
    ! [Z: real,C: fun_in6553099857819052212t_real,A: int,B: int] :
      ( aa_set_real_o(member_real2(Z),aa_int_set_real(aa_int4847878404602763637t_real(C,A),B))
     => aa_set_real_o(member_real2(Z),produc6452406959799940328t_real(C,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% mem_case_prodI
tff(fact_4676_mem__case__prodI,axiom,
    ! [Z: $o,C: fun_in4068701459849370962_set_o,A: int,B: int] :
      ( aa_set_o_o(member_o2((Z)),aa_int_set_o(aa_int_fun_int_set_o(C,A),B))
     => aa_set_o_o(member_o2((Z)),produc4257766111578684402_set_o(C,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% mem_case_prodI
tff(fact_4677_case__prodI2_H,axiom,
    ! [P2: product_prod_nat_nat,C: fun_na5914473681949881652_nat_o,Xb: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A3),B3) = P2 )
         => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,A3),B3),Xb) )
     => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(C),P2),Xb) ) ).

% case_prodI2'
tff(fact_4678_of__int__of__bool,axiom,
    ! [P: $o] : aa_int_real(ring_1_of_int_real,aa_o_int(zero_n2684676970156552555ol_int,(P))) = aa_o_real(zero_n3304061248610475627l_real,(P)) ).

% of_int_of_bool
tff(fact_4679_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_4680_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_4681_of__int__0,axiom,
    aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_4682_of__int__0,axiom,
    aa_int_complex(ring_17405671764205052669omplex,zero_zero_int) = zero_zero_complex ).

% of_int_0
tff(fact_4683_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_4684_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_4685_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_4686_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_4687_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_4688_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_4689_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_4690_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_4691_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_4692_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = numeral_numeral_int(Nb) )
    <=> ( Z = numeral_numeral_int(Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4693_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = numeral_numeral_real(Nb) )
    <=> ( Z = numeral_numeral_int(Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4694_of__int__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,numeral_numeral_int(K)) = numeral_numeral_int(K) ).

% of_int_numeral
tff(fact_4695_of__int__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,numeral_numeral_int(K)) = numeral_numeral_real(K) ).

% of_int_numeral
tff(fact_4696_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_4697_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_4698_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_4699_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_4700_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_4701_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_4702_of__int__1,axiom,
    aa_int_complex(ring_17405671764205052669omplex,one_one_int) = one_one_complex ).

% of_int_1
tff(fact_4703_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_4704_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_4705_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(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_4706_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_4707_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_4708_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(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_4709_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_4710_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_4711_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_4712_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(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_4713_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_4714_eq__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,Nb) )
    <=> ( aa_num_nat(pred_numeral,K) = Nb ) ) ).

% eq_numeral_Suc
tff(fact_4715_Suc__eq__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( ( aa_nat_nat(suc,Nb) = aa_num_nat(numeral_numeral_nat,K) )
    <=> ( Nb = aa_num_nat(pred_numeral,K) ) ) ).

% Suc_eq_numeral
tff(fact_4716_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W2: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Xb) = aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W2) )
    <=> ( Xb = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4717_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W2: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Xb) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2) )
    <=> ( Xb = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4718_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W2: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Xb) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W2) )
    <=> ( Xb = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4719_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,Xb: int] :
      ( ( aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W2) = aa_int_real(ring_1_of_int_real,Xb) )
    <=> ( aa_nat_int(power_power_int(B),W2) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4720_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,Xb: 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,Xb) )
    <=> ( aa_nat_int(power_power_int(B),W2) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4721_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,Xb: int] :
      ( ( aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W2) = aa_int_complex(ring_17405671764205052669omplex,Xb) )
    <=> ( aa_nat_int(power_power_int(B),W2) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4722_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,Z)),Nb) ).

% of_int_power
tff(fact_4723_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Z)),Nb) ).

% of_int_power
tff(fact_4724_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,Z)),Nb) ).

% of_int_power
tff(fact_4725_less__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(pred_numeral,K)) ) ).

% less_Suc_numeral
tff(fact_4726_less__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(pred_numeral,K)),Nb) ) ).

% less_numeral_Suc
tff(fact_4727_le__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),Nb) ) ).

% le_numeral_Suc
tff(fact_4728_le__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_num_nat(pred_numeral,K)) ) ).

% le_Suc_numeral
tff(fact_4729_diff__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(pred_numeral,K)) ).

% diff_Suc_numeral
tff(fact_4730_diff__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),Nb) ).

% diff_numeral_Suc
tff(fact_4731_max__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),aa_num_nat(pred_numeral,K))) ).

% max_Suc_numeral
tff(fact_4732_max__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),Nb)) ).

% max_numeral_Suc
tff(fact_4733_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_4734_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_4735_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_4736_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_4737_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_4738_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_4739_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_4740_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_4741_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),numeral_numeral_real(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),numeral_numeral_int(Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_4742_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),numeral_numeral_int(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),numeral_numeral_int(Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_4743_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Nb)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_4744_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(Nb)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_4745_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),numeral_numeral_int(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),numeral_numeral_int(Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_4746_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),numeral_numeral_real(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),numeral_numeral_int(Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_4747_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,numeral_numeral_int(Nb)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,numeral_numeral_int(Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_4748_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Nb)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,numeral_numeral_int(Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_4749_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_4750_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_4751_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_4752_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_4753_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_4754_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_4755_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_4756_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_4757_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W2)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W2)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4758_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,Xb: 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,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W2)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4759_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: 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,Xb)),aa_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,Xb),aa_nat_int(power_power_int(B),W2)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4760_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: 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,Xb)),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,Xb),aa_nat_int(power_power_int(B),W2)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4761_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xb)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4762_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4763_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(power_power_real(numeral_numeral_real(Xb)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4764_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xb)),Nb) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4765_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4766_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_real(power_power_real(numeral_numeral_real(Xb)),Nb) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4767_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(inc(Nb))) ).

% add_neg_numeral_special(5)
tff(fact_4768_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,numeral_numeral_int(Nb))) = aa_int_int(uminus_uminus_int,numeral_numeral_int(inc(Nb))) ).

% add_neg_numeral_special(5)
tff(fact_4769_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(inc(Nb))) ).

% add_neg_numeral_special(5)
tff(fact_4770_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(inc(Mb))) ).

% add_neg_numeral_special(6)
tff(fact_4771_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,numeral_numeral_int(Mb))),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,numeral_numeral_int(inc(Mb))) ).

% add_neg_numeral_special(6)
tff(fact_4772_add__neg__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(inc(Mb))) ).

% add_neg_numeral_special(6)
tff(fact_4773_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W2)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W2)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4774_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,Xb: 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,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W2)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4775_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Xb)),aa_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,Xb),aa_nat_int(power_power_int(B),W2)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4776_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Xb)),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,Xb),aa_nat_int(power_power_int(B),W2)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4777_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(numeral_numeral_real(Xb)),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4778_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4779_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(numeral_numeral_real(Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4780_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4781_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4782_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(numeral_numeral_real(Xb)),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4783_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4784_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(numeral_numeral_real(Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4785_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xb))),Nb) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Xb))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_4786_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Xb))),Nb) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Xb))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_4787_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(Xb))),Nb) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Xb))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_4788_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xb))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_4789_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Xb))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_4790_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(Xb))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_4791_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(Xb))),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Xb))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_4792_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Xb))),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Xb))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_4793_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_4794_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_4795_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Xb))),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Xb))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_4796_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(Xb))),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Xb))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_4797_ex__le__of__int,axiom,
    ! [Xb: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Z2)) ).

% ex_le_of_int
tff(fact_4798_ex__less__of__int,axiom,
    ! [Xb: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,Z2)) ).

% ex_less_of_int
tff(fact_4799_ex__of__int__less,axiom,
    ! [Xb: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),Xb) ).

% ex_of_int_less
tff(fact_4800_mult__of__int__commute,axiom,
    ! [Xb: int,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,Xb)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_int_int(ring_1_of_int_int,Xb)) ).

% mult_of_int_commute
tff(fact_4801_mult__of__int__commute,axiom,
    ! [Xb: int,Y: real] : aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,Xb)),Y) = aa_real_real(times_times_real(Y),aa_int_real(ring_1_of_int_real,Xb)) ).

% mult_of_int_commute
tff(fact_4802_mult__of__int__commute,axiom,
    ! [Xb: int,Y: complex] : aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,Xb)),Y) = aa_complex_complex(times_times_complex(Y),aa_int_complex(ring_17405671764205052669omplex,Xb)) ).

% mult_of_int_commute
tff(fact_4803_mem__case__prodE,axiom,
    ! [Z: real,C: fun_na312559527389416956t_real,P2: product_prod_nat_nat] :
      ( aa_set_real_o(member_real2(Z),produc3668448655016342576t_real(C,P2))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3) )
           => ~ aa_set_real_o(member_real2(Z),aa_nat_set_real(aa_nat3669977791431069885t_real(C,X4),Y3)) ) ) ).

% mem_case_prodE
tff(fact_4804_mem__case__prodE,axiom,
    ! [Z: $o,C: fun_na3100059926620910090_set_o,P2: product_prod_nat_nat] :
      ( aa_set_o_o(member_o2((Z)),produc59986286002894506_set_o(C,P2))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3) )
           => ~ aa_set_o_o(member_o2((Z)),aa_nat_set_o(aa_nat_fun_nat_set_o(C,X4),Y3)) ) ) ).

% mem_case_prodE
tff(fact_4805_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_na6128946301132549280et_nat,P2: product_prod_nat_nat] :
      ( aa_set_nat_o(member_nat2(Z),produc6189476227299908564et_nat(C,P2))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3) )
           => ~ aa_set_nat_o(member_nat2(Z),aa_nat_set_nat(aa_nat1818925681583257441et_nat(C,X4),Y3)) ) ) ).

% mem_case_prodE
tff(fact_4806_mem__case__prodE,axiom,
    ! [Z: int,C: fun_na1315397882344630524et_int,P2: product_prod_nat_nat] :
      ( aa_set_int_o(member_int2(Z),produc2011625207790711856et_int(C,P2))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3) )
           => ~ aa_set_int_o(member_int2(Z),aa_nat_set_int(aa_nat8552214573506306493et_int(C,X4),Y3)) ) ) ).

% mem_case_prodE
tff(fact_4807_mem__case__prodE,axiom,
    ! [Z: real,C: fun_VE3839197795564616566t_real,P2: produc9072475918466114483BT_nat] :
      ( aa_set_real_o(member_real2(Z),produc4370036051912753340t_real(C,P2))
     => ~ ! [X4: vEBT_VEBT,Y3: nat] :
            ( ( P2 = aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,X4),Y3) )
           => ~ aa_set_real_o(member_real2(Z),aa_nat_set_real(aa_VEB7442336959815234087t_real(C,X4),Y3)) ) ) ).

% mem_case_prodE
tff(fact_4808_mem__case__prodE,axiom,
    ! [Z: $o,C: fun_VE7913661350430487824_set_o,P2: produc9072475918466114483BT_nat] :
      ( aa_set_o_o(member_o2((Z)),produc162892877563308318_set_o(C,P2))
     => ~ ! [X4: vEBT_VEBT,Y3: nat] :
            ( ( P2 = aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,X4),Y3) )
           => ~ aa_set_o_o(member_o2((Z)),aa_nat_set_o(aa_VEB5857738340502637065_set_o(C,X4),Y3)) ) ) ).

% mem_case_prodE
tff(fact_4809_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_VE6544184183685019418et_nat,P2: produc9072475918466114483BT_nat] :
      ( aa_set_nat_o(member_nat2(Z),produc8779078400790687328et_nat(C,P2))
     => ~ ! [X4: vEBT_VEBT,Y3: nat] :
            ( ( P2 = aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,X4),Y3) )
           => ~ aa_set_nat_o(member_nat2(Z),aa_nat_set_nat(aa_VEB7217921854499810763et_nat(C,X4),Y3)) ) ) ).

% mem_case_prodE
tff(fact_4810_mem__case__prodE,axiom,
    ! [Z: int,C: fun_VE1730635764897100662et_int,P2: produc9072475918466114483BT_nat] :
      ( aa_set_int_o(member_int2(Z),produc4601227381281490620et_int(C,P2))
     => ~ ! [X4: vEBT_VEBT,Y3: nat] :
            ( ( P2 = aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,X4),Y3) )
           => ~ aa_set_int_o(member_int2(Z),aa_nat_set_int(aa_VEB4727838709568084007et_int(C,X4),Y3)) ) ) ).

% mem_case_prodE
tff(fact_4811_mem__case__prodE,axiom,
    ! [Z: real,C: fun_in6553099857819052212t_real,P2: product_prod_int_int] :
      ( aa_set_real_o(member_real2(Z),produc6452406959799940328t_real(C,P2))
     => ~ ! [X4: int,Y3: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3) )
           => ~ aa_set_real_o(member_real2(Z),aa_int_set_real(aa_int4847878404602763637t_real(C,X4),Y3)) ) ) ).

% mem_case_prodE
tff(fact_4812_mem__case__prodE,axiom,
    ! [Z: $o,C: fun_in4068701459849370962_set_o,P2: product_prod_int_int] :
      ( aa_set_o_o(member_o2((Z)),produc4257766111578684402_set_o(C,P2))
     => ~ ! [X4: int,Y3: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3) )
           => ~ aa_set_o_o(member_o2((Z)),aa_int_set_o(aa_int_fun_int_set_o(C,X4),Y3)) ) ) ).

% mem_case_prodE
tff(fact_4813_of__int__max,axiom,
    ! [Xb: int,Y: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y)) = aa_real_real(ord_max_real(aa_int_real(ring_1_of_int_real,Xb)),aa_int_real(ring_1_of_int_real,Y)) ).

% of_int_max
tff(fact_4814_of__int__max,axiom,
    ! [Xb: int,Y: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(ring_1_of_int_int,Xb)),aa_int_int(ring_1_of_int_int,Y)) ).

% of_int_max
tff(fact_4815_case__prodE,axiom,
    ! [C: fun_Pr2028404973800837934_nat_o,P2: produc859450856879609959at_nat] :
      ( aa_Pro602569247023830573_nat_o(produc6590410687421337004_nat_o(C),P2)
     => ~ ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( P2 = aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,X4),Y3) )
           => ~ aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(C,X4),Y3) ) ) ).

% case_prodE
tff(fact_4816_case__prodE,axiom,
    ! [C: fun_VE6480751511521678896_nat_o,P2: produc9072475918466114483BT_nat] :
      ( aa_Pro632845113475883111_nat_o(produc7574032145190910526_nat_o(C),P2)
     => ~ ! [X4: vEBT_VEBT,Y3: nat] :
            ( ( P2 = aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,X4),Y3) )
           => ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(C,X4),Y3) ) ) ).

% case_prodE
tff(fact_4817_case__prodE,axiom,
    ! [C: fun_VE3387248342596292540enat_o,P2: produc7272778201969148633d_enat] :
      ( aa_Pro1441479085436938555enat_o(produc1206667027950169146enat_o(C),P2)
     => ~ ! [X4: vEBT_VEBT,Y3: extended_enat] :
            ( ( P2 = aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,X4),Y3) )
           => ~ aa_Extended_enat_o(aa_VEB2705759856535948141enat_o(C,X4),Y3) ) ) ).

% case_prodE
tff(fact_4818_case__prodE,axiom,
    ! [C: fun_nat_fun_nat_o,P2: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(C),P2)
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(C,X4),Y3) ) ) ).

% case_prodE
tff(fact_4819_case__prodE,axiom,
    ! [C: fun_int_fun_int_o,P2: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P2)
     => ~ ! [X4: int,Y3: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3) )
           => ~ aa_int_o(aa_int_fun_int_o(C,X4),Y3) ) ) ).

% case_prodE
tff(fact_4820_case__prodD,axiom,
    ! [F: fun_Pr2028404973800837934_nat_o,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( aa_Pro602569247023830573_nat_o(produc6590410687421337004_nat_o(F),aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,A),B))
     => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(F,A),B) ) ).

% case_prodD
tff(fact_4821_case__prodD,axiom,
    ! [F: fun_VE6480751511521678896_nat_o,A: vEBT_VEBT,B: nat] :
      ( aa_Pro632845113475883111_nat_o(produc7574032145190910526_nat_o(F),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,A),B))
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(F,A),B) ) ).

% case_prodD
tff(fact_4822_case__prodD,axiom,
    ! [F: fun_VE3387248342596292540enat_o,A: vEBT_VEBT,B: extended_enat] :
      ( aa_Pro1441479085436938555enat_o(produc1206667027950169146enat_o(F),aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,A),B))
     => aa_Extended_enat_o(aa_VEB2705759856535948141enat_o(F,A),B) ) ).

% case_prodD
tff(fact_4823_case__prodD,axiom,
    ! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B))
     => aa_nat_o(aa_nat_fun_nat_o(F,A),B) ) ).

% case_prodD
tff(fact_4824_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_4825_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)
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3) )
           => ~ aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,X4),Y3),Z) ) ) ).

% case_prodE'
tff(fact_4826_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(aa_nat8803788312118491388at_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_4827_concat__bit__assoc,axiom,
    ! [Nb: nat,K: int,Mb: nat,L: int,R2: int] : aa_int_int(bit_concat_bit(Nb,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),Nb),aa_int_int(bit_concat_bit(Nb,K),L)),R2) ).

% concat_bit_assoc
tff(fact_4828_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_4829_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_4830_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_4831_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_4832_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_4833_floor__exists,axiom,
    ! [Xb: real] :
    ? [Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),Xb)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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_4834_floor__exists1,axiom,
    ! [Xb: real] :
    ? [X4: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X4)),Xb)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),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)),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int))) )
         => ( Y4 = X4 ) ) ) ).

% floor_exists1
tff(fact_4835_numeral__inc,axiom,
    ! [Xb: num] : numera6690914467698888265omplex(inc(Xb)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Xb)),one_one_complex) ).

% numeral_inc
tff(fact_4836_numeral__inc,axiom,
    ! [Xb: num] : aa_num_nat(numeral_numeral_nat,inc(Xb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Xb)),one_one_nat) ).

% numeral_inc
tff(fact_4837_numeral__inc,axiom,
    ! [Xb: num] : numera1916890842035813515d_enat(inc(Xb)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Xb)),one_on7984719198319812577d_enat) ).

% numeral_inc
tff(fact_4838_numeral__inc,axiom,
    ! [Xb: num] : numeral_numeral_int(inc(Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(Xb)),one_one_int) ).

% numeral_inc
tff(fact_4839_numeral__inc,axiom,
    ! [Xb: num] : numeral_numeral_real(inc(Xb)) = aa_real_real(plus_plus_real(numeral_numeral_real(Xb)),one_one_real) ).

% numeral_inc
tff(fact_4840_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(K))) = aa_int_int(uminus_uminus_int,numeral_numeral_int(K)) ).

% of_int_neg_numeral
tff(fact_4841_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,numeral_numeral_int(K))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(K)) ).

% of_int_neg_numeral
tff(fact_4842_divmod__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : divmod_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ).

% divmod_nat_def
tff(fact_4843_listrel1p__def,axiom,
    ! [R2: fun_nat_fun_nat_o,Xs: list_nat,Ys: list_nat] :
      ( aa_list_nat_o(listrel1p_nat(R2,Xs),Ys)
    <=> member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),listrel1_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(R2)))) ) ).

% listrel1p_def
tff(fact_4844_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_4845_round__unique,axiom,
    ! [Xb: real,Y: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(Xb),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))),aa_int_real(ring_1_of_int_real,Y))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y)),aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))))
       => ( aa_real_int(archim8280529875227126926d_real,Xb) = Y ) ) ) ).

% round_unique
tff(fact_4846_of__int__round__gt,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(Xb),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))) ).

% of_int_round_gt
tff(fact_4847_of__int__round__ge,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(Xb),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))) ).

% of_int_round_ge
tff(fact_4848_of__int__round__le,axiom,
    ! [Xb: 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,Xb))),aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))) ).

% of_int_round_le
tff(fact_4849_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_do(D))) ).

% int_ge_less_than2_def
tff(fact_4850_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_dp(D))) ).

% int_ge_less_than_def
tff(fact_4851_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_4852_round__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim8280529875227126926d_real,Xb)),aa_real_int(archim8280529875227126926d_real,Y)) ) ).

% round_mono
tff(fact_4853_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),K)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(suc,Nb))),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Nb)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_4854_mask__numeral,axiom,
    ! [Nb: num] : aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(pred_numeral,Nb)))) ).

% mask_numeral
tff(fact_4855_mask__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(pred_numeral,Nb)))) ).

% mask_numeral
tff(fact_4856_set__encode__insert,axiom,
    ! [A2: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat2(Nb),A2)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat2(Nb,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,bit0(one))),Nb)),aa_set_nat_nat(nat_set_encode,A2)) ) ) ) ).

% set_encode_insert
tff(fact_4857_Sum__Icc__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% Sum_Icc_nat
tff(fact_4858_neg__numeral__le__ceiling,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(V))),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)),Xb) ) ).

% neg_numeral_le_ceiling
tff(fact_4859_ceiling__less__neg__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_int_int(uminus_uminus_int,numeral_numeral_int(V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)) ) ).

% ceiling_less_neg_numeral
tff(fact_4860_arith__series__nat,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_dq(A),D)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),D)))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% arith_series_nat
tff(fact_4861_mask__nat__positive__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% mask_nat_positive_iff
tff(fact_4862_singletonI,axiom,
    ! [A: set_nat] : aa_set_set_nat_o(member_set_nat2(A),insert_set_nat2(A,bot_bot_set_set_nat)) ).

% singletonI
tff(fact_4863_singletonI,axiom,
    ! [A: real] : aa_set_real_o(member_real2(A),insert_real2(A,bot_bot_set_real)) ).

% singletonI
tff(fact_4864_singletonI,axiom,
    ! [A: $o] : aa_set_o_o(member_o2((A)),insert_o2((A),bot_bot_set_o)) ).

% singletonI
tff(fact_4865_singletonI,axiom,
    ! [A: nat] : aa_set_nat_o(member_nat2(A),insert_nat2(A,bot_bot_set_nat)) ).

% singletonI
tff(fact_4866_singletonI,axiom,
    ! [A: int] : aa_set_int_o(member_int2(A),insert_int2(A,bot_bot_set_int)) ).

% singletonI
tff(fact_4867_insert__subset,axiom,
    ! [Xb: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(insert_real2(Xb,A2)),B2)
    <=> ( aa_set_real_o(member_real2(Xb),B2)
        & aa_set_real_o(ord_less_eq_set_real(A2),B2) ) ) ).

% insert_subset
tff(fact_4868_insert__subset,axiom,
    ! [Xb: $o,A2: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(insert_o2((Xb),A2)),B2)
    <=> ( aa_set_o_o(member_o2((Xb)),B2)
        & aa_set_o_o(ord_less_eq_set_o(A2),B2) ) ) ).

% insert_subset
tff(fact_4869_insert__subset,axiom,
    ! [Xb: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(insert_set_nat2(Xb,A2)),B2)
    <=> ( aa_set_set_nat_o(member_set_nat2(Xb),B2)
        & aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2) ) ) ).

% insert_subset
tff(fact_4870_insert__subset,axiom,
    ! [Xb: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(insert_int2(Xb,A2)),B2)
    <=> ( aa_set_int_o(member_int2(Xb),B2)
        & aa_set_int_o(ord_less_eq_set_int(A2),B2) ) ) ).

% insert_subset
tff(fact_4871_insert__subset,axiom,
    ! [Xb: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,insert_nat2(Xb,A2)),B2)
    <=> ( aa_set_nat_o(member_nat2(Xb),B2)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2) ) ) ).

% insert_subset
tff(fact_4872_bit__0__eq,axiom,
    bit_se1146084159140164899it_int(zero_zero_int) = bot_bot_fun_nat_o ).

% bit_0_eq
tff(fact_4873_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_4874_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ax),A2) = zero_zero_nat ).

% sum.neutral_const
tff(fact_4875_sum_Oneutral__const,axiom,
    ! [A2: set_complex] : groups7754918857620584856omplex(aTP_Lamm_ba,A2) = zero_zero_complex ).

% sum.neutral_const
tff(fact_4876_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_dr),A2) = zero_zero_real ).

% sum.neutral_const
tff(fact_4877_singleton__conv,axiom,
    ! [A: list_nat] : collect_list_nat(aTP_Lamm_ds(A)) = insert_list_nat(A,bot_bot_set_list_nat) ).

% singleton_conv
tff(fact_4878_singleton__conv,axiom,
    ! [A: set_nat] : collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_dt,A)) = insert_set_nat2(A,bot_bot_set_set_nat) ).

% singleton_conv
tff(fact_4879_singleton__conv,axiom,
    ! [A: real] : collect_real(aa_real_fun_real_o(aTP_Lamm_du,A)) = insert_real2(A,bot_bot_set_real) ).

% singleton_conv
tff(fact_4880_singleton__conv,axiom,
    ! [A: $o] : collect_o(aa_o_fun_o_o(aTP_Lamm_dv,(A))) = insert_o2((A),bot_bot_set_o) ).

% singleton_conv
tff(fact_4881_singleton__conv,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dw,A)) = insert_nat2(A,bot_bot_set_nat) ).

% singleton_conv
tff(fact_4882_singleton__conv,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_o(aTP_Lamm_dx,A)) = insert_int2(A,bot_bot_set_int) ).

% singleton_conv
tff(fact_4883_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_4884_singleton__conv2,axiom,
    ! [A: set_nat] : collect_set_nat(aa_set7861427066675100071_nat_o(fequal_set_nat,A)) = insert_set_nat2(A,bot_bot_set_set_nat) ).

% singleton_conv2
tff(fact_4885_singleton__conv2,axiom,
    ! [A: real] : collect_real(aa_real_fun_real_o(fequal_real,A)) = insert_real2(A,bot_bot_set_real) ).

% singleton_conv2
tff(fact_4886_singleton__conv2,axiom,
    ! [A: $o] : collect_o(aa_o_fun_o_o(fequal_o,(A))) = insert_o2((A),bot_bot_set_o) ).

% singleton_conv2
tff(fact_4887_singleton__conv2,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_o(fequal_nat,A)) = insert_nat2(A,bot_bot_set_nat) ).

% singleton_conv2
tff(fact_4888_singleton__conv2,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_o(fequal_int,A)) = insert_int2(A,bot_bot_set_int) ).

% singleton_conv2
tff(fact_4889_of__int__sum,axiom,
    ! [F: fun_complex_int,A2: set_complex] : aa_int_complex(ring_17405671764205052669omplex,groups5690904116761175830ex_int(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_dy(F),A2) ).

% of_int_sum
tff(fact_4890_of__int__sum,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_real(ring_1_of_int_real,aa_set_nat_int(groups3539618377306564664at_int(F),A2)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_dz(F)),A2) ).

% of_int_sum
tff(fact_4891_sum_Oempty,axiom,
    ! [G: fun_real_nat] : groups1935376822645274424al_nat(G,bot_bot_set_real) = zero_zero_nat ).

% sum.empty
tff(fact_4892_sum_Oempty,axiom,
    ! [G: fun_real_real] : groups8097168146408367636l_real(G,bot_bot_set_real) = zero_zero_real ).

% sum.empty
tff(fact_4893_sum_Oempty,axiom,
    ! [G: fun_real_int] : groups1932886352136224148al_int(G,bot_bot_set_real) = zero_zero_int ).

% sum.empty
tff(fact_4894_sum_Oempty,axiom,
    ! [G: fun_real_complex] : groups5754745047067104278omplex(G,bot_bot_set_real) = zero_zero_complex ).

% sum.empty
tff(fact_4895_sum_Oempty,axiom,
    ! [G: fun_re6065469302365115155d_enat] : groups2800946370649118462d_enat(G,bot_bot_set_real) = zero_z5237406670263579293d_enat ).

% sum.empty
tff(fact_4896_sum_Oempty,axiom,
    ! [G: fun_o_nat] : groups8507830703676809646_o_nat(G,bot_bot_set_o) = zero_zero_nat ).

% sum.empty
tff(fact_4897_sum_Oempty,axiom,
    ! [G: fun_o_real] : groups8691415230153176458o_real(G,bot_bot_set_o) = zero_zero_real ).

% sum.empty
tff(fact_4898_sum_Oempty,axiom,
    ! [G: fun_o_int] : groups8505340233167759370_o_int(G,bot_bot_set_o) = zero_zero_int ).

% sum.empty
tff(fact_4899_sum_Oempty,axiom,
    ! [G: fun_o_complex] : groups5328290441151304332omplex(G,bot_bot_set_o) = zero_zero_complex ).

% sum.empty
tff(fact_4900_sum_Oempty,axiom,
    ! [G: fun_o_Extended_enat] : groups7198740251461348360d_enat(G,bot_bot_set_o) = zero_z5237406670263579293d_enat ).

% sum.empty
tff(fact_4901_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_4902_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_4903_sum_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( groups2027974829824023292at_nat(G,A2) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_4904_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_4905_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_4906_sum_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( groups4148127829035722712t_real(G,A2) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_4907_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_int] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),A2) = zero_zero_int ) ) ).

% sum.infinite
tff(fact_4908_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_int] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5690904116761175830ex_int(G,A2) = zero_zero_int ) ) ).

% sum.infinite
tff(fact_4909_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_int] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups4538972089207619220nt_int(G,A2) = zero_zero_int ) ) ).

% sum.infinite
tff(fact_4910_sum_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( groups2025484359314973016at_int(G,A2) = zero_zero_int ) ) ).

% sum.infinite
tff(fact_4911_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 )
      <=> ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),F3)
           => ( aa_complex_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4912_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 )
      <=> ! [X3: int] :
            ( aa_set_int_o(member_int2(X3),F3)
           => ( aa_int_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4913_sum__eq__0__iff,axiom,
    ! [F3: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F3)
     => ( ( groups2027974829824023292at_nat(F,F3) = zero_zero_nat )
      <=> ! [X3: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X3),F3)
           => ( aa_Extended_enat_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4914_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F: fun_na8797796813480665199d_enat] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( ( groups7108830773950497114d_enat(F,F3) = zero_z5237406670263579293d_enat )
      <=> ! [X3: nat] :
            ( aa_set_nat_o(member_nat2(X3),F3)
           => ( aa_nat_Extended_enat(F,X3) = zero_z5237406670263579293d_enat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4915_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F: fun_co6496365337113880337d_enat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F3)
     => ( ( groups1752964319039525884d_enat(F,F3) = zero_z5237406670263579293d_enat )
      <=> ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),F3)
           => ( aa_com3683005661447327866d_enat(F,X3) = zero_z5237406670263579293d_enat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4916_sum__eq__0__iff,axiom,
    ! [F3: set_int,F: fun_in5914218760682845459d_enat] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( ( groups4225252721152677374d_enat(F,F3) = zero_z5237406670263579293d_enat )
      <=> ! [X3: int] :
            ( aa_set_int_o(member_int2(X3),F3)
           => ( aa_int_Extended_enat(F,X3) = zero_z5237406670263579293d_enat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4917_sum__eq__0__iff,axiom,
    ! [F3: set_Extended_enat,F: fun_Ex5852502975177633731d_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F3)
     => ( ( groups2433450451889696826d_enat(F,F3) = zero_z5237406670263579293d_enat )
      <=> ! [X3: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X3),F3)
           => ( aa_Ext7969882492699152700d_enat(F,X3) = zero_z5237406670263579293d_enat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4918_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 )
      <=> ! [X3: nat] :
            ( aa_set_nat_o(member_nat2(X3),F3)
           => ( aa_nat_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4919_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A2: set_real,B: real] :
      ( ( insert_real2(A,A2) = insert_real2(B,bot_bot_set_real) )
    <=> ( ( A = B )
        & aa_set_real_o(ord_less_eq_set_real(A2),insert_real2(B,bot_bot_set_real)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4920_singleton__insert__inj__eq_H,axiom,
    ! [A: $o,A2: set_o,B: $o] :
      ( ( insert_o2((A),A2) = insert_o2((B),bot_bot_set_o) )
    <=> ( ( (A)
        <=> (B) )
        & aa_set_o_o(ord_less_eq_set_o(A2),insert_o2((B),bot_bot_set_o)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4921_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A2: set_int,B: int] :
      ( ( insert_int2(A,A2) = insert_int2(B,bot_bot_set_int) )
    <=> ( ( A = B )
        & aa_set_int_o(ord_less_eq_set_int(A2),insert_int2(B,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4922_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B: nat] :
      ( ( insert_nat2(A,A2) = insert_nat2(B,bot_bot_set_nat) )
    <=> ( ( A = B )
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat2(B,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4923_singleton__insert__inj__eq,axiom,
    ! [B: real,A: real,A2: set_real] :
      ( ( insert_real2(B,bot_bot_set_real) = insert_real2(A,A2) )
    <=> ( ( A = B )
        & aa_set_real_o(ord_less_eq_set_real(A2),insert_real2(B,bot_bot_set_real)) ) ) ).

% singleton_insert_inj_eq
tff(fact_4924_singleton__insert__inj__eq,axiom,
    ! [B: $o,A: $o,A2: set_o] :
      ( ( insert_o2((B),bot_bot_set_o) = insert_o2((A),A2) )
    <=> ( ( (A)
        <=> (B) )
        & aa_set_o_o(ord_less_eq_set_o(A2),insert_o2((B),bot_bot_set_o)) ) ) ).

% singleton_insert_inj_eq
tff(fact_4925_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A2: set_int] :
      ( ( insert_int2(B,bot_bot_set_int) = insert_int2(A,A2) )
    <=> ( ( A = B )
        & aa_set_int_o(ord_less_eq_set_int(A2),insert_int2(B,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq
tff(fact_4926_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A2: set_nat] :
      ( ( insert_nat2(B,bot_bot_set_nat) = insert_nat2(A,A2) )
    <=> ( ( A = B )
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat2(B,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_4927_atLeastAtMost__singleton,axiom,
    ! [A: $o] : set_or8904488021354931149Most_o((A),(A)) = insert_o2((A),bot_bot_set_o) ).

% atLeastAtMost_singleton
tff(fact_4928_atLeastAtMost__singleton,axiom,
    ! [A: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),A) = insert_nat2(A,bot_bot_set_nat) ).

% atLeastAtMost_singleton
tff(fact_4929_atLeastAtMost__singleton,axiom,
    ! [A: int] : set_or1266510415728281911st_int(A,A) = insert_int2(A,bot_bot_set_int) ).

% atLeastAtMost_singleton
tff(fact_4930_atLeastAtMost__singleton,axiom,
    ! [A: real] : set_or1222579329274155063t_real(A,A) = insert_real2(A,bot_bot_set_real) ).

% atLeastAtMost_singleton
tff(fact_4931_atLeastAtMost__singleton__iff,axiom,
    ! [A: $o,B: $o,C: $o] :
      ( ( set_or8904488021354931149Most_o((A),(B)) = insert_o2((C),bot_bot_set_o) )
    <=> ( ( (A)
        <=> (B) )
        & ( (B)
        <=> (C) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_4932_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B) = insert_nat2(C,bot_bot_set_nat) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_4933_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( set_or1266510415728281911st_int(A,B) = insert_int2(C,bot_bot_set_int) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_4934_atLeastAtMost__singleton__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( set_or1222579329274155063t_real(A,B) = insert_real2(C,bot_bot_set_real) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_4935_insert__Diff__single,axiom,
    ! [A: real,A2: set_real] : insert_real2(A,minus_minus_set_real(A2,insert_real2(A,bot_bot_set_real))) = insert_real2(A,A2) ).

% insert_Diff_single
tff(fact_4936_insert__Diff__single,axiom,
    ! [A: $o,A2: set_o] : insert_o2((A),minus_minus_set_o(A2,insert_o2((A),bot_bot_set_o))) = insert_o2((A),A2) ).

% insert_Diff_single
tff(fact_4937_insert__Diff__single,axiom,
    ! [A: int,A2: set_int] : insert_int2(A,minus_minus_set_int(A2,insert_int2(A,bot_bot_set_int))) = insert_int2(A,A2) ).

% insert_Diff_single
tff(fact_4938_insert__Diff__single,axiom,
    ! [A: nat,A2: set_nat] : insert_nat2(A,minus_minus_set_nat(A2,insert_nat2(A,bot_bot_set_nat))) = insert_nat2(A,A2) ).

% insert_Diff_single
tff(fact_4939_ceiling__zero,axiom,
    aa_real_int(archim7802044766580827645g_real,zero_zero_real) = zero_zero_int ).

% ceiling_zero
tff(fact_4940_mask__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,zero_zero_nat) = zero_zero_int ).

% mask_0
tff(fact_4941_mask__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,zero_zero_nat) = zero_zero_nat ).

% mask_0
tff(fact_4942_mask__eq__0__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_int(bit_se2000444600071755411sk_int,Nb) = zero_zero_int )
    <=> ( Nb = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_4943_mask__eq__0__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) = zero_zero_nat )
    <=> ( Nb = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_4944_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ea(A),B),S3) = $ite(aa_set_real_o(member_real2(A),S3),aa_real_nat(B,A),zero_zero_nat) ) ) ).

% sum.delta
tff(fact_4945_sum_Odelta,axiom,
    ! [S3: set_o,A: $o,B: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S3)
     => ( groups8507830703676809646_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_eb((A)),B),S3) = $ite(aa_set_o_o(member_o2((A)),S3),aa_o_nat(B,(A)),zero_zero_nat) ) ) ).

% sum.delta
tff(fact_4946_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_ec(A),B),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_complex_nat(B,A),zero_zero_nat) ) ) ).

% sum.delta
tff(fact_4947_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ed(A),B),S3) = $ite(aa_set_int_o(member_int2(A),S3),aa_int_nat(B,A),zero_zero_nat) ) ) ).

% sum.delta
tff(fact_4948_sum_Odelta,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
     => ( groups2027974829824023292at_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_ee(A),B),S3) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(A),S3),aa_Extended_enat_nat(B,A),zero_zero_nat) ) ) ).

% sum.delta
tff(fact_4949_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_ef(A),B),S3) = $ite(aa_set_real_o(member_real2(A),S3),aa_real_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_4950_sum_Odelta,axiom,
    ! [S3: set_o,A: $o,B: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,S3)
     => ( groups8691415230153176458o_real(aa_fun688608109223240316o_real(aTP_Lamm_eg((A)),B),S3) = $ite(aa_set_o_o(member_o2((A)),S3),aa_o_real(B,(A)),zero_zero_real) ) ) ).

% sum.delta
tff(fact_4951_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_eh(A),B),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_complex_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_4952_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_ei(A),B),S3) = $ite(aa_set_int_o(member_int2(A),S3),aa_int_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_4953_sum_Odelta,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
     => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aTP_Lamm_ej(A),B),S3) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(A),S3),aa_Ext2541572767299617110t_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_4954_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ek(A),B),S3) = $ite(aa_set_real_o(member_real2(A),S3),aa_real_nat(B,A),zero_zero_nat) ) ) ).

% sum.delta'
tff(fact_4955_sum_Odelta_H,axiom,
    ! [S3: set_o,A: $o,B: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S3)
     => ( groups8507830703676809646_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_el((A)),B),S3) = $ite(aa_set_o_o(member_o2((A)),S3),aa_o_nat(B,(A)),zero_zero_nat) ) ) ).

% sum.delta'
tff(fact_4956_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_em(A),B),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_complex_nat(B,A),zero_zero_nat) ) ) ).

% sum.delta'
tff(fact_4957_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_en(A),B),S3) = $ite(aa_set_int_o(member_int2(A),S3),aa_int_nat(B,A),zero_zero_nat) ) ) ).

% sum.delta'
tff(fact_4958_sum_Odelta_H,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
     => ( groups2027974829824023292at_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_eo(A),B),S3) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(A),S3),aa_Extended_enat_nat(B,A),zero_zero_nat) ) ) ).

% sum.delta'
tff(fact_4959_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_ep(A),B),S3) = $ite(aa_set_real_o(member_real2(A),S3),aa_real_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_4960_sum_Odelta_H,axiom,
    ! [S3: set_o,A: $o,B: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,S3)
     => ( groups8691415230153176458o_real(aa_fun688608109223240316o_real(aTP_Lamm_eq((A)),B),S3) = $ite(aa_set_o_o(member_o2((A)),S3),aa_o_real(B,(A)),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_4961_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_er(A),B),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_complex_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_4962_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_es(A),B),S3) = $ite(aa_set_int_o(member_int2(A),S3),aa_int_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_4963_sum_Odelta_H,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
     => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aTP_Lamm_et(A),B),S3) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(A),S3),aa_Ext2541572767299617110t_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_4964_List_Oset__insert,axiom,
    ! [Xb: real,Xs: list_real] : set_real2(insert_real(Xb,Xs)) = insert_real2(Xb,set_real2(Xs)) ).

% List.set_insert
tff(fact_4965_List_Oset__insert,axiom,
    ! [Xb: $o,Xs: list_o] : set_o2(insert_o((Xb),Xs)) = insert_o2((Xb),set_o2(Xs)) ).

% List.set_insert
tff(fact_4966_List_Oset__insert,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] : set_VEBT_VEBT2(insert_VEBT_VEBT(Xb,Xs)) = insert_VEBT_VEBT2(Xb,set_VEBT_VEBT2(Xs)) ).

% List.set_insert
tff(fact_4967_List_Oset__insert,axiom,
    ! [Xb: int,Xs: list_int] : set_int2(insert_int(Xb,Xs)) = insert_int2(Xb,set_int2(Xs)) ).

% List.set_insert
tff(fact_4968_List_Oset__insert,axiom,
    ! [Xb: nat,Xs: list_nat] : set_nat2(insert_nat(Xb,Xs)) = insert_nat2(Xb,set_nat2(Xs)) ).

% List.set_insert
tff(fact_4969_sum_Oinsert,axiom,
    ! [A2: set_real,Xb: real,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ~ aa_set_real_o(member_real2(Xb),A2)
       => ( groups1935376822645274424al_nat(G,insert_real2(Xb,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,Xb)),groups1935376822645274424al_nat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_4970_sum_Oinsert,axiom,
    ! [A2: set_o,Xb: $o,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ~ aa_set_o_o(member_o2((Xb)),A2)
       => ( groups8507830703676809646_o_nat(G,insert_o2((Xb),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(G,(Xb))),groups8507830703676809646_o_nat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_4971_sum_Oinsert,axiom,
    ! [A2: set_complex,Xb: complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ~ aa_set_complex_o(member_complex(Xb),A2)
       => ( groups5693394587270226106ex_nat(G,insert_complex(Xb,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,Xb)),groups5693394587270226106ex_nat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_4972_sum_Oinsert,axiom,
    ! [A2: set_int,Xb: int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int2(Xb),A2)
       => ( groups4541462559716669496nt_nat(G,insert_int2(Xb,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(G,Xb)),groups4541462559716669496nt_nat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_4973_sum_Oinsert,axiom,
    ! [A2: set_Extended_enat,Xb: extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(Xb),A2)
       => ( groups2027974829824023292at_nat(G,insert_Extended_enat(Xb,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Extended_enat_nat(G,Xb)),groups2027974829824023292at_nat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_4974_sum_Oinsert,axiom,
    ! [A2: set_real,Xb: real,G: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ~ aa_set_real_o(member_real2(Xb),A2)
       => ( groups1932886352136224148al_int(G,insert_real2(Xb,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(G,Xb)),groups1932886352136224148al_int(G,A2)) ) ) ) ).

% sum.insert
tff(fact_4975_sum_Oinsert,axiom,
    ! [A2: set_o,Xb: $o,G: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ~ aa_set_o_o(member_o2((Xb)),A2)
       => ( groups8505340233167759370_o_int(G,insert_o2((Xb),A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(G,(Xb))),groups8505340233167759370_o_int(G,A2)) ) ) ) ).

% sum.insert
tff(fact_4976_sum_Oinsert,axiom,
    ! [A2: set_nat,Xb: nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat2(Xb),A2)
       => ( aa_set_nat_int(groups3539618377306564664at_int(G),insert_nat2(Xb,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Xb)),aa_set_nat_int(groups3539618377306564664at_int(G),A2)) ) ) ) ).

% sum.insert
tff(fact_4977_sum_Oinsert,axiom,
    ! [A2: set_complex,Xb: complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ~ aa_set_complex_o(member_complex(Xb),A2)
       => ( groups5690904116761175830ex_int(G,insert_complex(Xb,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,Xb)),groups5690904116761175830ex_int(G,A2)) ) ) ) ).

% sum.insert
tff(fact_4978_sum_Oinsert,axiom,
    ! [A2: set_int,Xb: int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int2(Xb),A2)
       => ( groups4538972089207619220nt_int(G,insert_int2(Xb,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(G,Xb)),groups4538972089207619220nt_int(G,A2)) ) ) ) ).

% sum.insert
tff(fact_4979_bit__numeral__Bit0__Suc__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(numeral_numeral_int(bit0(Mb))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(numeral_numeral_int(Mb)),Nb) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_4980_bit__numeral__Bit0__Suc__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,bit0(Mb))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_4981_bit__numeral__Bit1__Suc__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(numeral_numeral_int(aa_num_num(bit1,Mb))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(numeral_numeral_int(Mb)),Nb) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_4982_bit__numeral__Bit1__Suc__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_4983_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_4984_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_4985_subset__Compl__singleton,axiom,
    ! [A2: set_set_nat,B: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),uminus613421341184616069et_nat(insert_set_nat2(B,bot_bot_set_set_nat)))
    <=> ~ aa_set_set_nat_o(member_set_nat2(B),A2) ) ).

% subset_Compl_singleton
tff(fact_4986_subset__Compl__singleton,axiom,
    ! [A2: set_real,B: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),uminus612125837232591019t_real(insert_real2(B,bot_bot_set_real)))
    <=> ~ aa_set_real_o(member_real2(B),A2) ) ).

% subset_Compl_singleton
tff(fact_4987_subset__Compl__singleton,axiom,
    ! [A2: set_o,B: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),uminus_uminus_set_o(insert_o2((B),bot_bot_set_o)))
    <=> ~ aa_set_o_o(member_o2((B)),A2) ) ).

% subset_Compl_singleton
tff(fact_4988_subset__Compl__singleton,axiom,
    ! [A2: set_int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),uminus1532241313380277803et_int(insert_int2(B,bot_bot_set_int)))
    <=> ~ aa_set_int_o(member_int2(B),A2) ) ).

% subset_Compl_singleton
tff(fact_4989_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_nat2(B,bot_bot_set_nat)))
    <=> ~ aa_set_nat_o(member_nat2(B),A2) ) ).

% subset_Compl_singleton
tff(fact_4990_ceiling__add__of__int,axiom,
    ! [Xb: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(plus_plus_real(Xb),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,Xb)),Z) ).

% ceiling_add_of_int
tff(fact_4991_ceiling__le__zero,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% ceiling_le_zero
tff(fact_4992_zero__less__ceiling,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% zero_less_ceiling
tff(fact_4993_ceiling__le__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),numeral_numeral_int(V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),numeral_numeral_real(V)) ) ).

% ceiling_le_numeral
tff(fact_4994_ceiling__less__one,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% ceiling_less_one
tff(fact_4995_one__le__ceiling,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% one_le_ceiling
tff(fact_4996_numeral__less__ceiling,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,numeral_numeral_int(V)),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(V)),Xb) ) ).

% numeral_less_ceiling
tff(fact_4997_ceiling__le__one,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ).

% ceiling_le_one
tff(fact_4998_one__less__ceiling,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ).

% one_less_ceiling
tff(fact_4999_set__replicate,axiom,
    ! [Nb: nat,Xb: vEBT_VEBT] :
      ( ( Nb != zero_zero_nat )
     => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Xb)) = insert_VEBT_VEBT2(Xb,bot_bo8194388402131092736T_VEBT) ) ) ).

% set_replicate
tff(fact_5000_set__replicate,axiom,
    ! [Nb: nat,Xb: real] :
      ( ( Nb != zero_zero_nat )
     => ( set_real2(replicate_real(Nb,Xb)) = insert_real2(Xb,bot_bot_set_real) ) ) ).

% set_replicate
tff(fact_5001_set__replicate,axiom,
    ! [Nb: nat,Xb: $o] :
      ( ( Nb != zero_zero_nat )
     => ( set_o2(replicate_o(Nb,(Xb))) = insert_o2((Xb),bot_bot_set_o) ) ) ).

% set_replicate
tff(fact_5002_set__replicate,axiom,
    ! [Nb: nat,Xb: nat] :
      ( ( Nb != zero_zero_nat )
     => ( set_nat2(replicate_nat(Nb,Xb)) = insert_nat2(Xb,bot_bot_set_nat) ) ) ).

% set_replicate
tff(fact_5003_set__replicate,axiom,
    ! [Nb: nat,Xb: int] :
      ( ( Nb != zero_zero_nat )
     => ( set_int2(replicate_int(Nb,Xb)) = insert_int2(Xb,bot_bot_set_int) ) ) ).

% set_replicate
tff(fact_5004_ceiling__add__numeral,axiom,
    ! [Xb: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(plus_plus_real(Xb),numeral_numeral_real(V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),numeral_numeral_int(V)) ).

% ceiling_add_numeral
tff(fact_5005_ceiling__add__one,axiom,
    ! [Xb: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(plus_plus_real(Xb),one_one_real)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int) ).

% ceiling_add_one
tff(fact_5006_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W2: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(bit0(W2)))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(W2))),Nb) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_5007_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W2: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(aa_num_num(bit1,W2)))),aa_nat_nat(suc,Nb))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(numeral_numeral_int(W2)),Nb) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_5008_bit__0,axiom,
    ! [A: int] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),zero_zero_nat)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A) ) ).

% bit_0
tff(fact_5009_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,bit0(one))),A) ) ).

% bit_0
tff(fact_5010_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] :
      aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_5011_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_complex,Mb: nat,Nb: nat] :
      groups2073611262835488442omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_complex,aa_complex_complex(plus_plus_complex(groups2073611262835488442omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_5012_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_na8797796813480665199d_enat,Mb: nat,Nb: nat] :
      groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_z5237406670263579293d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_nat_Extended_enat(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_5013_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_5014_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_real,aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_5015_ceiling__less__zero,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% ceiling_less_zero
tff(fact_5016_zero__le__ceiling,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb) ) ).

% zero_le_ceiling
tff(fact_5017_sum__zero__power,axiom,
    ! [C: fun_nat_complex,A2: set_nat] :
      groups2073611262835488442omplex(aTP_Lamm_eu(C),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat2(zero_zero_nat),A2) ),
        aa_nat_complex(C,zero_zero_nat),
        zero_zero_complex ) ).

% sum_zero_power
tff(fact_5018_sum__zero__power,axiom,
    ! [C: fun_nat_real,A2: set_nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ev(C)),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat2(zero_zero_nat),A2) ),
        aa_nat_real(C,zero_zero_nat),
        zero_zero_real ) ).

% sum_zero_power
tff(fact_5019_bit__mod__2__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(modulo_modulo_int(A),numeral_numeral_int(bit0(one)))),Nb)
    <=> ( ( Nb = zero_zero_nat )
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A) ) ) ).

% bit_mod_2_iff
tff(fact_5020_bit__mod__2__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),Nb)
    <=> ( ( Nb = zero_zero_nat )
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A) ) ) ).

% bit_mod_2_iff
tff(fact_5021_ceiling__less__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),numeral_numeral_int(V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(minus_minus_real(numeral_numeral_real(V)),one_one_real)) ) ).

% ceiling_less_numeral
tff(fact_5022_numeral__le__ceiling,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(V)),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(numeral_numeral_real(V)),one_one_real)),Xb) ) ).

% numeral_le_ceiling
tff(fact_5023_ceiling__le__neg__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_int_int(uminus_uminus_int,numeral_numeral_int(V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) ) ).

% ceiling_le_neg_numeral
tff(fact_5024_neg__numeral__less__ceiling,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(V))),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),Xb) ) ).

% neg_numeral_less_ceiling
tff(fact_5025_sum__zero__power_H,axiom,
    ! [C: fun_nat_complex,D: fun_nat_complex,A2: set_nat] :
      groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ew(C),D),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat2(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_5026_sum__zero__power_H,axiom,
    ! [C: fun_nat_real,D: fun_nat_real,A2: set_nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ex(C),D)),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat2(zero_zero_nat),A2) ),
        aa_real_real(divide_divide_real(aa_nat_real(C,zero_zero_nat)),aa_nat_real(D,zero_zero_nat)),
        zero_zero_real ) ).

% sum_zero_power'
tff(fact_5027_sum_Oinsert__if,axiom,
    ! [A2: set_real,G: fun_real_nat,Xb: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1935376822645274424al_nat(G,insert_real2(Xb,A2)) = $ite(aa_set_real_o(member_real2(Xb),A2),groups1935376822645274424al_nat(G,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,Xb)),groups1935376822645274424al_nat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_5028_sum_Oinsert__if,axiom,
    ! [A2: set_o,G: fun_o_nat,Xb: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups8507830703676809646_o_nat(G,insert_o2((Xb),A2)) = $ite(aa_set_o_o(member_o2((Xb)),A2),groups8507830703676809646_o_nat(G,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(G,(Xb))),groups8507830703676809646_o_nat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_5029_sum_Oinsert__if,axiom,
    ! [A2: set_complex,G: fun_complex_nat,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5693394587270226106ex_nat(G,insert_complex(Xb,A2)) = $ite(aa_set_complex_o(member_complex(Xb),A2),groups5693394587270226106ex_nat(G,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,Xb)),groups5693394587270226106ex_nat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_5030_sum_Oinsert__if,axiom,
    ! [A2: set_int,G: fun_int_nat,Xb: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(G,insert_int2(Xb,A2)) = $ite(aa_set_int_o(member_int2(Xb),A2),groups4541462559716669496nt_nat(G,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(G,Xb)),groups4541462559716669496nt_nat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_5031_sum_Oinsert__if,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,Xb: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( groups2027974829824023292at_nat(G,insert_Extended_enat(Xb,A2)) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(Xb),A2),groups2027974829824023292at_nat(G,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Extended_enat_nat(G,Xb)),groups2027974829824023292at_nat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_5032_sum_Oinsert__if,axiom,
    ! [A2: set_real,G: fun_real_int,Xb: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1932886352136224148al_int(G,insert_real2(Xb,A2)) = $ite(aa_set_real_o(member_real2(Xb),A2),groups1932886352136224148al_int(G,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(G,Xb)),groups1932886352136224148al_int(G,A2))) ) ) ).

% sum.insert_if
tff(fact_5033_sum_Oinsert__if,axiom,
    ! [A2: set_o,G: fun_o_int,Xb: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups8505340233167759370_o_int(G,insert_o2((Xb),A2)) = $ite(aa_set_o_o(member_o2((Xb)),A2),groups8505340233167759370_o_int(G,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(G,(Xb))),groups8505340233167759370_o_int(G,A2))) ) ) ).

% sum.insert_if
tff(fact_5034_sum_Oinsert__if,axiom,
    ! [A2: set_nat,G: fun_nat_int,Xb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),insert_nat2(Xb,A2)) = $ite(aa_set_nat_o(member_nat2(Xb),A2),aa_set_nat_int(groups3539618377306564664at_int(G),A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Xb)),aa_set_nat_int(groups3539618377306564664at_int(G),A2))) ) ) ).

% sum.insert_if
tff(fact_5035_sum_Oinsert__if,axiom,
    ! [A2: set_complex,G: fun_complex_int,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5690904116761175830ex_int(G,insert_complex(Xb,A2)) = $ite(aa_set_complex_o(member_complex(Xb),A2),groups5690904116761175830ex_int(G,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,Xb)),groups5690904116761175830ex_int(G,A2))) ) ) ).

% sum.insert_if
tff(fact_5036_sum_Oinsert__if,axiom,
    ! [A2: set_int,G: fun_int_int,Xb: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4538972089207619220nt_int(G,insert_int2(Xb,A2)) = $ite(aa_set_int_o(member_int2(Xb),A2),groups4538972089207619220nt_int(G,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(G,Xb)),groups4538972089207619220nt_int(G,A2))) ) ) ).

% sum.insert_if
tff(fact_5037_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat2(X4),A2)
         => ( aa_nat_nat(G,X4) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_5038_sum_Oneutral,axiom,
    ! [A2: set_complex,G: fun_complex_complex] :
      ( ! [X4: complex] :
          ( aa_set_complex_o(member_complex(X4),A2)
         => ( aa_complex_complex(G,X4) = zero_zero_complex ) )
     => ( groups7754918857620584856omplex(G,A2) = zero_zero_complex ) ) ).

% sum.neutral
tff(fact_5039_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_real] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat2(X4),A2)
         => ( aa_nat_real(G,X4) = zero_zero_real ) )
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),A2) = zero_zero_real ) ) ).

% sum.neutral
tff(fact_5040_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_real2(A3),A2)
           => ( aa_real_nat(G,A3) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5041_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_o_nat,A2: set_o] :
      ( ( groups8507830703676809646_o_nat(G,A2) != zero_zero_nat )
     => ~ ! [A3: $o] :
            ( aa_set_o_o(member_o2((A3)),A2)
           => ( aa_o_nat(G,(A3)) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5042_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_int2(A3),A2)
           => ( aa_int_nat(G,A3) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5043_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_real2(A3),A2)
           => ( aa_real_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5044_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_o_real,A2: set_o] :
      ( ( groups8691415230153176458o_real(G,A2) != zero_zero_real )
     => ~ ! [A3: $o] :
            ( aa_set_o_o(member_o2((A3)),A2)
           => ( aa_o_real(G,(A3)) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5045_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_int2(A3),A2)
           => ( aa_int_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5046_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_int,A2: set_real] :
      ( ( groups1932886352136224148al_int(G,A2) != zero_zero_int )
     => ~ ! [A3: real] :
            ( aa_set_real_o(member_real2(A3),A2)
           => ( aa_real_int(G,A3) = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5047_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_o_int,A2: set_o] :
      ( ( groups8505340233167759370_o_int(G,A2) != zero_zero_int )
     => ~ ! [A3: $o] :
            ( aa_set_o_o(member_o2((A3)),A2)
           => ( aa_o_int(G,(A3)) = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5048_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_int,A2: set_nat] :
      ( ( aa_set_nat_int(groups3539618377306564664at_int(G),A2) != zero_zero_int )
     => ~ ! [A3: nat] :
            ( aa_set_nat_o(member_nat2(A3),A2)
           => ( aa_nat_int(G,A3) = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5049_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_int,A2: set_int] :
      ( ( groups4538972089207619220nt_int(G,A2) != zero_zero_int )
     => ~ ! [A3: int] :
            ( aa_set_int_o(member_int2(A3),A2)
           => ( aa_int_int(G,A3) = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5050_bit__disjunctive__add__iff,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(bit_se1146084159140164899it_int(A),N)
          | ~ aa_nat_o(bit_se1146084159140164899it_int(B),N) )
     => ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Nb)
      <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
          | aa_nat_o(bit_se1146084159140164899it_int(B),Nb) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_5051_bit__disjunctive__add__iff,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N)
          | ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),N) )
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Nb)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
          | aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),Nb) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_5052_singletonD,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_set_nat_o(member_set_nat2(B),insert_set_nat2(A,bot_bot_set_set_nat))
     => ( B = A ) ) ).

% singletonD
tff(fact_5053_singletonD,axiom,
    ! [B: real,A: real] :
      ( aa_set_real_o(member_real2(B),insert_real2(A,bot_bot_set_real))
     => ( B = A ) ) ).

% singletonD
tff(fact_5054_singletonD,axiom,
    ! [B: $o,A: $o] :
      ( aa_set_o_o(member_o2((B)),insert_o2((A),bot_bot_set_o))
     => ( (B)
      <=> (A) ) ) ).

% singletonD
tff(fact_5055_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( aa_set_nat_o(member_nat2(B),insert_nat2(A,bot_bot_set_nat))
     => ( B = A ) ) ).

% singletonD
tff(fact_5056_singletonD,axiom,
    ! [B: int,A: int] :
      ( aa_set_int_o(member_int2(B),insert_int2(A,bot_bot_set_int))
     => ( B = A ) ) ).

% singletonD
tff(fact_5057_singleton__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_set_nat_o(member_set_nat2(B),insert_set_nat2(A,bot_bot_set_set_nat))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_5058_singleton__iff,axiom,
    ! [B: real,A: real] :
      ( aa_set_real_o(member_real2(B),insert_real2(A,bot_bot_set_real))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_5059_singleton__iff,axiom,
    ! [B: $o,A: $o] :
      ( aa_set_o_o(member_o2((B)),insert_o2((A),bot_bot_set_o))
    <=> ( (B)
      <=> (A) ) ) ).

% singleton_iff
tff(fact_5060_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_set_nat_o(member_nat2(B),insert_nat2(A,bot_bot_set_nat))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_5061_singleton__iff,axiom,
    ! [B: int,A: int] :
      ( aa_set_int_o(member_int2(B),insert_int2(A,bot_bot_set_int))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_5062_doubleton__eq__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( insert_real2(A,insert_real2(B,bot_bot_set_real)) = insert_real2(C,insert_real2(D,bot_bot_set_real)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_5063_doubleton__eq__iff,axiom,
    ! [A: $o,B: $o,C: $o,D: $o] :
      ( ( insert_o2((A),insert_o2((B),bot_bot_set_o)) = insert_o2((C),insert_o2((D),bot_bot_set_o)) )
    <=> ( ( ( (A)
          <=> (C) )
          & ( (B)
          <=> (D) ) )
        | ( ( (A)
          <=> (D) )
          & ( (B)
          <=> (C) ) ) ) ) ).

% doubleton_eq_iff
tff(fact_5064_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( insert_nat2(A,insert_nat2(B,bot_bot_set_nat)) = insert_nat2(C,insert_nat2(D,bot_bot_set_nat)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_5065_doubleton__eq__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( insert_int2(A,insert_int2(B,bot_bot_set_int)) = insert_int2(C,insert_int2(D,bot_bot_set_int)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_5066_insert__not__empty,axiom,
    ! [A: real,A2: set_real] : insert_real2(A,A2) != bot_bot_set_real ).

% insert_not_empty
tff(fact_5067_insert__not__empty,axiom,
    ! [A: $o,A2: set_o] : insert_o2((A),A2) != bot_bot_set_o ).

% insert_not_empty
tff(fact_5068_insert__not__empty,axiom,
    ! [A: nat,A2: set_nat] : insert_nat2(A,A2) != bot_bot_set_nat ).

% insert_not_empty
tff(fact_5069_insert__not__empty,axiom,
    ! [A: int,A2: set_int] : insert_int2(A,A2) != bot_bot_set_int ).

% insert_not_empty
tff(fact_5070_singleton__inject,axiom,
    ! [A: real,B: real] :
      ( ( insert_real2(A,bot_bot_set_real) = insert_real2(B,bot_bot_set_real) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_5071_singleton__inject,axiom,
    ! [A: $o,B: $o] :
      ( ( insert_o2((A),bot_bot_set_o) = insert_o2((B),bot_bot_set_o) )
     => ( (A)
      <=> (B) ) ) ).

% singleton_inject
tff(fact_5072_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( insert_nat2(A,bot_bot_set_nat) = insert_nat2(B,bot_bot_set_nat) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_5073_singleton__inject,axiom,
    ! [A: int,B: int] :
      ( ( insert_int2(A,bot_bot_set_int) = insert_int2(B,bot_bot_set_int) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_5074_insert__subsetI,axiom,
    ! [Xb: real,A2: set_real,X5: set_real] :
      ( aa_set_real_o(member_real2(Xb),A2)
     => ( aa_set_real_o(ord_less_eq_set_real(X5),A2)
       => aa_set_real_o(ord_less_eq_set_real(insert_real2(Xb,X5)),A2) ) ) ).

% insert_subsetI
tff(fact_5075_insert__subsetI,axiom,
    ! [Xb: $o,A2: set_o,X5: set_o] :
      ( aa_set_o_o(member_o2((Xb)),A2)
     => ( aa_set_o_o(ord_less_eq_set_o(X5),A2)
       => aa_set_o_o(ord_less_eq_set_o(insert_o2((Xb),X5)),A2) ) ) ).

% insert_subsetI
tff(fact_5076_insert__subsetI,axiom,
    ! [Xb: set_nat,A2: set_set_nat,X5: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat2(Xb),A2)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(X5),A2)
       => aa_set_set_nat_o(ord_le6893508408891458716et_nat(insert_set_nat2(Xb,X5)),A2) ) ) ).

% insert_subsetI
tff(fact_5077_insert__subsetI,axiom,
    ! [Xb: int,A2: set_int,X5: set_int] :
      ( aa_set_int_o(member_int2(Xb),A2)
     => ( aa_set_int_o(ord_less_eq_set_int(X5),A2)
       => aa_set_int_o(ord_less_eq_set_int(insert_int2(Xb,X5)),A2) ) ) ).

% insert_subsetI
tff(fact_5078_insert__subsetI,axiom,
    ! [Xb: nat,A2: set_nat,X5: set_nat] :
      ( aa_set_nat_o(member_nat2(Xb),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_nat2(Xb,X5)),A2) ) ) ).

% insert_subsetI
tff(fact_5079_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_int2(A,C4)),insert_int2(A,D5)) ) ).

% insert_mono
tff(fact_5080_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_real2(A,C4)),insert_real2(A,D5)) ) ).

% insert_mono
tff(fact_5081_insert__mono,axiom,
    ! [C4: set_o,D5: set_o,A: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(C4),D5)
     => aa_set_o_o(ord_less_eq_set_o(insert_o2((A),C4)),insert_o2((A),D5)) ) ).

% insert_mono
tff(fact_5082_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_nat2(A,C4)),insert_nat2(A,D5)) ) ).

% insert_mono
tff(fact_5083_subset__insert,axiom,
    ! [Xb: real,A2: set_real,B2: set_real] :
      ( ~ aa_set_real_o(member_real2(Xb),A2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),insert_real2(Xb,B2))
      <=> aa_set_real_o(ord_less_eq_set_real(A2),B2) ) ) ).

% subset_insert
tff(fact_5084_subset__insert,axiom,
    ! [Xb: $o,A2: set_o,B2: set_o] :
      ( ~ aa_set_o_o(member_o2((Xb)),A2)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),insert_o2((Xb),B2))
      <=> aa_set_o_o(ord_less_eq_set_o(A2),B2) ) ) ).

% subset_insert
tff(fact_5085_subset__insert,axiom,
    ! [Xb: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat2(Xb),A2)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),insert_set_nat2(Xb,B2))
      <=> aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2) ) ) ).

% subset_insert
tff(fact_5086_subset__insert,axiom,
    ! [Xb: int,A2: set_int,B2: set_int] :
      ( ~ aa_set_int_o(member_int2(Xb),A2)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),insert_int2(Xb,B2))
      <=> aa_set_int_o(ord_less_eq_set_int(A2),B2) ) ) ).

% subset_insert
tff(fact_5087_subset__insert,axiom,
    ! [Xb: nat,A2: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat2(Xb),A2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat2(Xb,B2))
      <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2) ) ) ).

% subset_insert
tff(fact_5088_subset__insertI,axiom,
    ! [B2: set_int,A: int] : aa_set_int_o(ord_less_eq_set_int(B2),insert_int2(A,B2)) ).

% subset_insertI
tff(fact_5089_subset__insertI,axiom,
    ! [B2: set_real,A: real] : aa_set_real_o(ord_less_eq_set_real(B2),insert_real2(A,B2)) ).

% subset_insertI
tff(fact_5090_subset__insertI,axiom,
    ! [B2: set_o,A: $o] : aa_set_o_o(ord_less_eq_set_o(B2),insert_o2((A),B2)) ).

% subset_insertI
tff(fact_5091_subset__insertI,axiom,
    ! [B2: set_nat,A: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),insert_nat2(A,B2)) ).

% subset_insertI
tff(fact_5092_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_int2(B,B2)) ) ).

% subset_insertI2
tff(fact_5093_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_real2(B,B2)) ) ).

% subset_insertI2
tff(fact_5094_subset__insertI2,axiom,
    ! [A2: set_o,B2: set_o,B: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
     => aa_set_o_o(ord_less_eq_set_o(A2),insert_o2((B),B2)) ) ).

% subset_insertI2
tff(fact_5095_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_nat2(B,B2)) ) ).

% subset_insertI2
tff(fact_5096_less__eq__mask,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ).

% less_eq_mask
tff(fact_5097_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_real,G: fun_real_real] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real2(I2),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I2)),aa_real_real(G,I2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,K5)),groups8097168146408367636l_real(G,K5)) ) ).

% sum_mono
tff(fact_5098_sum__mono,axiom,
    ! [K5: set_o,F: fun_o_real,G: fun_o_real] :
      ( ! [I2: $o] :
          ( aa_set_o_o(member_o2((I2)),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(F,(I2))),aa_o_real(G,(I2))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8691415230153176458o_real(F,K5)),groups8691415230153176458o_real(G,K5)) ) ).

% sum_mono
tff(fact_5099_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_real,G: fun_int_real] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int2(I2),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I2)),aa_int_real(G,I2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,K5)),groups8778361861064173332t_real(G,K5)) ) ).

% sum_mono
tff(fact_5100_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real2(I2),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,K5)),groups1935376822645274424al_nat(G,K5)) ) ).

% sum_mono
tff(fact_5101_sum__mono,axiom,
    ! [K5: set_o,F: fun_o_nat,G: fun_o_nat] :
      ( ! [I2: $o] :
          ( aa_set_o_o(member_o2((I2)),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(I2))),aa_o_nat(G,(I2))) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups8507830703676809646_o_nat(F,K5)),groups8507830703676809646_o_nat(G,K5)) ) ).

% sum_mono
tff(fact_5102_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int2(I2),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,K5)),groups4541462559716669496nt_nat(G,K5)) ) ).

% sum_mono
tff(fact_5103_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real2(I2),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I2)),aa_real_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,K5)),groups1932886352136224148al_int(G,K5)) ) ).

% sum_mono
tff(fact_5104_sum__mono,axiom,
    ! [K5: set_o,F: fun_o_int,G: fun_o_int] :
      ( ! [I2: $o] :
          ( aa_set_o_o(member_o2((I2)),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,(I2))),aa_o_int(G,(I2))) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups8505340233167759370_o_int(F,K5)),groups8505340233167759370_o_int(G,K5)) ) ).

% sum_mono
tff(fact_5105_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat2(I2),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I2)),aa_nat_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_nat_int(groups3539618377306564664at_int(F),K5)),aa_set_nat_int(groups3539618377306564664at_int(G),K5)) ) ).

% sum_mono
tff(fact_5106_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_int,G: fun_int_int] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int2(I2),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,I2)),aa_int_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4538972089207619220nt_int(F,K5)),groups4538972089207619220nt_int(G,K5)) ) ).

% sum_mono
tff(fact_5107_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_ey(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_5108_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_ez(G),H),A2) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,A2)),groups7754918857620584856omplex(H,A2)) ).

% sum.distrib
tff(fact_5109_sum_Odistrib,axiom,
    ! [G: fun_nat_real,H: fun_nat_real,A2: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_fa(G),H)),A2) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),A2)),aa_set_nat_real(groups6591440286371151544t_real(H),A2)) ).

% sum.distrib
tff(fact_5110_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_nat2(A,bot_bot_set_set_nat))) = $ite(aa_set_set_nat_o(member_set_nat2(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_5111_sum__diff1__nat,axiom,
    ! [F: fun_real_nat,A2: set_real,A: real] :
      groups1935376822645274424al_nat(F,minus_minus_set_real(A2,insert_real2(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real2(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_5112_sum__diff1__nat,axiom,
    ! [F: fun_o_nat,A2: set_o,A: $o] :
      groups8507830703676809646_o_nat(F,minus_minus_set_o(A2,insert_o2((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o2((A)),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8507830703676809646_o_nat(F,A2)),aa_o_nat(F,(A))),groups8507830703676809646_o_nat(F,A2)) ).

% sum_diff1_nat
tff(fact_5113_sum__diff1__nat,axiom,
    ! [F: fun_int_nat,A2: set_int,A: int] :
      groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,insert_int2(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int2(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_5114_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_nat2(A,bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat2(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_5115_Collect__conv__if,axiom,
    ! [A: list_nat,P: fun_list_nat_o] :
      collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_fb(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_5116_Collect__conv__if,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_fc(A),P)) = $ite(aa_set_nat_o(P,A),insert_set_nat2(A,bot_bot_set_set_nat),bot_bot_set_set_nat) ).

% Collect_conv_if
tff(fact_5117_Collect__conv__if,axiom,
    ! [A: real,P: fun_real_o] :
      collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_fd(A),P)) = $ite(aa_real_o(P,A),insert_real2(A,bot_bot_set_real),bot_bot_set_real) ).

% Collect_conv_if
tff(fact_5118_Collect__conv__if,axiom,
    ! [A: $o,P: fun_o_o] :
      collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_fe((A)),P)) = $ite(aa_o_o(P,(A)),insert_o2((A),bot_bot_set_o),bot_bot_set_o) ).

% Collect_conv_if
tff(fact_5119_Collect__conv__if,axiom,
    ! [A: nat,P: fun_nat_o] :
      collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ff(A),P)) = $ite(aa_nat_o(P,A),insert_nat2(A,bot_bot_set_nat),bot_bot_set_nat) ).

% Collect_conv_if
tff(fact_5120_Collect__conv__if,axiom,
    ! [A: int,P: fun_int_o] :
      collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_fg(A),P)) = $ite(aa_int_o(P,A),insert_int2(A,bot_bot_set_int),bot_bot_set_int) ).

% Collect_conv_if
tff(fact_5121_Collect__conv__if2,axiom,
    ! [A: list_nat,P: fun_list_nat_o] :
      collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_fh(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_5122_Collect__conv__if2,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_fi(A),P)) = $ite(aa_set_nat_o(P,A),insert_set_nat2(A,bot_bot_set_set_nat),bot_bot_set_set_nat) ).

% Collect_conv_if2
tff(fact_5123_Collect__conv__if2,axiom,
    ! [A: real,P: fun_real_o] :
      collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_fj(A),P)) = $ite(aa_real_o(P,A),insert_real2(A,bot_bot_set_real),bot_bot_set_real) ).

% Collect_conv_if2
tff(fact_5124_Collect__conv__if2,axiom,
    ! [A: $o,P: fun_o_o] :
      collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_fk((A)),P)) = $ite(aa_o_o(P,(A)),insert_o2((A),bot_bot_set_o),bot_bot_set_o) ).

% Collect_conv_if2
tff(fact_5125_Collect__conv__if2,axiom,
    ! [A: nat,P: fun_nat_o] :
      collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_fl(A),P)) = $ite(aa_nat_o(P,A),insert_nat2(A,bot_bot_set_nat),bot_bot_set_nat) ).

% Collect_conv_if2
tff(fact_5126_Collect__conv__if2,axiom,
    ! [A: int,P: fun_int_o] :
      collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_fm(A),P)) = $ite(aa_int_o(P,A),insert_int2(A,bot_bot_set_int),bot_bot_set_int) ).

% Collect_conv_if2
tff(fact_5127_sum_Oremove,axiom,
    ! [A2: set_complex,Xb: complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(Xb),A2)
       => ( groups5693394587270226106ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,Xb)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(Xb,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_5128_sum_Oremove,axiom,
    ! [A2: set_Extended_enat,Xb: extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Xb),A2)
       => ( groups2027974829824023292at_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Extended_enat_nat(G,Xb)),groups2027974829824023292at_nat(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(Xb,bot_bo7653980558646680370d_enat)))) ) ) ) ).

% sum.remove
tff(fact_5129_sum_Oremove,axiom,
    ! [A2: set_complex,Xb: complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(Xb),A2)
       => ( groups5690904116761175830ex_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,Xb)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,insert_complex(Xb,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_5130_sum_Oremove,axiom,
    ! [A2: set_Extended_enat,Xb: extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Xb),A2)
       => ( groups2025484359314973016at_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Extended_enat_int(G,Xb)),groups2025484359314973016at_int(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(Xb,bot_bo7653980558646680370d_enat)))) ) ) ) ).

% sum.remove
tff(fact_5131_sum_Oremove,axiom,
    ! [A2: set_complex,Xb: complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(Xb),A2)
       => ( groups5808333547571424918x_real(G,A2) = aa_real_real(plus_plus_real(aa_complex_real(G,Xb)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,insert_complex(Xb,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_5132_sum_Oremove,axiom,
    ! [A2: set_Extended_enat,Xb: extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Xb),A2)
       => ( groups4148127829035722712t_real(G,A2) = aa_real_real(plus_plus_real(aa_Ext2541572767299617110t_real(G,Xb)),groups4148127829035722712t_real(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(Xb,bot_bo7653980558646680370d_enat)))) ) ) ) ).

% sum.remove
tff(fact_5133_sum_Oremove,axiom,
    ! [A2: set_complex,Xb: complex,G: fun_co6496365337113880337d_enat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(Xb),A2)
       => ( groups1752964319039525884d_enat(G,A2) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_com3683005661447327866d_enat(G,Xb)),groups1752964319039525884d_enat(G,minus_811609699411566653omplex(A2,insert_complex(Xb,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_5134_sum_Oremove,axiom,
    ! [A2: set_Extended_enat,Xb: extended_enat,G: fun_Ex5852502975177633731d_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Xb),A2)
       => ( groups2433450451889696826d_enat(G,A2) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(G,Xb)),groups2433450451889696826d_enat(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(Xb,bot_bo7653980558646680370d_enat)))) ) ) ) ).

% sum.remove
tff(fact_5135_sum_Oremove,axiom,
    ! [A2: set_real,Xb: real,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real2(Xb),A2)
       => ( groups1935376822645274424al_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,Xb)),groups1935376822645274424al_nat(G,minus_minus_set_real(A2,insert_real2(Xb,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_5136_sum_Oremove,axiom,
    ! [A2: set_real,Xb: real,G: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real2(Xb),A2)
       => ( groups1932886352136224148al_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(G,Xb)),groups1932886352136224148al_int(G,minus_minus_set_real(A2,insert_real2(Xb,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_5137_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_nat,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5693394587270226106ex_nat(G,insert_complex(Xb,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,Xb)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(Xb,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_5138_sum_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,Xb: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( groups2027974829824023292at_nat(G,insert_Extended_enat(Xb,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Extended_enat_nat(G,Xb)),groups2027974829824023292at_nat(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(Xb,bot_bo7653980558646680370d_enat)))) ) ) ).

% sum.insert_remove
tff(fact_5139_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_int,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5690904116761175830ex_int(G,insert_complex(Xb,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,Xb)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,insert_complex(Xb,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_5140_sum_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex3482447378695272101at_int,Xb: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( groups2025484359314973016at_int(G,insert_Extended_enat(Xb,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Extended_enat_int(G,Xb)),groups2025484359314973016at_int(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(Xb,bot_bo7653980558646680370d_enat)))) ) ) ).

% sum.insert_remove
tff(fact_5141_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_real,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(G,insert_complex(Xb,A2)) = aa_real_real(plus_plus_real(aa_complex_real(G,Xb)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,insert_complex(Xb,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_5142_sum_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex1451620480498224165t_real,Xb: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( groups4148127829035722712t_real(G,insert_Extended_enat(Xb,A2)) = aa_real_real(plus_plus_real(aa_Ext2541572767299617110t_real(G,Xb)),groups4148127829035722712t_real(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(Xb,bot_bo7653980558646680370d_enat)))) ) ) ).

% sum.insert_remove
tff(fact_5143_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_co6496365337113880337d_enat,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups1752964319039525884d_enat(G,insert_complex(Xb,A2)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_com3683005661447327866d_enat(G,Xb)),groups1752964319039525884d_enat(G,minus_811609699411566653omplex(A2,insert_complex(Xb,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_5144_sum_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex5852502975177633731d_enat,Xb: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( groups2433450451889696826d_enat(G,insert_Extended_enat(Xb,A2)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(G,Xb)),groups2433450451889696826d_enat(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(Xb,bot_bo7653980558646680370d_enat)))) ) ) ).

% sum.insert_remove
tff(fact_5145_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_nat,Xb: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1935376822645274424al_nat(G,insert_real2(Xb,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,Xb)),groups1935376822645274424al_nat(G,minus_minus_set_real(A2,insert_real2(Xb,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_5146_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_int,Xb: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1932886352136224148al_int(G,insert_real2(Xb,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(G,Xb)),groups1932886352136224148al_int(G,minus_minus_set_real(A2,insert_real2(Xb,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_5147_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_5148_sum__diff1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex3482447378695272101at_int,A: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( groups2025484359314973016at_int(F,minus_925952699566721837d_enat(A2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(A),A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups2025484359314973016at_int(F,A2)),aa_Extended_enat_int(F,A)),groups2025484359314973016at_int(F,A2)) ) ) ).

% sum_diff1
tff(fact_5149_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_real2(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real2(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_5150_sum__diff1,axiom,
    ! [A2: set_o,F: fun_o_int,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups8505340233167759370_o_int(F,minus_minus_set_o(A2,insert_o2((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o2((A)),A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups8505340233167759370_o_int(F,A2)),aa_o_int(F,(A))),groups8505340233167759370_o_int(F,A2)) ) ) ).

% sum_diff1
tff(fact_5151_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_int2(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int2(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_5152_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(minus_minus_real(groups5808333547571424918x_real(F,A2)),aa_complex_real(F,A)),groups5808333547571424918x_real(F,A2)) ) ) ).

% sum_diff1
tff(fact_5153_sum__diff1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real,A: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( groups4148127829035722712t_real(F,minus_925952699566721837d_enat(A2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(A),A2),aa_real_real(minus_minus_real(groups4148127829035722712t_real(F,A2)),aa_Ext2541572767299617110t_real(F,A)),groups4148127829035722712t_real(F,A2)) ) ) ).

% sum_diff1
tff(fact_5154_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_real2(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real2(A),A2),aa_real_real(minus_minus_real(groups8097168146408367636l_real(F,A2)),aa_real_real(F,A)),groups8097168146408367636l_real(F,A2)) ) ) ).

% sum_diff1
tff(fact_5155_sum__diff1,axiom,
    ! [A2: set_o,F: fun_o_real,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups8691415230153176458o_real(F,minus_minus_set_o(A2,insert_o2((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o2((A)),A2),aa_real_real(minus_minus_real(groups8691415230153176458o_real(F,A2)),aa_o_real(F,(A))),groups8691415230153176458o_real(F,A2)) ) ) ).

% sum_diff1
tff(fact_5156_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_int2(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int2(A),A2),aa_real_real(minus_minus_real(groups8778361861064173332t_real(F,A2)),aa_int_real(F,A)),groups8778361861064173332t_real(F,A2)) ) ) ).

% sum_diff1
tff(fact_5157_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_re6065469302365115155d_enat] :
      ( ! [X4: real] :
          ( aa_set_real_o(member_real2(X4),A2)
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,X4)) )
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),groups2800946370649118462d_enat(F,A2)) ) ).

% sum_nonneg
tff(fact_5158_sum__nonneg,axiom,
    ! [A2: set_o,F: fun_o_Extended_enat] :
      ( ! [X4: $o] :
          ( aa_set_o_o(member_o2((X4)),A2)
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_o_Extended_enat(F,(X4))) )
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),groups7198740251461348360d_enat(F,A2)) ) ).

% sum_nonneg
tff(fact_5159_sum__nonneg,axiom,
    ! [A2: set_nat,F: fun_na8797796813480665199d_enat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat2(X4),A2)
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(F,X4)) )
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),groups7108830773950497114d_enat(F,A2)) ) ).

% sum_nonneg
tff(fact_5160_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_in5914218760682845459d_enat] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int2(X4),A2)
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(F,X4)) )
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),groups4225252721152677374d_enat(F,A2)) ) ).

% sum_nonneg
tff(fact_5161_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( aa_set_real_o(member_real2(X4),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A2)) ) ).

% sum_nonneg
tff(fact_5162_sum__nonneg,axiom,
    ! [A2: set_o,F: fun_o_real] :
      ( ! [X4: $o] :
          ( aa_set_o_o(member_o2((X4)),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(X4))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8691415230153176458o_real(F,A2)) ) ).

% sum_nonneg
tff(fact_5163_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int2(X4),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A2)) ) ).

% sum_nonneg
tff(fact_5164_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( aa_set_real_o(member_real2(X4),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A2)) ) ).

% sum_nonneg
tff(fact_5165_sum__nonneg,axiom,
    ! [A2: set_o,F: fun_o_nat] :
      ( ! [X4: $o] :
          ( aa_set_o_o(member_o2((X4)),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(F,(X4))) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups8507830703676809646_o_nat(F,A2)) ) ).

% sum_nonneg
tff(fact_5166_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int2(X4),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A2)) ) ).

% sum_nonneg
tff(fact_5167_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_re6065469302365115155d_enat] :
      ( ! [X4: real] :
          ( aa_set_real_o(member_real2(X4),A2)
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_rea1194391308913012860d_enat(F,X4)),zero_z5237406670263579293d_enat) )
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups2800946370649118462d_enat(F,A2)),zero_z5237406670263579293d_enat) ) ).

% sum_nonpos
tff(fact_5168_sum__nonpos,axiom,
    ! [A2: set_o,F: fun_o_Extended_enat] :
      ( ! [X4: $o] :
          ( aa_set_o_o(member_o2((X4)),A2)
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_o_Extended_enat(F,(X4))),zero_z5237406670263579293d_enat) )
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups7198740251461348360d_enat(F,A2)),zero_z5237406670263579293d_enat) ) ).

% sum_nonpos
tff(fact_5169_sum__nonpos,axiom,
    ! [A2: set_nat,F: fun_na8797796813480665199d_enat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat2(X4),A2)
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,X4)),zero_z5237406670263579293d_enat) )
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups7108830773950497114d_enat(F,A2)),zero_z5237406670263579293d_enat) ) ).

% sum_nonpos
tff(fact_5170_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_in5914218760682845459d_enat] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int2(X4),A2)
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_int_Extended_enat(F,X4)),zero_z5237406670263579293d_enat) )
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups4225252721152677374d_enat(F,A2)),zero_z5237406670263579293d_enat) ) ).

% sum_nonpos
tff(fact_5171_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( aa_set_real_o(member_real2(X4),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X4)),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_5172_sum__nonpos,axiom,
    ! [A2: set_o,F: fun_o_real] :
      ( ! [X4: $o] :
          ( aa_set_o_o(member_o2((X4)),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(F,(X4))),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8691415230153176458o_real(F,A2)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_5173_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int2(X4),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X4)),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_5174_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( aa_set_real_o(member_real2(X4),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X4)),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_5175_sum__nonpos,axiom,
    ! [A2: set_o,F: fun_o_nat] :
      ( ! [X4: $o] :
          ( aa_set_o_o(member_o2((X4)),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(X4))),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups8507830703676809646_o_nat(F,A2)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_5176_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int2(X4),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X4)),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_5177_not__bit__1__Suc,axiom,
    ! [Nb: nat] : ~ aa_nat_o(bit_se1146084159140164899it_int(one_one_int),aa_nat_nat(suc,Nb)) ).

% not_bit_1_Suc
tff(fact_5178_not__bit__1__Suc,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),aa_nat_nat(suc,Nb)) ).

% not_bit_1_Suc
tff(fact_5179_sum__mono__inv,axiom,
    ! [F: fun_real_real,I5: set_real,G: fun_real_real,Ia: real] :
      ( ( groups8097168146408367636l_real(F,I5) = groups8097168146408367636l_real(G,I5) )
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real2(I2),I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I2)),aa_real_real(G,I2)) )
       => ( aa_set_real_o(member_real2(Ia),I5)
         => ( aa_set_real_o(finite_finite_real,I5)
           => ( aa_real_real(F,Ia) = aa_real_real(G,Ia) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5180_sum__mono__inv,axiom,
    ! [F: fun_o_real,I5: set_o,G: fun_o_real,Ia: $o] :
      ( ( groups8691415230153176458o_real(F,I5) = groups8691415230153176458o_real(G,I5) )
     => ( ! [I2: $o] :
            ( aa_set_o_o(member_o2((I2)),I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(F,(I2))),aa_o_real(G,(I2))) )
       => ( aa_set_o_o(member_o2((Ia)),I5)
         => ( aa_set_o_o(finite_finite_o,I5)
           => ( aa_o_real(F,(Ia)) = aa_o_real(G,(Ia)) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5181_sum__mono__inv,axiom,
    ! [F: fun_complex_real,I5: set_complex,G: fun_complex_real,Ia: complex] :
      ( ( groups5808333547571424918x_real(F,I5) = groups5808333547571424918x_real(G,I5) )
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I2)),aa_complex_real(G,I2)) )
       => ( aa_set_complex_o(member_complex(Ia),I5)
         => ( aa_set_complex_o(finite3207457112153483333omplex,I5)
           => ( aa_complex_real(F,Ia) = aa_complex_real(G,Ia) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5182_sum__mono__inv,axiom,
    ! [F: fun_int_real,I5: set_int,G: fun_int_real,Ia: int] :
      ( ( groups8778361861064173332t_real(F,I5) = groups8778361861064173332t_real(G,I5) )
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int2(I2),I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I2)),aa_int_real(G,I2)) )
       => ( aa_set_int_o(member_int2(Ia),I5)
         => ( aa_set_int_o(finite_finite_int,I5)
           => ( aa_int_real(F,Ia) = aa_int_real(G,Ia) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5183_sum__mono__inv,axiom,
    ! [F: fun_Ex1451620480498224165t_real,I5: set_Extended_enat,G: fun_Ex1451620480498224165t_real,Ia: extended_enat] :
      ( ( groups4148127829035722712t_real(F,I5) = groups4148127829035722712t_real(G,I5) )
     => ( ! [I2: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(I2),I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,I2)),aa_Ext2541572767299617110t_real(G,I2)) )
       => ( aa_set3337841544853333262enat_o(member_Extended_enat(Ia),I5)
         => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,I5)
           => ( aa_Ext2541572767299617110t_real(F,Ia) = aa_Ext2541572767299617110t_real(G,Ia) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5184_sum__mono__inv,axiom,
    ! [F: fun_real_nat,I5: set_real,G: fun_real_nat,Ia: real] :
      ( ( groups1935376822645274424al_nat(F,I5) = groups1935376822645274424al_nat(G,I5) )
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real2(I2),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2)) )
       => ( aa_set_real_o(member_real2(Ia),I5)
         => ( aa_set_real_o(finite_finite_real,I5)
           => ( aa_real_nat(F,Ia) = aa_real_nat(G,Ia) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5185_sum__mono__inv,axiom,
    ! [F: fun_o_nat,I5: set_o,G: fun_o_nat,Ia: $o] :
      ( ( groups8507830703676809646_o_nat(F,I5) = groups8507830703676809646_o_nat(G,I5) )
     => ( ! [I2: $o] :
            ( aa_set_o_o(member_o2((I2)),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(I2))),aa_o_nat(G,(I2))) )
       => ( aa_set_o_o(member_o2((Ia)),I5)
         => ( aa_set_o_o(finite_finite_o,I5)
           => ( aa_o_nat(F,(Ia)) = aa_o_nat(G,(Ia)) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5186_sum__mono__inv,axiom,
    ! [F: fun_complex_nat,I5: set_complex,G: fun_complex_nat,Ia: complex] :
      ( ( groups5693394587270226106ex_nat(F,I5) = groups5693394587270226106ex_nat(G,I5) )
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2)) )
       => ( aa_set_complex_o(member_complex(Ia),I5)
         => ( aa_set_complex_o(finite3207457112153483333omplex,I5)
           => ( aa_complex_nat(F,Ia) = aa_complex_nat(G,Ia) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5187_sum__mono__inv,axiom,
    ! [F: fun_int_nat,I5: set_int,G: fun_int_nat,Ia: int] :
      ( ( groups4541462559716669496nt_nat(F,I5) = groups4541462559716669496nt_nat(G,I5) )
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int2(I2),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2)) )
       => ( aa_set_int_o(member_int2(Ia),I5)
         => ( aa_set_int_o(finite_finite_int,I5)
           => ( aa_int_nat(F,Ia) = aa_int_nat(G,Ia) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5188_sum__mono__inv,axiom,
    ! [F: fun_Ex7660298398204468809at_nat,I5: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,Ia: extended_enat] :
      ( ( groups2027974829824023292at_nat(F,I5) = groups2027974829824023292at_nat(G,I5) )
     => ( ! [I2: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(I2),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Extended_enat_nat(F,I2)),aa_Extended_enat_nat(G,I2)) )
       => ( aa_set3337841544853333262enat_o(member_Extended_enat(Ia),I5)
         => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,I5)
           => ( aa_Extended_enat_nat(F,Ia) = aa_Extended_enat_nat(G,Ia) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5189_bit__1__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(one_one_int),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_5190_bit__1__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_5191_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ aa_set_nat_o(member_nat2(zero_zero_nat),A2)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat2(aa_nat_nat(suc,X4)),A2)
           => ( aa_nat_nat(F,aa_nat_nat(suc,X4)) = aa_nat_nat(G,aa_nat_nat(suc,X4)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) ) ) ) ).

% sum_cong_Suc
tff(fact_5192_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ aa_set_nat_o(member_nat2(zero_zero_nat),A2)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat2(aa_nat_nat(suc,X4)),A2)
           => ( aa_nat_real(F,aa_nat_nat(suc,X4)) = aa_nat_real(G,aa_nat_nat(suc,X4)) ) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),A2) = aa_set_nat_real(groups6591440286371151544t_real(G),A2) ) ) ) ).

% sum_cong_Suc
tff(fact_5193_bit__take__bit__iff,axiom,
    ! [Mb: nat,A: nat,Nb: 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)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb) ) ) ).

% bit_take_bit_iff
tff(fact_5194_bit__take__bit__iff,axiom,
    ! [Mb: nat,A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se2923211474154528505it_int(Mb),A)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        & aa_nat_o(bit_se1146084159140164899it_int(A),Nb) ) ) ).

% bit_take_bit_iff
tff(fact_5195_ceiling__mono,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Y)),aa_real_int(archim7802044766580827645g_real,Xb)) ) ).

% ceiling_mono
tff(fact_5196_le__of__int__ceiling,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb))) ).

% le_of_int_ceiling
tff(fact_5197_ceiling__less__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_real_int(archim7802044766580827645g_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% ceiling_less_cancel
tff(fact_5198_bit__of__bool__iff,axiom,
    ! [B: $o,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_o_int(zero_n2684676970156552555ol_int,(B))),Nb)
    <=> ( (B)
        & ( Nb = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_5199_bit__of__bool__iff,axiom,
    ! [B: $o,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B))),Nb)
    <=> ( (B)
        & ( Nb = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_5200_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_fn(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_5201_sum_Odelta__remove,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: fun_Ex7660298398204468809at_nat,C: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
     => ( groups2027974829824023292at_nat(aa_fun1810565495048290044at_nat(aa_fun5812849712608808208at_nat(aTP_Lamm_fo(A),B),C),S3) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(A),S3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Extended_enat_nat(B,A)),groups2027974829824023292at_nat(C,minus_925952699566721837d_enat(S3,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))),groups2027974829824023292at_nat(C,minus_925952699566721837d_enat(S3,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))) ) ) ).

% sum.delta_remove
tff(fact_5202_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_fp(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_5203_sum_Odelta__remove,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: fun_Ex3482447378695272101at_int,C: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
     => ( groups2025484359314973016at_int(aa_fun1695976788104708604at_int(aa_fun8556764835550362732at_int(aTP_Lamm_fq(A),B),C),S3) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(A),S3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Extended_enat_int(B,A)),groups2025484359314973016at_int(C,minus_925952699566721837d_enat(S3,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))),groups2025484359314973016at_int(C,minus_925952699566721837d_enat(S3,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))) ) ) ).

% sum.delta_remove
tff(fact_5204_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_fr(A),B),C),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_real_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_5205_sum_Odelta__remove,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: fun_Ex1451620480498224165t_real,C: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
     => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aa_fun967516747568145260t_real(aTP_Lamm_fs(A),B),C),S3) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(A),S3),aa_real_real(plus_plus_real(aa_Ext2541572767299617110t_real(B,A)),groups4148127829035722712t_real(C,minus_925952699566721837d_enat(S3,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))),groups4148127829035722712t_real(C,minus_925952699566721837d_enat(S3,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))) ) ) ).

% sum.delta_remove
tff(fact_5206_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: fun_co6496365337113880337d_enat,C: fun_co6496365337113880337d_enat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups1752964319039525884d_enat(aa_fun4954905782667315452d_enat(aa_fun2568869125039355352d_enat(aTP_Lamm_ft(A),B),C),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_com3683005661447327866d_enat(B,A)),groups1752964319039525884d_enat(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))),groups1752964319039525884d_enat(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_5207_sum_Odelta__remove,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: fun_Ex5852502975177633731d_enat,C: fun_Ex5852502975177633731d_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
     => ( groups2433450451889696826d_enat(aa_fun7865826685190571204d_enat(aa_fun6128326792922203780d_enat(aTP_Lamm_fu(A),B),C),S3) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(A),S3),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(B,A)),groups2433450451889696826d_enat(C,minus_925952699566721837d_enat(S3,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))),groups2433450451889696826d_enat(C,minus_925952699566721837d_enat(S3,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))) ) ) ).

% sum.delta_remove
tff(fact_5208_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_fv(A),B),C),S3) = $ite(aa_set_real_o(member_real2(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_real2(A,bot_bot_set_real)))),groups1935376822645274424al_nat(C,minus_minus_set_real(S3,insert_real2(A,bot_bot_set_real)))) ) ) ).

% sum.delta_remove
tff(fact_5209_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_fw(A),B),C),S3) = $ite(aa_set_real_o(member_real2(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_real2(A,bot_bot_set_real)))),groups1932886352136224148al_int(C,minus_minus_set_real(S3,insert_real2(A,bot_bot_set_real)))) ) ) ).

% sum.delta_remove
tff(fact_5210_finite_Ocases,axiom,
    ! [A: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ( A != bot_bot_set_complex )
       => ~ ! [A6: set_complex] :
              ( ? [A3: complex] : A = insert_complex(A3,A6)
             => ~ aa_set_complex_o(finite3207457112153483333omplex,A6) ) ) ) ).

% finite.cases
tff(fact_5211_finite_Ocases,axiom,
    ! [A: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A)
     => ( ( A != bot_bo7653980558646680370d_enat )
       => ~ ! [A6: set_Extended_enat] :
              ( ? [A3: extended_enat] : A = insert_Extended_enat(A3,A6)
             => ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A6) ) ) ) ).

% finite.cases
tff(fact_5212_finite_Ocases,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( ( A != bot_bot_set_real )
       => ~ ! [A6: set_real] :
              ( ? [A3: real] : A = insert_real2(A3,A6)
             => ~ aa_set_real_o(finite_finite_real,A6) ) ) ) ).

% finite.cases
tff(fact_5213_finite_Ocases,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ~ ! [A6: set_o] :
              ( ? [A3: $o] : A = insert_o2((A3),A6)
             => ~ aa_set_o_o(finite_finite_o,A6) ) ) ) ).

% finite.cases
tff(fact_5214_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A6: set_nat] :
              ( ? [A3: nat] : A = insert_nat2(A3,A6)
             => ~ aa_set_nat_o(finite_finite_nat,A6) ) ) ) ).

% finite.cases
tff(fact_5215_finite_Ocases,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ~ ! [A6: set_int] :
              ( ? [A3: int] : A = insert_int2(A3,A6)
             => ~ aa_set_int_o(finite_finite_int,A6) ) ) ) ).

% finite.cases
tff(fact_5216_finite_Osimps,axiom,
    ! [A: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
    <=> ( ( A = bot_bot_set_complex )
        | ? [A7: set_complex,A5: complex] :
            ( ( A = insert_complex(A5,A7) )
            & aa_set_complex_o(finite3207457112153483333omplex,A7) ) ) ) ).

% finite.simps
tff(fact_5217_finite_Osimps,axiom,
    ! [A: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A)
    <=> ( ( A = bot_bo7653980558646680370d_enat )
        | ? [A7: set_Extended_enat,A5: extended_enat] :
            ( ( A = insert_Extended_enat(A5,A7) )
            & aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A7) ) ) ) ).

% finite.simps
tff(fact_5218_finite_Osimps,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(finite_finite_real,A)
    <=> ( ( A = bot_bot_set_real )
        | ? [A7: set_real,A5: real] :
            ( ( A = insert_real2(A5,A7) )
            & aa_set_real_o(finite_finite_real,A7) ) ) ) ).

% finite.simps
tff(fact_5219_finite_Osimps,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
    <=> ( ( A = bot_bot_set_o )
        | ? [A7: set_o,A5: $o] :
            ( ( A = insert_o2((A5),A7) )
            & aa_set_o_o(finite_finite_o,A7) ) ) ) ).

% finite.simps
tff(fact_5220_finite_Osimps,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
    <=> ( ( A = bot_bot_set_nat )
        | ? [A7: set_nat,A5: nat] :
            ( ( A = insert_nat2(A5,A7) )
            & aa_set_nat_o(finite_finite_nat,A7) ) ) ) ).

% finite.simps
tff(fact_5221_finite_Osimps,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
    <=> ( ( A = bot_bot_set_int )
        | ? [A7: set_int,A5: int] :
            ( ( A = insert_int2(A5,A7) )
            & aa_set_int_o(finite_finite_int,A7) ) ) ) ).

% finite.simps
tff(fact_5222_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)
       => ( ! [X4: set_nat,F4: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
             => ( ~ aa_set_set_nat_o(member_set_nat2(X4),F4)
               => ( aa_set_set_nat_o(P,F4)
                 => aa_set_set_nat_o(P,insert_set_nat2(X4,F4)) ) ) )
         => aa_set_set_nat_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_5223_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)
       => ( ! [X4: complex,F4: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,F4)
             => ( ~ aa_set_complex_o(member_complex(X4),F4)
               => ( aa_set_complex_o(P,F4)
                 => aa_set_complex_o(P,insert_complex(X4,F4)) ) ) )
         => aa_set_complex_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_5224_finite__induct,axiom,
    ! [F3: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F3)
     => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
       => ( ! [X4: extended_enat,F4: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F4)
             => ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(X4),F4)
               => ( aa_set3337841544853333262enat_o(P,F4)
                 => aa_set3337841544853333262enat_o(P,insert_Extended_enat(X4,F4)) ) ) )
         => aa_set3337841544853333262enat_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_5225_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)
       => ( ! [X4: real,F4: set_real] :
              ( aa_set_real_o(finite_finite_real,F4)
             => ( ~ aa_set_real_o(member_real2(X4),F4)
               => ( aa_set_real_o(P,F4)
                 => aa_set_real_o(P,insert_real2(X4,F4)) ) ) )
         => aa_set_real_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_5226_finite__induct,axiom,
    ! [F3: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F3)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X4: $o,F4: set_o] :
              ( aa_set_o_o(finite_finite_o,F4)
             => ( ~ aa_set_o_o(member_o2((X4)),F4)
               => ( aa_set_o_o(P,F4)
                 => aa_set_o_o(P,insert_o2((X4),F4)) ) ) )
         => aa_set_o_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_5227_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)
       => ( ! [X4: nat,F4: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,F4)
             => ( ~ aa_set_nat_o(member_nat2(X4),F4)
               => ( aa_set_nat_o(P,F4)
                 => aa_set_nat_o(P,insert_nat2(X4,F4)) ) ) )
         => aa_set_nat_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_5228_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)
       => ( ! [X4: int,F4: set_int] :
              ( aa_set_int_o(finite_finite_int,F4)
             => ( ~ aa_set_int_o(member_int2(X4),F4)
               => ( aa_set_int_o(P,F4)
                 => aa_set_int_o(P,insert_int2(X4,F4)) ) ) )
         => aa_set_int_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_5229_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 )
       => ( ! [X4: set_nat] : aa_set_set_nat_o(P,insert_set_nat2(X4,bot_bot_set_set_nat))
         => ( ! [X4: 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_nat2(X4),F4)
                   => ( aa_set_set_nat_o(P,F4)
                     => aa_set_set_nat_o(P,insert_set_nat2(X4,F4)) ) ) ) )
           => aa_set_set_nat_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_5230_finite__ne__induct,axiom,
    ! [F3: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F3)
     => ( ( F3 != bot_bot_set_complex )
       => ( ! [X4: complex] : aa_set_complex_o(P,insert_complex(X4,bot_bot_set_complex))
         => ( ! [X4: complex,F4: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F4)
               => ( ( F4 != bot_bot_set_complex )
                 => ( ~ aa_set_complex_o(member_complex(X4),F4)
                   => ( aa_set_complex_o(P,F4)
                     => aa_set_complex_o(P,insert_complex(X4,F4)) ) ) ) )
           => aa_set_complex_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_5231_finite__ne__induct,axiom,
    ! [F3: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F3)
     => ( ( F3 != bot_bo7653980558646680370d_enat )
       => ( ! [X4: extended_enat] : aa_set3337841544853333262enat_o(P,insert_Extended_enat(X4,bot_bo7653980558646680370d_enat))
         => ( ! [X4: extended_enat,F4: set_Extended_enat] :
                ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F4)
               => ( ( F4 != bot_bo7653980558646680370d_enat )
                 => ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(X4),F4)
                   => ( aa_set3337841544853333262enat_o(P,F4)
                     => aa_set3337841544853333262enat_o(P,insert_Extended_enat(X4,F4)) ) ) ) )
           => aa_set3337841544853333262enat_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_5232_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 )
       => ( ! [X4: real] : aa_set_real_o(P,insert_real2(X4,bot_bot_set_real))
         => ( ! [X4: real,F4: set_real] :
                ( aa_set_real_o(finite_finite_real,F4)
               => ( ( F4 != bot_bot_set_real )
                 => ( ~ aa_set_real_o(member_real2(X4),F4)
                   => ( aa_set_real_o(P,F4)
                     => aa_set_real_o(P,insert_real2(X4,F4)) ) ) ) )
           => aa_set_real_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_5233_finite__ne__induct,axiom,
    ! [F3: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F3)
     => ( ( F3 != bot_bot_set_o )
       => ( ! [X4: $o] : aa_set_o_o(P,insert_o2((X4),bot_bot_set_o))
         => ( ! [X4: $o,F4: set_o] :
                ( aa_set_o_o(finite_finite_o,F4)
               => ( ( F4 != bot_bot_set_o )
                 => ( ~ aa_set_o_o(member_o2((X4)),F4)
                   => ( aa_set_o_o(P,F4)
                     => aa_set_o_o(P,insert_o2((X4),F4)) ) ) ) )
           => aa_set_o_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_5234_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 )
       => ( ! [X4: nat] : aa_set_nat_o(P,insert_nat2(X4,bot_bot_set_nat))
         => ( ! [X4: nat,F4: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F4)
               => ( ( F4 != bot_bot_set_nat )
                 => ( ~ aa_set_nat_o(member_nat2(X4),F4)
                   => ( aa_set_nat_o(P,F4)
                     => aa_set_nat_o(P,insert_nat2(X4,F4)) ) ) ) )
           => aa_set_nat_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_5235_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 )
       => ( ! [X4: int] : aa_set_int_o(P,insert_int2(X4,bot_bot_set_int))
         => ( ! [X4: int,F4: set_int] :
                ( aa_set_int_o(finite_finite_int,F4)
               => ( ( F4 != bot_bot_set_int )
                 => ( ~ aa_set_int_o(member_int2(X4),F4)
                   => ( aa_set_int_o(P,F4)
                     => aa_set_int_o(P,insert_int2(X4,F4)) ) ) ) )
           => aa_set_int_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_5236_infinite__finite__induct,axiom,
    ! [P: fun_set_set_nat_o,A2: set_set_nat] :
      ( ! [A6: set_set_nat] :
          ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A6)
         => aa_set_set_nat_o(P,A6) )
     => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
       => ( ! [X4: set_nat,F4: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
             => ( ~ aa_set_set_nat_o(member_set_nat2(X4),F4)
               => ( aa_set_set_nat_o(P,F4)
                 => aa_set_set_nat_o(P,insert_set_nat2(X4,F4)) ) ) )
         => aa_set_set_nat_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_5237_infinite__finite__induct,axiom,
    ! [P: fun_set_complex_o,A2: set_complex] :
      ( ! [A6: set_complex] :
          ( ~ aa_set_complex_o(finite3207457112153483333omplex,A6)
         => aa_set_complex_o(P,A6) )
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X4: complex,F4: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,F4)
             => ( ~ aa_set_complex_o(member_complex(X4),F4)
               => ( aa_set_complex_o(P,F4)
                 => aa_set_complex_o(P,insert_complex(X4,F4)) ) ) )
         => aa_set_complex_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_5238_infinite__finite__induct,axiom,
    ! [P: fun_se8123728892507317397enat_o,A2: set_Extended_enat] :
      ( ! [A6: set_Extended_enat] :
          ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A6)
         => aa_set3337841544853333262enat_o(P,A6) )
     => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
       => ( ! [X4: extended_enat,F4: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F4)
             => ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(X4),F4)
               => ( aa_set3337841544853333262enat_o(P,F4)
                 => aa_set3337841544853333262enat_o(P,insert_Extended_enat(X4,F4)) ) ) )
         => aa_set3337841544853333262enat_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_5239_infinite__finite__induct,axiom,
    ! [P: fun_set_real_o,A2: set_real] :
      ( ! [A6: set_real] :
          ( ~ aa_set_real_o(finite_finite_real,A6)
         => aa_set_real_o(P,A6) )
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X4: real,F4: set_real] :
              ( aa_set_real_o(finite_finite_real,F4)
             => ( ~ aa_set_real_o(member_real2(X4),F4)
               => ( aa_set_real_o(P,F4)
                 => aa_set_real_o(P,insert_real2(X4,F4)) ) ) )
         => aa_set_real_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_5240_infinite__finite__induct,axiom,
    ! [P: fun_set_o_o,A2: set_o] :
      ( ! [A6: set_o] :
          ( ~ aa_set_o_o(finite_finite_o,A6)
         => aa_set_o_o(P,A6) )
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X4: $o,F4: set_o] :
              ( aa_set_o_o(finite_finite_o,F4)
             => ( ~ aa_set_o_o(member_o2((X4)),F4)
               => ( aa_set_o_o(P,F4)
                 => aa_set_o_o(P,insert_o2((X4),F4)) ) ) )
         => aa_set_o_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_5241_infinite__finite__induct,axiom,
    ! [P: fun_set_nat_o,A2: set_nat] :
      ( ! [A6: set_nat] :
          ( ~ aa_set_nat_o(finite_finite_nat,A6)
         => aa_set_nat_o(P,A6) )
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X4: nat,F4: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,F4)
             => ( ~ aa_set_nat_o(member_nat2(X4),F4)
               => ( aa_set_nat_o(P,F4)
                 => aa_set_nat_o(P,insert_nat2(X4,F4)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_5242_infinite__finite__induct,axiom,
    ! [P: fun_set_int_o,A2: set_int] :
      ( ! [A6: set_int] :
          ( ~ aa_set_int_o(finite_finite_int,A6)
         => aa_set_int_o(P,A6) )
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X4: int,F4: set_int] :
              ( aa_set_int_o(finite_finite_int,F4)
             => ( ~ aa_set_int_o(member_int2(X4),F4)
               => ( aa_set_int_o(P,F4)
                 => aa_set_int_o(P,insert_int2(X4,F4)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_5243_subset__singleton__iff,axiom,
    ! [X5: set_real,A: real] :
      ( aa_set_real_o(ord_less_eq_set_real(X5),insert_real2(A,bot_bot_set_real))
    <=> ( ( X5 = bot_bot_set_real )
        | ( X5 = insert_real2(A,bot_bot_set_real) ) ) ) ).

% subset_singleton_iff
tff(fact_5244_subset__singleton__iff,axiom,
    ! [X5: set_o,A: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(X5),insert_o2((A),bot_bot_set_o))
    <=> ( ( X5 = bot_bot_set_o )
        | ( X5 = insert_o2((A),bot_bot_set_o) ) ) ) ).

% subset_singleton_iff
tff(fact_5245_subset__singleton__iff,axiom,
    ! [X5: set_int,A: int] :
      ( aa_set_int_o(ord_less_eq_set_int(X5),insert_int2(A,bot_bot_set_int))
    <=> ( ( X5 = bot_bot_set_int )
        | ( X5 = insert_int2(A,bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_5246_subset__singleton__iff,axiom,
    ! [X5: set_nat,A: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X5),insert_nat2(A,bot_bot_set_nat))
    <=> ( ( X5 = bot_bot_set_nat )
        | ( X5 = insert_nat2(A,bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_5247_subset__singletonD,axiom,
    ! [A2: set_real,Xb: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),insert_real2(Xb,bot_bot_set_real))
     => ( ( A2 = bot_bot_set_real )
        | ( A2 = insert_real2(Xb,bot_bot_set_real) ) ) ) ).

% subset_singletonD
tff(fact_5248_subset__singletonD,axiom,
    ! [A2: set_o,Xb: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),insert_o2((Xb),bot_bot_set_o))
     => ( ( A2 = bot_bot_set_o )
        | ( A2 = insert_o2((Xb),bot_bot_set_o) ) ) ) ).

% subset_singletonD
tff(fact_5249_subset__singletonD,axiom,
    ! [A2: set_int,Xb: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),insert_int2(Xb,bot_bot_set_int))
     => ( ( A2 = bot_bot_set_int )
        | ( A2 = insert_int2(Xb,bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_5250_subset__singletonD,axiom,
    ! [A2: set_nat,Xb: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat2(Xb,bot_bot_set_nat))
     => ( ( A2 = bot_bot_set_nat )
        | ( A2 = insert_nat2(Xb,bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_5251_atLeastAtMost__singleton_H,axiom,
    ! [A: $o,B: $o] :
      ( ( (A)
      <=> (B) )
     => ( set_or8904488021354931149Most_o((A),(B)) = insert_o2((A),bot_bot_set_o) ) ) ).

% atLeastAtMost_singleton'
tff(fact_5252_atLeastAtMost__singleton_H,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B) = insert_nat2(A,bot_bot_set_nat) ) ) ).

% atLeastAtMost_singleton'
tff(fact_5253_atLeastAtMost__singleton_H,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( set_or1266510415728281911st_int(A,B) = insert_int2(A,bot_bot_set_int) ) ) ).

% atLeastAtMost_singleton'
tff(fact_5254_atLeastAtMost__singleton_H,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( set_or1222579329274155063t_real(A,B) = insert_real2(A,bot_bot_set_real) ) ) ).

% atLeastAtMost_singleton'
tff(fact_5255_Diff__insert__absorb,axiom,
    ! [Xb: set_nat,A2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat2(Xb),A2)
     => ( minus_2163939370556025621et_nat(insert_set_nat2(Xb,A2),insert_set_nat2(Xb,bot_bot_set_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5256_Diff__insert__absorb,axiom,
    ! [Xb: real,A2: set_real] :
      ( ~ aa_set_real_o(member_real2(Xb),A2)
     => ( minus_minus_set_real(insert_real2(Xb,A2),insert_real2(Xb,bot_bot_set_real)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5257_Diff__insert__absorb,axiom,
    ! [Xb: $o,A2: set_o] :
      ( ~ aa_set_o_o(member_o2((Xb)),A2)
     => ( minus_minus_set_o(insert_o2((Xb),A2),insert_o2((Xb),bot_bot_set_o)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5258_Diff__insert__absorb,axiom,
    ! [Xb: int,A2: set_int] :
      ( ~ aa_set_int_o(member_int2(Xb),A2)
     => ( minus_minus_set_int(insert_int2(Xb,A2),insert_int2(Xb,bot_bot_set_int)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5259_Diff__insert__absorb,axiom,
    ! [Xb: nat,A2: set_nat] :
      ( ~ aa_set_nat_o(member_nat2(Xb),A2)
     => ( minus_minus_set_nat(insert_nat2(Xb,A2),insert_nat2(Xb,bot_bot_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_5260_Diff__insert2,axiom,
    ! [A2: set_real,A: real,B2: set_real] : minus_minus_set_real(A2,insert_real2(A,B2)) = minus_minus_set_real(minus_minus_set_real(A2,insert_real2(A,bot_bot_set_real)),B2) ).

% Diff_insert2
tff(fact_5261_Diff__insert2,axiom,
    ! [A2: set_o,A: $o,B2: set_o] : minus_minus_set_o(A2,insert_o2((A),B2)) = minus_minus_set_o(minus_minus_set_o(A2,insert_o2((A),bot_bot_set_o)),B2) ).

% Diff_insert2
tff(fact_5262_Diff__insert2,axiom,
    ! [A2: set_int,A: int,B2: set_int] : minus_minus_set_int(A2,insert_int2(A,B2)) = minus_minus_set_int(minus_minus_set_int(A2,insert_int2(A,bot_bot_set_int)),B2) ).

% Diff_insert2
tff(fact_5263_Diff__insert2,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] : minus_minus_set_nat(A2,insert_nat2(A,B2)) = minus_minus_set_nat(minus_minus_set_nat(A2,insert_nat2(A,bot_bot_set_nat)),B2) ).

% Diff_insert2
tff(fact_5264_insert__Diff,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat2(A),A2)
     => ( insert_set_nat2(A,minus_2163939370556025621et_nat(A2,insert_set_nat2(A,bot_bot_set_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_5265_insert__Diff,axiom,
    ! [A: real,A2: set_real] :
      ( aa_set_real_o(member_real2(A),A2)
     => ( insert_real2(A,minus_minus_set_real(A2,insert_real2(A,bot_bot_set_real))) = A2 ) ) ).

% insert_Diff
tff(fact_5266_insert__Diff,axiom,
    ! [A: $o,A2: set_o] :
      ( aa_set_o_o(member_o2((A)),A2)
     => ( insert_o2((A),minus_minus_set_o(A2,insert_o2((A),bot_bot_set_o))) = A2 ) ) ).

% insert_Diff
tff(fact_5267_insert__Diff,axiom,
    ! [A: int,A2: set_int] :
      ( aa_set_int_o(member_int2(A),A2)
     => ( insert_int2(A,minus_minus_set_int(A2,insert_int2(A,bot_bot_set_int))) = A2 ) ) ).

% insert_Diff
tff(fact_5268_insert__Diff,axiom,
    ! [A: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat2(A),A2)
     => ( insert_nat2(A,minus_minus_set_nat(A2,insert_nat2(A,bot_bot_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_5269_Diff__insert,axiom,
    ! [A2: set_real,A: real,B2: set_real] : minus_minus_set_real(A2,insert_real2(A,B2)) = minus_minus_set_real(minus_minus_set_real(A2,B2),insert_real2(A,bot_bot_set_real)) ).

% Diff_insert
tff(fact_5270_Diff__insert,axiom,
    ! [A2: set_o,A: $o,B2: set_o] : minus_minus_set_o(A2,insert_o2((A),B2)) = minus_minus_set_o(minus_minus_set_o(A2,B2),insert_o2((A),bot_bot_set_o)) ).

% Diff_insert
tff(fact_5271_Diff__insert,axiom,
    ! [A2: set_int,A: int,B2: set_int] : minus_minus_set_int(A2,insert_int2(A,B2)) = minus_minus_set_int(minus_minus_set_int(A2,B2),insert_int2(A,bot_bot_set_int)) ).

% Diff_insert
tff(fact_5272_Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] : minus_minus_set_nat(A2,insert_nat2(A,B2)) = minus_minus_set_nat(minus_minus_set_nat(A2,B2),insert_nat2(A,bot_bot_set_nat)) ).

% Diff_insert
tff(fact_5273_subset__Diff__insert,axiom,
    ! [A2: set_real,B2: set_real,Xb: real,C4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),minus_minus_set_real(B2,insert_real2(Xb,C4)))
    <=> ( aa_set_real_o(ord_less_eq_set_real(A2),minus_minus_set_real(B2,C4))
        & ~ aa_set_real_o(member_real2(Xb),A2) ) ) ).

% subset_Diff_insert
tff(fact_5274_subset__Diff__insert,axiom,
    ! [A2: set_o,B2: set_o,Xb: $o,C4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),minus_minus_set_o(B2,insert_o2((Xb),C4)))
    <=> ( aa_set_o_o(ord_less_eq_set_o(A2),minus_minus_set_o(B2,C4))
        & ~ aa_set_o_o(member_o2((Xb)),A2) ) ) ).

% subset_Diff_insert
tff(fact_5275_subset__Diff__insert,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,Xb: set_nat,C4: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),minus_2163939370556025621et_nat(B2,insert_set_nat2(Xb,C4)))
    <=> ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),minus_2163939370556025621et_nat(B2,C4))
        & ~ aa_set_set_nat_o(member_set_nat2(Xb),A2) ) ) ).

% subset_Diff_insert
tff(fact_5276_subset__Diff__insert,axiom,
    ! [A2: set_int,B2: set_int,Xb: int,C4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),minus_minus_set_int(B2,insert_int2(Xb,C4)))
    <=> ( aa_set_int_o(ord_less_eq_set_int(A2),minus_minus_set_int(B2,C4))
        & ~ aa_set_int_o(member_int2(Xb),A2) ) ) ).

% subset_Diff_insert
tff(fact_5277_subset__Diff__insert,axiom,
    ! [A2: set_nat,B2: set_nat,Xb: nat,C4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),minus_minus_set_nat(B2,insert_nat2(Xb,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_nat2(Xb),A2) ) ) ).

% subset_Diff_insert
tff(fact_5278_sum_Ointer__filter,axiom,
    ! [A2: set_o,G: fun_o_nat,P: fun_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups8507830703676809646_o_nat(G,collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_al(A2),P))) = groups8507830703676809646_o_nat(aa_fun_o_o_fun_o_nat(aTP_Lamm_fx(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5279_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_nat,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1935376822645274424al_nat(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_am(A2),P))) = groups1935376822645274424al_nat(aa_fun5513133441921015042al_nat(aTP_Lamm_fy(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5280_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_nat,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5693394587270226106ex_nat(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_fz(A2),P))) = groups5693394587270226106ex_nat(aa_fun7518244829075205762ex_nat(aTP_Lamm_ga(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5281_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_nat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ap(A2),P))) = groups4541462559716669496nt_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_gb(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5282_sum_Ointer__filter,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,P: fun_Extended_enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( groups2027974829824023292at_nat(G,collec4429806609662206161d_enat(aa_fun3141604562678283816enat_o(aTP_Lamm_gc(A2),P))) = groups2027974829824023292at_nat(aa_fun1905096444444700630at_nat(aTP_Lamm_gd(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5283_sum_Ointer__filter,axiom,
    ! [A2: set_o,G: fun_o_real,P: fun_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups8691415230153176458o_real(G,collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_al(A2),P))) = groups8691415230153176458o_real(aa_fun1851579542500809110o_real(aTP_Lamm_ge(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5284_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_am(A2),P))) = groups8097168146408367636l_real(aa_fun4448911596306054750l_real(aTP_Lamm_gf(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5285_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_fz(A2),P))) = groups5808333547571424918x_real(aa_fun8261057609414086622x_real(aTP_Lamm_gg(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5286_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_ap(A2),P))) = groups8778361861064173332t_real(aa_fun7506669883404309598t_real(aTP_Lamm_gh(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5287_sum_Ointer__filter,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex1451620480498224165t_real,P: fun_Extended_enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( groups4148127829035722712t_real(G,collec4429806609662206161d_enat(aa_fun3141604562678283816enat_o(aTP_Lamm_gc(A2),P))) = groups4148127829035722712t_real(aa_fun6735295395714377522t_real(aTP_Lamm_gi(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5288_member__le__sum,axiom,
    ! [Ia: complex,A2: set_complex,F: fun_co6496365337113880337d_enat] :
      ( aa_set_complex_o(member_complex(Ia),A2)
     => ( ! [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(A2,insert_complex(Ia,bot_bot_set_complex)))
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(F,X4)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_com3683005661447327866d_enat(F,Ia)),groups1752964319039525884d_enat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_5289_member__le__sum,axiom,
    ! [Ia: extended_enat,A2: set_Extended_enat,F: fun_Ex5852502975177633731d_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(Ia),A2)
     => ( ! [X4: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(A2,insert_Extended_enat(Ia,bot_bo7653980558646680370d_enat)))
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(F,X4)) )
       => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(F,Ia)),groups2433450451889696826d_enat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_5290_member__le__sum,axiom,
    ! [Ia: real,A2: set_real,F: fun_re6065469302365115155d_enat] :
      ( aa_set_real_o(member_real2(Ia),A2)
     => ( ! [X4: real] :
            ( aa_set_real_o(member_real2(X4),minus_minus_set_real(A2,insert_real2(Ia,bot_bot_set_real)))
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,X4)) )
       => ( aa_set_real_o(finite_finite_real,A2)
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_rea1194391308913012860d_enat(F,Ia)),groups2800946370649118462d_enat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_5291_member__le__sum,axiom,
    ! [Ia: $o,A2: set_o,F: fun_o_Extended_enat] :
      ( aa_set_o_o(member_o2((Ia)),A2)
     => ( ! [X4: $o] :
            ( aa_set_o_o(member_o2((X4)),minus_minus_set_o(A2,insert_o2((Ia),bot_bot_set_o)))
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_o_Extended_enat(F,(X4))) )
       => ( aa_set_o_o(finite_finite_o,A2)
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_o_Extended_enat(F,(Ia))),groups7198740251461348360d_enat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_5292_member__le__sum,axiom,
    ! [Ia: int,A2: set_int,F: fun_in5914218760682845459d_enat] :
      ( aa_set_int_o(member_int2(Ia),A2)
     => ( ! [X4: int] :
            ( aa_set_int_o(member_int2(X4),minus_minus_set_int(A2,insert_int2(Ia,bot_bot_set_int)))
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(F,X4)) )
       => ( aa_set_int_o(finite_finite_int,A2)
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_int_Extended_enat(F,Ia)),groups4225252721152677374d_enat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_5293_member__le__sum,axiom,
    ! [Ia: nat,A2: set_nat,F: fun_na8797796813480665199d_enat] :
      ( aa_set_nat_o(member_nat2(Ia),A2)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),minus_minus_set_nat(A2,insert_nat2(Ia,bot_bot_set_nat)))
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(F,X4)) )
       => ( aa_set_nat_o(finite_finite_nat,A2)
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,Ia)),groups7108830773950497114d_enat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_5294_member__le__sum,axiom,
    ! [Ia: complex,A2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(member_complex(Ia),A2)
     => ( ! [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(A2,insert_complex(Ia,bot_bot_set_complex)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,Ia)),groups5808333547571424918x_real(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_5295_member__le__sum,axiom,
    ! [Ia: extended_enat,A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(Ia),A2)
     => ( ! [X4: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(A2,insert_Extended_enat(Ia,bot_bo7653980558646680370d_enat)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,X4)) )
       => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,Ia)),groups4148127829035722712t_real(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_5296_member__le__sum,axiom,
    ! [Ia: real,A2: set_real,F: fun_real_real] :
      ( aa_set_real_o(member_real2(Ia),A2)
     => ( ! [X4: real] :
            ( aa_set_real_o(member_real2(X4),minus_minus_set_real(A2,insert_real2(Ia,bot_bot_set_real)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4)) )
       => ( aa_set_real_o(finite_finite_real,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Ia)),groups8097168146408367636l_real(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_5297_member__le__sum,axiom,
    ! [Ia: $o,A2: set_o,F: fun_o_real] :
      ( aa_set_o_o(member_o2((Ia)),A2)
     => ( ! [X4: $o] :
            ( aa_set_o_o(member_o2((X4)),minus_minus_set_o(A2,insert_o2((Ia),bot_bot_set_o)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(X4))) )
       => ( aa_set_o_o(finite_finite_o,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(F,(Ia))),groups8691415230153176458o_real(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_5298_sum__subtractf__nat,axiom,
    ! [A2: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X4: real] :
          ( aa_set_real_o(member_real2(X4),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(G,X4)),aa_real_nat(F,X4)) )
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_gj(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_5299_sum__subtractf__nat,axiom,
    ! [A2: set_o,G: fun_o_nat,F: fun_o_nat] :
      ( ! [X4: $o] :
          ( aa_set_o_o(member_o2((X4)),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(G,(X4))),aa_o_nat(F,(X4))) )
     => ( groups8507830703676809646_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_gk(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8507830703676809646_o_nat(F,A2)),groups8507830703676809646_o_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_5300_sum__subtractf__nat,axiom,
    ! [A2: set_set_nat,G: fun_set_nat_nat,F: fun_set_nat_nat] :
      ( ! [X4: set_nat] :
          ( aa_set_set_nat_o(member_set_nat2(X4),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(G,X4)),aa_set_nat_nat(F,X4)) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_gl(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_5301_sum__subtractf__nat,axiom,
    ! [A2: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int2(X4),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(G,X4)),aa_int_nat(F,X4)) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_gm(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_5302_sum__subtractf__nat,axiom,
    ! [A2: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat2(X4),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(G,X4)),aa_nat_nat(F,X4)) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_gn(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_5303_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_go(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_5304_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gp(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_5305_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,K: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(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,Nb),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gq(G),K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_5306_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,Mb: nat,K: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(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,Nb),K))) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_gr(G),K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_5307_sum__le__included,axiom,
    ! [S: set_nat,Ta: set_nat,G: fun_na8797796813480665199d_enat,Ia: fun_nat_nat,F: fun_na8797796813480665199d_enat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat2(X4),Ta)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(G,X4)) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(member_nat2(X4),S)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat2(Xa),Ta)
                    & ( aa_nat_nat(Ia,Xa) = X4 )
                    & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,X4)),aa_nat_Extended_enat(G,Xa)) ) )
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups7108830773950497114d_enat(F,S)),groups7108830773950497114d_enat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_5308_sum__le__included,axiom,
    ! [S: set_nat,Ta: set_complex,G: fun_co6496365337113880337d_enat,Ia: fun_complex_nat,F: fun_na8797796813480665199d_enat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),Ta)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(G,X4)) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(member_nat2(X4),S)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_nat(Ia,Xa) = X4 )
                    & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,X4)),aa_com3683005661447327866d_enat(G,Xa)) ) )
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups7108830773950497114d_enat(F,S)),groups1752964319039525884d_enat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_5309_sum__le__included,axiom,
    ! [S: set_nat,Ta: set_int,G: fun_in5914218760682845459d_enat,Ia: fun_int_nat,F: fun_na8797796813480665199d_enat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),Ta)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(G,X4)) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(member_nat2(X4),S)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int2(Xa),Ta)
                    & ( aa_int_nat(Ia,Xa) = X4 )
                    & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,X4)),aa_int_Extended_enat(G,Xa)) ) )
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups7108830773950497114d_enat(F,S)),groups4225252721152677374d_enat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_5310_sum__le__included,axiom,
    ! [S: set_nat,Ta: set_Extended_enat,G: fun_Ex5852502975177633731d_enat,Ia: fun_Ex7660298398204468809at_nat,F: fun_na8797796813480665199d_enat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,Ta)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),Ta)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(G,X4)) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(member_nat2(X4),S)
               => ? [Xa: extended_enat] :
                    ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),Ta)
                    & ( aa_Extended_enat_nat(Ia,Xa) = X4 )
                    & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,X4)),aa_Ext7969882492699152700d_enat(G,Xa)) ) )
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups7108830773950497114d_enat(F,S)),groups2433450451889696826d_enat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_5311_sum__le__included,axiom,
    ! [S: set_complex,Ta: set_nat,G: fun_na8797796813480665199d_enat,Ia: fun_nat_complex,F: fun_co6496365337113880337d_enat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat2(X4),Ta)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(G,X4)) )
         => ( ! [X4: complex] :
                ( aa_set_complex_o(member_complex(X4),S)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat2(Xa),Ta)
                    & ( aa_nat_complex(Ia,Xa) = X4 )
                    & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_com3683005661447327866d_enat(F,X4)),aa_nat_Extended_enat(G,Xa)) ) )
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups1752964319039525884d_enat(F,S)),groups7108830773950497114d_enat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_5312_sum__le__included,axiom,
    ! [S: set_complex,Ta: set_complex,G: fun_co6496365337113880337d_enat,Ia: fun_complex_complex,F: fun_co6496365337113880337d_enat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),Ta)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(G,X4)) )
         => ( ! [X4: complex] :
                ( aa_set_complex_o(member_complex(X4),S)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_complex(Ia,Xa) = X4 )
                    & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_com3683005661447327866d_enat(F,X4)),aa_com3683005661447327866d_enat(G,Xa)) ) )
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups1752964319039525884d_enat(F,S)),groups1752964319039525884d_enat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_5313_sum__le__included,axiom,
    ! [S: set_complex,Ta: set_int,G: fun_in5914218760682845459d_enat,Ia: fun_int_complex,F: fun_co6496365337113880337d_enat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),Ta)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(G,X4)) )
         => ( ! [X4: complex] :
                ( aa_set_complex_o(member_complex(X4),S)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int2(Xa),Ta)
                    & ( aa_int_complex(Ia,Xa) = X4 )
                    & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_com3683005661447327866d_enat(F,X4)),aa_int_Extended_enat(G,Xa)) ) )
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups1752964319039525884d_enat(F,S)),groups4225252721152677374d_enat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_5314_sum__le__included,axiom,
    ! [S: set_complex,Ta: set_Extended_enat,G: fun_Ex5852502975177633731d_enat,Ia: fun_Ex3741802288643809319omplex,F: fun_co6496365337113880337d_enat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,Ta)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),Ta)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(G,X4)) )
         => ( ! [X4: complex] :
                ( aa_set_complex_o(member_complex(X4),S)
               => ? [Xa: extended_enat] :
                    ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa),Ta)
                    & ( aa_Ext8748583412541189208omplex(Ia,Xa) = X4 )
                    & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_com3683005661447327866d_enat(F,X4)),aa_Ext7969882492699152700d_enat(G,Xa)) ) )
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups1752964319039525884d_enat(F,S)),groups2433450451889696826d_enat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_5315_sum__le__included,axiom,
    ! [S: set_int,Ta: set_nat,G: fun_na8797796813480665199d_enat,Ia: fun_nat_int,F: fun_in5914218760682845459d_enat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat2(X4),Ta)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(G,X4)) )
         => ( ! [X4: int] :
                ( aa_set_int_o(member_int2(X4),S)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat2(Xa),Ta)
                    & ( aa_nat_int(Ia,Xa) = X4 )
                    & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_int_Extended_enat(F,X4)),aa_nat_Extended_enat(G,Xa)) ) )
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups4225252721152677374d_enat(F,S)),groups7108830773950497114d_enat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_5316_sum__le__included,axiom,
    ! [S: set_int,Ta: set_complex,G: fun_co6496365337113880337d_enat,Ia: fun_complex_int,F: fun_in5914218760682845459d_enat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),Ta)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(G,X4)) )
         => ( ! [X4: int] :
                ( aa_set_int_o(member_int2(X4),S)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_int(Ia,Xa) = X4 )
                    & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_int_Extended_enat(F,X4)),aa_com3683005661447327866d_enat(G,Xa)) ) )
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups4225252721152677374d_enat(F,S)),groups1752964319039525884d_enat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_5317_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_re6065469302365115155d_enat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ! [X4: real] :
            ( aa_set_real_o(member_real2(X4),A2)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,X4)) )
       => ( ( groups2800946370649118462d_enat(F,A2) = zero_z5237406670263579293d_enat )
        <=> ! [X3: real] :
              ( aa_set_real_o(member_real2(X3),A2)
             => ( aa_rea1194391308913012860d_enat(F,X3) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5318_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_o,F: fun_o_Extended_enat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ! [X4: $o] :
            ( aa_set_o_o(member_o2((X4)),A2)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_o_Extended_enat(F,(X4))) )
       => ( ( groups7198740251461348360d_enat(F,A2) = zero_z5237406670263579293d_enat )
        <=> ! [X3: $o] :
              ( aa_set_o_o(member_o2((X3)),A2)
             => ( aa_o_Extended_enat(F,(X3)) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5319_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: fun_na8797796813480665199d_enat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),A2)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(F,X4)) )
       => ( ( groups7108830773950497114d_enat(F,A2) = zero_z5237406670263579293d_enat )
        <=> ! [X3: nat] :
              ( aa_set_nat_o(member_nat2(X3),A2)
             => ( aa_nat_Extended_enat(F,X3) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5320_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_co6496365337113880337d_enat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A2)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(F,X4)) )
       => ( ( groups1752964319039525884d_enat(F,A2) = zero_z5237406670263579293d_enat )
        <=> ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A2)
             => ( aa_com3683005661447327866d_enat(F,X3) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5321_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_in5914218760682845459d_enat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X4: int] :
            ( aa_set_int_o(member_int2(X4),A2)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(F,X4)) )
       => ( ( groups4225252721152677374d_enat(F,A2) = zero_z5237406670263579293d_enat )
        <=> ! [X3: int] :
              ( aa_set_int_o(member_int2(X3),A2)
             => ( aa_int_Extended_enat(F,X3) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5322_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex5852502975177633731d_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ! [X4: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),A2)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(F,X4)) )
       => ( ( groups2433450451889696826d_enat(F,A2) = zero_z5237406670263579293d_enat )
        <=> ! [X3: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X3),A2)
             => ( aa_Ext7969882492699152700d_enat(F,X3) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5323_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ! [X4: real] :
            ( aa_set_real_o(member_real2(X4),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4)) )
       => ( ( groups8097168146408367636l_real(F,A2) = zero_zero_real )
        <=> ! [X3: real] :
              ( aa_set_real_o(member_real2(X3),A2)
             => ( aa_real_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5324_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ! [X4: $o] :
            ( aa_set_o_o(member_o2((X4)),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(X4))) )
       => ( ( groups8691415230153176458o_real(F,A2) = zero_zero_real )
        <=> ! [X3: $o] :
              ( aa_set_o_o(member_o2((X3)),A2)
             => ( aa_o_real(F,(X3)) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5325_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4)) )
       => ( ( groups5808333547571424918x_real(F,A2) = zero_zero_real )
        <=> ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A2)
             => ( aa_complex_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5326_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X4: int] :
            ( aa_set_int_o(member_int2(X4),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4)) )
       => ( ( groups8778361861064173332t_real(F,A2) = zero_zero_real )
        <=> ! [X3: int] :
              ( aa_set_int_o(member_int2(X3),A2)
             => ( aa_int_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5327_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X4)),aa_complex_real(G,X4)) )
       => ( ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),A2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X)),aa_complex_real(G,X)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5328_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X4: int] :
            ( aa_set_int_o(member_int2(X4),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(G,X4)) )
       => ( ? [X: int] :
              ( aa_set_int_o(member_int2(X),A2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X)),aa_int_real(G,X)) )
         => 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_5329_sum__strict__mono__ex1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ! [X4: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(G,X4)) )
       => ( ? [X: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X),A2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Ext2541572767299617110t_real(F,X)),aa_Ext2541572767299617110t_real(G,X)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups4148127829035722712t_real(F,A2)),groups4148127829035722712t_real(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5330_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X4)),aa_complex_nat(G,X4)) )
       => ( ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),A2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,X)),aa_complex_nat(G,X)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5331_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X4: int] :
            ( aa_set_int_o(member_int2(X4),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X4)),aa_int_nat(G,X4)) )
       => ( ? [X: int] :
              ( aa_set_int_o(member_int2(X),A2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X)),aa_int_nat(G,X)) )
         => 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_5332_sum__strict__mono__ex1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ! [X4: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Extended_enat_nat(F,X4)),aa_Extended_enat_nat(G,X4)) )
       => ( ? [X: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X),A2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Extended_enat_nat(F,X)),aa_Extended_enat_nat(G,X)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups2027974829824023292at_nat(F,A2)),groups2027974829824023292at_nat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5333_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),A2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X4)),aa_nat_int(G,X4)) )
       => ( ? [X: nat] :
              ( aa_set_nat_o(member_nat2(X),A2)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X)),aa_nat_int(G,X)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_set_nat_int(groups3539618377306564664at_int(F),A2)),aa_set_nat_int(groups3539618377306564664at_int(G),A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5334_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,X4)),aa_complex_int(G,X4)) )
       => ( ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),A2)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_complex_int(F,X)),aa_complex_int(G,X)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5335_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X4: int] :
            ( aa_set_int_o(member_int2(X4),A2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X4)),aa_int_int(G,X4)) )
       => ( ? [X: int] :
              ( aa_set_int_o(member_int2(X),A2)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,X)),aa_int_int(G,X)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5336_sum__strict__mono__ex1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex3482447378695272101at_int,G: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ! [X4: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),A2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Extended_enat_int(F,X4)),aa_Extended_enat_int(G,X4)) )
       => ( ? [X: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X),A2)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Extended_enat_int(F,X)),aa_Extended_enat_int(G,X)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups2025484359314973016at_int(F,A2)),groups2025484359314973016at_int(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5337_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)
         => ( ! [X4: complex] :
                ( aa_set_complex_o(member_complex(X4),S3)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_complex_nat(H,X4)),aa_complex_nat(G,X4)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,groups5693394587270226106ex_nat(H,S3)),groups5693394587270226106ex_nat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_5338_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)
         => ( ! [X4: int] :
                ( aa_set_int_o(member_int2(X4),S3)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_int_nat(H,X4)),aa_int_nat(G,X4)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,groups4541462559716669496nt_nat(H,S3)),groups4541462559716669496nt_nat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_5339_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S3: set_Extended_enat,H: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_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_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
         => ( ! [X4: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),S3)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_Extended_enat_nat(H,X4)),aa_Extended_enat_nat(G,X4)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,groups2027974829824023292at_nat(H,S3)),groups2027974829824023292at_nat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_5340_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(plus_plus_real(X15),Y15)),aa_real_real(plus_plus_real(X23),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S3)
         => ( ! [X4: complex] :
                ( aa_set_complex_o(member_complex(X4),S3)
               => aa_real_o(aa_real_fun_real_o(R,aa_complex_real(H,X4)),aa_complex_real(G,X4)) )
           => aa_real_o(aa_real_fun_real_o(R,groups5808333547571424918x_real(H,S3)),groups5808333547571424918x_real(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_5341_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(plus_plus_real(X15),Y15)),aa_real_real(plus_plus_real(X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S3)
         => ( ! [X4: int] :
                ( aa_set_int_o(member_int2(X4),S3)
               => aa_real_o(aa_real_fun_real_o(R,aa_int_real(H,X4)),aa_int_real(G,X4)) )
           => aa_real_o(aa_real_fun_real_o(R,groups8778361861064173332t_real(H,S3)),groups8778361861064173332t_real(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_5342_sum_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S3: set_Extended_enat,H: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_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(plus_plus_real(X15),Y15)),aa_real_real(plus_plus_real(X23),Y23)) )
       => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
         => ( ! [X4: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),S3)
               => aa_real_o(aa_real_fun_real_o(R,aa_Ext2541572767299617110t_real(H,X4)),aa_Ext2541572767299617110t_real(G,X4)) )
           => aa_real_o(aa_real_fun_real_o(R,groups4148127829035722712t_real(H,S3)),groups4148127829035722712t_real(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_5343_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)
         => ( ! [X4: nat] :
                ( aa_set_nat_o(member_nat2(X4),S3)
               => aa_int_o(aa_int_fun_int_o(R,aa_nat_int(H,X4)),aa_nat_int(G,X4)) )
           => aa_int_o(aa_int_fun_int_o(R,aa_set_nat_int(groups3539618377306564664at_int(H),S3)),aa_set_nat_int(groups3539618377306564664at_int(G),S3)) ) ) ) ) ).

% sum.related
tff(fact_5344_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S3: set_complex,H: fun_complex_int,G: fun_complex_int] :
      ( aa_int_o(aa_int_fun_int_o(R,zero_zero_int),zero_zero_int)
     => ( ! [X15: int,Y15: int,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_complex_o(finite3207457112153483333omplex,S3)
         => ( ! [X4: complex] :
                ( aa_set_complex_o(member_complex(X4),S3)
               => aa_int_o(aa_int_fun_int_o(R,aa_complex_int(H,X4)),aa_complex_int(G,X4)) )
           => aa_int_o(aa_int_fun_int_o(R,groups5690904116761175830ex_int(H,S3)),groups5690904116761175830ex_int(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_5345_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S3: set_int,H: fun_int_int,G: fun_int_int] :
      ( aa_int_o(aa_int_fun_int_o(R,zero_zero_int),zero_zero_int)
     => ( ! [X15: int,Y15: int,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_int_o(finite_finite_int,S3)
         => ( ! [X4: int] :
                ( aa_set_int_o(member_int2(X4),S3)
               => aa_int_o(aa_int_fun_int_o(R,aa_int_int(H,X4)),aa_int_int(G,X4)) )
           => aa_int_o(aa_int_fun_int_o(R,groups4538972089207619220nt_int(H,S3)),groups4538972089207619220nt_int(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_5346_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S3: set_Extended_enat,H: fun_Ex3482447378695272101at_int,G: fun_Ex3482447378695272101at_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_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
         => ( ! [X4: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),S3)
               => aa_int_o(aa_int_fun_int_o(R,aa_Extended_enat_int(H,X4)),aa_Extended_enat_int(G,X4)) )
           => aa_int_o(aa_int_fun_int_o(R,groups2025484359314973016at_int(H,S3)),groups2025484359314973016at_int(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_5347_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 )
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,X4)),aa_complex_nat(G,X4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_5348_sum__strict__mono,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Extended_enat_nat(F,X4)),aa_Extended_enat_nat(G,X4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups2027974829824023292at_nat(F,A2)),groups2027974829824023292at_nat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_5349_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 )
       => ( ! [X4: real] :
              ( aa_set_real_o(member_real2(X4),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(G,X4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_5350_sum__strict__mono,axiom,
    ! [A2: set_o,F: fun_o_nat,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ! [X4: $o] :
              ( aa_set_o_o(member_o2((X4)),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(F,(X4))),aa_o_nat(G,(X4))) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups8507830703676809646_o_nat(F,A2)),groups8507830703676809646_o_nat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_5351_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 )
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X4)),aa_int_nat(G,X4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_5352_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_co6496365337113880337d_enat,G: fun_co6496365337113880337d_enat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),A2)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_com3683005661447327866d_enat(F,X4)),aa_com3683005661447327866d_enat(G,X4)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,groups1752964319039525884d_enat(F,A2)),groups1752964319039525884d_enat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_5353_sum__strict__mono,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex5852502975177633731d_enat,G: fun_Ex5852502975177633731d_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),A2)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(F,X4)),aa_Ext7969882492699152700d_enat(G,X4)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,groups2433450451889696826d_enat(F,A2)),groups2433450451889696826d_enat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_5354_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_re6065469302365115155d_enat,G: fun_re6065469302365115155d_enat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( aa_set_real_o(member_real2(X4),A2)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X4)),aa_rea1194391308913012860d_enat(G,X4)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,groups2800946370649118462d_enat(F,A2)),groups2800946370649118462d_enat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_5355_sum__strict__mono,axiom,
    ! [A2: set_o,F: fun_o_Extended_enat,G: fun_o_Extended_enat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ! [X4: $o] :
              ( aa_set_o_o(member_o2((X4)),A2)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_o_Extended_enat(F,(X4))),aa_o_Extended_enat(G,(X4))) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,groups7198740251461348360d_enat(F,A2)),groups7198740251461348360d_enat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_5356_sum__strict__mono,axiom,
    ! [A2: set_nat,F: fun_na8797796813480665199d_enat,G: fun_na8797796813480665199d_enat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat2(X4),A2)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,X4)),aa_nat_Extended_enat(G,X4)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,groups7108830773950497114d_enat(F,A2)),groups7108830773950497114d_enat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_5357_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_real,S3: set_real,Ia: fun_real_real,J: fun_real_real,T2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,S5)
     => ( aa_set_real_o(finite_finite_real,T3)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real2(A3),minus_minus_set_real(S3,S5))
             => ( aa_real_real(Ia,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real2(A3),minus_minus_set_real(S3,S5))
               => aa_set_real_o(member_real2(aa_real_real(J,A3)),minus_minus_set_real(T2,T3)) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real2(B3),minus_minus_set_real(T2,T3))
                 => ( aa_real_real(J,aa_real_real(Ia,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( aa_set_real_o(member_real2(B3),minus_minus_set_real(T2,T3))
                   => aa_set_real_o(member_real2(aa_real_real(Ia,B3)),minus_minus_set_real(S3,S5)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real2(A3),S5)
                     => ( aa_real_nat(G,A3) = zero_zero_nat ) )
                 => ( ! [B3: real] :
                        ( aa_set_real_o(member_real2(B3),T3)
                       => ( aa_real_nat(H,B3) = zero_zero_nat ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real2(A3),S3)
                         => ( aa_real_nat(H,aa_real_real(J,A3)) = aa_real_nat(G,A3) ) )
                     => ( groups1935376822645274424al_nat(G,S3) = groups1935376822645274424al_nat(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5358_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_o,S3: set_real,Ia: fun_o_real,J: fun_real_o,T2: set_o,G: fun_real_nat,H: fun_o_nat] :
      ( aa_set_real_o(finite_finite_real,S5)
     => ( aa_set_o_o(finite_finite_o,T3)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real2(A3),minus_minus_set_real(S3,S5))
             => ( aa_o_real(Ia,aa_real_o(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real2(A3),minus_minus_set_real(S3,S5))
               => aa_set_o_o(member_o2(aa_real_o(J,A3)),minus_minus_set_o(T2,T3)) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o2((B3)),minus_minus_set_o(T2,T3))
                 => ( aa_real_o(J,aa_o_real(Ia,(B3)))
                  <=> (B3) ) )
             => ( ! [B3: $o] :
                    ( aa_set_o_o(member_o2((B3)),minus_minus_set_o(T2,T3))
                   => aa_set_real_o(member_real2(aa_o_real(Ia,(B3))),minus_minus_set_real(S3,S5)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real2(A3),S5)
                     => ( aa_real_nat(G,A3) = zero_zero_nat ) )
                 => ( ! [B3: $o] :
                        ( aa_set_o_o(member_o2((B3)),T3)
                       => ( aa_o_nat(H,(B3)) = zero_zero_nat ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real2(A3),S3)
                         => ( aa_o_nat(H,aa_real_o(J,A3)) = aa_real_nat(G,A3) ) )
                     => ( groups1935376822645274424al_nat(G,S3) = groups8507830703676809646_o_nat(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5359_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T3: set_real,S3: set_o,Ia: fun_real_o,J: fun_o_real,T2: set_real,G: fun_o_nat,H: fun_real_nat] :
      ( aa_set_o_o(finite_finite_o,S5)
     => ( aa_set_real_o(finite_finite_real,T3)
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o2((A3)),minus_minus_set_o(S3,S5))
             => ( aa_real_o(Ia,aa_o_real(J,(A3)))
              <=> (A3) ) )
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o2((A3)),minus_minus_set_o(S3,S5))
               => aa_set_real_o(member_real2(aa_o_real(J,(A3))),minus_minus_set_real(T2,T3)) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real2(B3),minus_minus_set_real(T2,T3))
                 => ( aa_o_real(J,aa_real_o(Ia,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( aa_set_real_o(member_real2(B3),minus_minus_set_real(T2,T3))
                   => aa_set_o_o(member_o2(aa_real_o(Ia,B3)),minus_minus_set_o(S3,S5)) )
               => ( ! [A3: $o] :
                      ( aa_set_o_o(member_o2((A3)),S5)
                     => ( aa_o_nat(G,(A3)) = zero_zero_nat ) )
                 => ( ! [B3: real] :
                        ( aa_set_real_o(member_real2(B3),T3)
                       => ( aa_real_nat(H,B3) = zero_zero_nat ) )
                   => ( ! [A3: $o] :
                          ( aa_set_o_o(member_o2((A3)),S3)
                         => ( aa_real_nat(H,aa_o_real(J,(A3))) = aa_o_nat(G,(A3)) ) )
                     => ( groups8507830703676809646_o_nat(G,S3) = groups1935376822645274424al_nat(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5360_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T3: set_o,S3: set_o,Ia: fun_o_o,J: fun_o_o,T2: set_o,G: fun_o_nat,H: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S5)
     => ( aa_set_o_o(finite_finite_o,T3)
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o2((A3)),minus_minus_set_o(S3,S5))
             => ( aa_o_o(Ia,aa_o_o(J,(A3)))
              <=> (A3) ) )
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o2((A3)),minus_minus_set_o(S3,S5))
               => aa_set_o_o(member_o2(aa_o_o(J,(A3))),minus_minus_set_o(T2,T3)) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o2((B3)),minus_minus_set_o(T2,T3))
                 => ( aa_o_o(J,aa_o_o(Ia,(B3)))
                  <=> (B3) ) )
             => ( ! [B3: $o] :
                    ( aa_set_o_o(member_o2((B3)),minus_minus_set_o(T2,T3))
                   => aa_set_o_o(member_o2(aa_o_o(Ia,(B3))),minus_minus_set_o(S3,S5)) )
               => ( ! [A3: $o] :
                      ( aa_set_o_o(member_o2((A3)),S5)
                     => ( aa_o_nat(G,(A3)) = zero_zero_nat ) )
                 => ( ! [B3: $o] :
                        ( aa_set_o_o(member_o2((B3)),T3)
                       => ( aa_o_nat(H,(B3)) = zero_zero_nat ) )
                   => ( ! [A3: $o] :
                          ( aa_set_o_o(member_o2((A3)),S3)
                         => ( aa_o_nat(H,aa_o_o(J,(A3))) = aa_o_nat(G,(A3)) ) )
                     => ( groups8507830703676809646_o_nat(G,S3) = groups8507830703676809646_o_nat(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5361_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_complex,S3: set_real,Ia: fun_complex_real,J: fun_real_complex,T2: set_complex,G: fun_real_nat,H: fun_complex_nat] :
      ( aa_set_real_o(finite_finite_real,S5)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T3)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real2(A3),minus_minus_set_real(S3,S5))
             => ( aa_complex_real(Ia,aa_real_complex(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real2(A3),minus_minus_set_real(S3,S5))
               => aa_set_complex_o(member_complex(aa_real_complex(J,A3)),minus_811609699411566653omplex(T2,T3)) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T2,T3))
                 => ( aa_real_complex(J,aa_complex_real(Ia,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T2,T3))
                   => aa_set_real_o(member_real2(aa_complex_real(Ia,B3)),minus_minus_set_real(S3,S5)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real2(A3),S5)
                     => ( aa_real_nat(G,A3) = zero_zero_nat ) )
                 => ( ! [B3: complex] :
                        ( aa_set_complex_o(member_complex(B3),T3)
                       => ( aa_complex_nat(H,B3) = zero_zero_nat ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real2(A3),S3)
                         => ( aa_complex_nat(H,aa_real_complex(J,A3)) = aa_real_nat(G,A3) ) )
                     => ( groups1935376822645274424al_nat(G,S3) = groups5693394587270226106ex_nat(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5362_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T3: set_complex,S3: set_o,Ia: fun_complex_o,J: fun_o_complex,T2: set_complex,G: fun_o_nat,H: fun_complex_nat] :
      ( aa_set_o_o(finite_finite_o,S5)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T3)
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o2((A3)),minus_minus_set_o(S3,S5))
             => ( aa_complex_o(Ia,aa_o_complex(J,(A3)))
              <=> (A3) ) )
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o2((A3)),minus_minus_set_o(S3,S5))
               => aa_set_complex_o(member_complex(aa_o_complex(J,(A3))),minus_811609699411566653omplex(T2,T3)) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T2,T3))
                 => ( aa_o_complex(J,aa_complex_o(Ia,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T2,T3))
                   => aa_set_o_o(member_o2(aa_complex_o(Ia,B3)),minus_minus_set_o(S3,S5)) )
               => ( ! [A3: $o] :
                      ( aa_set_o_o(member_o2((A3)),S5)
                     => ( aa_o_nat(G,(A3)) = zero_zero_nat ) )
                 => ( ! [B3: complex] :
                        ( aa_set_complex_o(member_complex(B3),T3)
                       => ( aa_complex_nat(H,B3) = zero_zero_nat ) )
                   => ( ! [A3: $o] :
                          ( aa_set_o_o(member_o2((A3)),S3)
                         => ( aa_complex_nat(H,aa_o_complex(J,(A3))) = aa_o_nat(G,(A3)) ) )
                     => ( groups8507830703676809646_o_nat(G,S3) = groups5693394587270226106ex_nat(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5363_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_int,S3: set_real,Ia: fun_int_real,J: fun_real_int,T2: set_int,G: fun_real_nat,H: fun_int_nat] :
      ( aa_set_real_o(finite_finite_real,S5)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real2(A3),minus_minus_set_real(S3,S5))
             => ( aa_int_real(Ia,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real2(A3),minus_minus_set_real(S3,S5))
               => aa_set_int_o(member_int2(aa_real_int(J,A3)),minus_minus_set_int(T2,T3)) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int2(B3),minus_minus_set_int(T2,T3))
                 => ( aa_real_int(J,aa_int_real(Ia,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( aa_set_int_o(member_int2(B3),minus_minus_set_int(T2,T3))
                   => aa_set_real_o(member_real2(aa_int_real(Ia,B3)),minus_minus_set_real(S3,S5)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real2(A3),S5)
                     => ( aa_real_nat(G,A3) = zero_zero_nat ) )
                 => ( ! [B3: int] :
                        ( aa_set_int_o(member_int2(B3),T3)
                       => ( aa_int_nat(H,B3) = zero_zero_nat ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real2(A3),S3)
                         => ( aa_int_nat(H,aa_real_int(J,A3)) = aa_real_nat(G,A3) ) )
                     => ( groups1935376822645274424al_nat(G,S3) = groups4541462559716669496nt_nat(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5364_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T3: set_int,S3: set_o,Ia: fun_int_o,J: fun_o_int,T2: set_int,G: fun_o_nat,H: fun_int_nat] :
      ( aa_set_o_o(finite_finite_o,S5)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o2((A3)),minus_minus_set_o(S3,S5))
             => ( aa_int_o(Ia,aa_o_int(J,(A3)))
              <=> (A3) ) )
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o2((A3)),minus_minus_set_o(S3,S5))
               => aa_set_int_o(member_int2(aa_o_int(J,(A3))),minus_minus_set_int(T2,T3)) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int2(B3),minus_minus_set_int(T2,T3))
                 => ( aa_o_int(J,aa_int_o(Ia,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( aa_set_int_o(member_int2(B3),minus_minus_set_int(T2,T3))
                   => aa_set_o_o(member_o2(aa_int_o(Ia,B3)),minus_minus_set_o(S3,S5)) )
               => ( ! [A3: $o] :
                      ( aa_set_o_o(member_o2((A3)),S5)
                     => ( aa_o_nat(G,(A3)) = zero_zero_nat ) )
                 => ( ! [B3: int] :
                        ( aa_set_int_o(member_int2(B3),T3)
                       => ( aa_int_nat(H,B3) = zero_zero_nat ) )
                   => ( ! [A3: $o] :
                          ( aa_set_o_o(member_o2((A3)),S3)
                         => ( aa_int_nat(H,aa_o_int(J,(A3))) = aa_o_nat(G,(A3)) ) )
                     => ( groups8507830703676809646_o_nat(G,S3) = groups4541462559716669496nt_nat(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5365_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_Extended_enat,S3: set_real,Ia: fun_Ex1451620480498224165t_real,J: fun_re6065469302365115155d_enat,T2: set_Extended_enat,G: fun_real_nat,H: fun_Ex7660298398204468809at_nat] :
      ( aa_set_real_o(finite_finite_real,S5)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T3)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real2(A3),minus_minus_set_real(S3,S5))
             => ( aa_Ext2541572767299617110t_real(Ia,aa_rea1194391308913012860d_enat(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real2(A3),minus_minus_set_real(S3,S5))
               => aa_set3337841544853333262enat_o(member_Extended_enat(aa_rea1194391308913012860d_enat(J,A3)),minus_925952699566721837d_enat(T2,T3)) )
           => ( ! [B3: extended_enat] :
                  ( aa_set3337841544853333262enat_o(member_Extended_enat(B3),minus_925952699566721837d_enat(T2,T3))
                 => ( aa_rea1194391308913012860d_enat(J,aa_Ext2541572767299617110t_real(Ia,B3)) = B3 ) )
             => ( ! [B3: extended_enat] :
                    ( aa_set3337841544853333262enat_o(member_Extended_enat(B3),minus_925952699566721837d_enat(T2,T3))
                   => aa_set_real_o(member_real2(aa_Ext2541572767299617110t_real(Ia,B3)),minus_minus_set_real(S3,S5)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real2(A3),S5)
                     => ( aa_real_nat(G,A3) = zero_zero_nat ) )
                 => ( ! [B3: extended_enat] :
                        ( aa_set3337841544853333262enat_o(member_Extended_enat(B3),T3)
                       => ( aa_Extended_enat_nat(H,B3) = zero_zero_nat ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real2(A3),S3)
                         => ( aa_Extended_enat_nat(H,aa_rea1194391308913012860d_enat(J,A3)) = aa_real_nat(G,A3) ) )
                     => ( groups1935376822645274424al_nat(G,S3) = groups2027974829824023292at_nat(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5366_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T3: set_Extended_enat,S3: set_o,Ia: fun_Extended_enat_o,J: fun_o_Extended_enat,T2: set_Extended_enat,G: fun_o_nat,H: fun_Ex7660298398204468809at_nat] :
      ( aa_set_o_o(finite_finite_o,S5)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T3)
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o2((A3)),minus_minus_set_o(S3,S5))
             => ( aa_Extended_enat_o(Ia,aa_o_Extended_enat(J,(A3)))
              <=> (A3) ) )
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o2((A3)),minus_minus_set_o(S3,S5))
               => aa_set3337841544853333262enat_o(member_Extended_enat(aa_o_Extended_enat(J,(A3))),minus_925952699566721837d_enat(T2,T3)) )
           => ( ! [B3: extended_enat] :
                  ( aa_set3337841544853333262enat_o(member_Extended_enat(B3),minus_925952699566721837d_enat(T2,T3))
                 => ( aa_o_Extended_enat(J,aa_Extended_enat_o(Ia,B3)) = B3 ) )
             => ( ! [B3: extended_enat] :
                    ( aa_set3337841544853333262enat_o(member_Extended_enat(B3),minus_925952699566721837d_enat(T2,T3))
                   => aa_set_o_o(member_o2(aa_Extended_enat_o(Ia,B3)),minus_minus_set_o(S3,S5)) )
               => ( ! [A3: $o] :
                      ( aa_set_o_o(member_o2((A3)),S5)
                     => ( aa_o_nat(G,(A3)) = zero_zero_nat ) )
                 => ( ! [B3: extended_enat] :
                        ( aa_set3337841544853333262enat_o(member_Extended_enat(B3),T3)
                       => ( aa_Extended_enat_nat(H,B3) = zero_zero_nat ) )
                   => ( ! [A3: $o] :
                          ( aa_set_o_o(member_o2((A3)),S3)
                         => ( aa_Extended_enat_nat(H,aa_o_Extended_enat(J,(A3))) = aa_o_nat(G,(A3)) ) )
                     => ( groups8507830703676809646_o_nat(G,S3) = groups2027974829824023292at_nat(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5367_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) )
      <=> ? [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
            & ( aa_complex_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa2: complex] :
                ( aa_set_complex_o(member_complex(Xa2),A2)
               => ( ( X3 != Xa2 )
                 => ( aa_complex_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_5368_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) )
      <=> ? [X3: int] :
            ( aa_set_int_o(member_int2(X3),A2)
            & ( aa_int_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa2: int] :
                ( aa_set_int_o(member_int2(Xa2),A2)
               => ( ( X3 != Xa2 )
                 => ( aa_int_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_5369_sum__eq__Suc0__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( groups2027974829824023292at_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X3),A2)
            & ( aa_Extended_enat_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa2: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa2),A2)
               => ( ( X3 != Xa2 )
                 => ( aa_Extended_enat_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_5370_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) )
      <=> ? [X3: nat] :
            ( aa_set_nat_o(member_nat2(X3),A2)
            & ( aa_nat_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa2: nat] :
                ( aa_set_nat_o(member_nat2(Xa2),A2)
               => ( ( X3 != Xa2 )
                 => ( aa_nat_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_5371_sum__SucD,axiom,
    ! [F: fun_nat_nat,A2: set_nat,Nb: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,Nb) )
     => ? [X4: nat] :
          ( aa_set_nat_o(member_nat2(X4),A2)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4)) ) ) ).

% sum_SucD
tff(fact_5372_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 )
      <=> ? [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
            & ( aa_complex_nat(F,X3) = one_one_nat )
            & ! [Xa2: complex] :
                ( aa_set_complex_o(member_complex(Xa2),A2)
               => ( ( X3 != Xa2 )
                 => ( aa_complex_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_5373_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 )
      <=> ? [X3: int] :
            ( aa_set_int_o(member_int2(X3),A2)
            & ( aa_int_nat(F,X3) = one_one_nat )
            & ! [Xa2: int] :
                ( aa_set_int_o(member_int2(Xa2),A2)
               => ( ( X3 != Xa2 )
                 => ( aa_int_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_5374_sum__eq__1__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( groups2027974829824023292at_nat(F,A2) = one_one_nat )
      <=> ? [X3: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X3),A2)
            & ( aa_Extended_enat_nat(F,X3) = one_one_nat )
            & ! [Xa2: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa2),A2)
               => ( ( X3 != Xa2 )
                 => ( aa_Extended_enat_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_5375_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 )
      <=> ? [X3: nat] :
            ( aa_set_nat_o(member_nat2(X3),A2)
            & ( aa_nat_nat(F,X3) = one_one_nat )
            & ! [Xa2: nat] :
                ( aa_set_nat_o(member_nat2(Xa2),A2)
               => ( ( X3 != Xa2 )
                 => ( aa_nat_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_5376_ceiling__le,axiom,
    ! [Xb: real,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),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,Xb)),A) ) ).

% ceiling_le
tff(fact_5377_ceiling__le__iff,axiom,
    ! [Xb: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Z)) ) ).

% ceiling_le_iff
tff(fact_5378_less__ceiling__iff,axiom,
    ! [Z: int,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),Xb) ) ).

% less_ceiling_iff
tff(fact_5379_ceiling__add__le,axiom,
    ! [Xb: real,Y: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(plus_plus_real(Xb),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_real_int(archim7802044766580827645g_real,Y))) ).

% ceiling_add_le
tff(fact_5380_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: fun_set_complex_o,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X4: complex,S6: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S6)
             => ( ! [Y4: complex] :
                    ( aa_set_complex_o(member_complex(Y4),S6)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,Y4)),aa_complex_real(F,X4)) )
               => ( aa_set_complex_o(P,S6)
                 => aa_set_complex_o(P,insert_complex(X4,S6)) ) ) )
         => aa_set_complex_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_5381_finite__ranking__induct,axiom,
    ! [S3: set_Extended_enat,P: fun_se8123728892507317397enat_o,F: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
     => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
       => ( ! [X4: extended_enat,S6: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S6)
             => ( ! [Y4: extended_enat] :
                    ( aa_set3337841544853333262enat_o(member_Extended_enat(Y4),S6)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,Y4)),aa_Ext2541572767299617110t_real(F,X4)) )
               => ( aa_set3337841544853333262enat_o(P,S6)
                 => aa_set3337841544853333262enat_o(P,insert_Extended_enat(X4,S6)) ) ) )
         => aa_set3337841544853333262enat_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_5382_finite__ranking__induct,axiom,
    ! [S3: set_real,P: fun_set_real_o,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X4: real,S6: set_real] :
              ( aa_set_real_o(finite_finite_real,S6)
             => ( ! [Y4: real] :
                    ( aa_set_real_o(member_real2(Y4),S6)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Y4)),aa_real_real(F,X4)) )
               => ( aa_set_real_o(P,S6)
                 => aa_set_real_o(P,insert_real2(X4,S6)) ) ) )
         => aa_set_real_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_5383_finite__ranking__induct,axiom,
    ! [S3: set_o,P: fun_set_o_o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,S3)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X4: $o,S6: set_o] :
              ( aa_set_o_o(finite_finite_o,S6)
             => ( ! [Y4: $o] :
                    ( aa_set_o_o(member_o2((Y4)),S6)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(F,(Y4))),aa_o_real(F,(X4))) )
               => ( aa_set_o_o(P,S6)
                 => aa_set_o_o(P,insert_o2((X4),S6)) ) ) )
         => aa_set_o_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_5384_finite__ranking__induct,axiom,
    ! [S3: set_nat,P: fun_set_nat_o,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X4: nat,S6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S6)
             => ( ! [Y4: nat] :
                    ( aa_set_nat_o(member_nat2(Y4),S6)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,Y4)),aa_nat_real(F,X4)) )
               => ( aa_set_nat_o(P,S6)
                 => aa_set_nat_o(P,insert_nat2(X4,S6)) ) ) )
         => aa_set_nat_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_5385_finite__ranking__induct,axiom,
    ! [S3: set_int,P: fun_set_int_o,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X4: int,S6: set_int] :
              ( aa_set_int_o(finite_finite_int,S6)
             => ( ! [Y4: int] :
                    ( aa_set_int_o(member_int2(Y4),S6)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,Y4)),aa_int_real(F,X4)) )
               => ( aa_set_int_o(P,S6)
                 => aa_set_int_o(P,insert_int2(X4,S6)) ) ) )
         => aa_set_int_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_5386_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)
       => ( ! [X4: complex,S6: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S6)
             => ( ! [Y4: complex] :
                    ( aa_set_complex_o(member_complex(Y4),S6)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,Y4)),aa_complex_nat(F,X4)) )
               => ( aa_set_complex_o(P,S6)
                 => aa_set_complex_o(P,insert_complex(X4,S6)) ) ) )
         => aa_set_complex_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_5387_finite__ranking__induct,axiom,
    ! [S3: set_Extended_enat,P: fun_se8123728892507317397enat_o,F: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
     => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
       => ( ! [X4: extended_enat,S6: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S6)
             => ( ! [Y4: extended_enat] :
                    ( aa_set3337841544853333262enat_o(member_Extended_enat(Y4),S6)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Extended_enat_nat(F,Y4)),aa_Extended_enat_nat(F,X4)) )
               => ( aa_set3337841544853333262enat_o(P,S6)
                 => aa_set3337841544853333262enat_o(P,insert_Extended_enat(X4,S6)) ) ) )
         => aa_set3337841544853333262enat_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_5388_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)
       => ( ! [X4: real,S6: set_real] :
              ( aa_set_real_o(finite_finite_real,S6)
             => ( ! [Y4: real] :
                    ( aa_set_real_o(member_real2(Y4),S6)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,Y4)),aa_real_nat(F,X4)) )
               => ( aa_set_real_o(P,S6)
                 => aa_set_real_o(P,insert_real2(X4,S6)) ) ) )
         => aa_set_real_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_5389_finite__ranking__induct,axiom,
    ! [S3: set_o,P: fun_set_o_o,F: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S3)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X4: $o,S6: set_o] :
              ( aa_set_o_o(finite_finite_o,S6)
             => ( ! [Y4: $o] :
                    ( aa_set_o_o(member_o2((Y4)),S6)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(Y4))),aa_o_nat(F,(X4))) )
               => ( aa_set_o_o(P,S6)
                 => aa_set_o_o(P,insert_o2((X4),S6)) ) ) )
         => aa_set_o_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_5390_finite__linorder__max__induct,axiom,
    ! [A2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [B3: $o,A6: set_o] :
              ( aa_set_o_o(finite_finite_o,A6)
             => ( ! [X: $o] :
                    ( aa_set_o_o(member_o2((X)),A6)
                   => aa_o_o(aa_o_fun_o_o(ord_less_o,(X)),(B3)) )
               => ( aa_set_o_o(P,A6)
                 => aa_set_o_o(P,insert_o2((B3),A6)) ) ) )
         => aa_set_o_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_5391_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,A6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A6)
             => ( ! [X: nat] :
                    ( aa_set_nat_o(member_nat2(X),A6)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),B3) )
               => ( aa_set_nat_o(P,A6)
                 => aa_set_nat_o(P,insert_nat2(B3,A6)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_5392_finite__linorder__max__induct,axiom,
    ! [A2: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
       => ( ! [B3: extended_enat,A6: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A6)
             => ( ! [X: extended_enat] :
                    ( aa_set3337841544853333262enat_o(member_Extended_enat(X),A6)
                   => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),B3) )
               => ( aa_set3337841544853333262enat_o(P,A6)
                 => aa_set3337841544853333262enat_o(P,insert_Extended_enat(B3,A6)) ) ) )
         => aa_set3337841544853333262enat_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_5393_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,A6: set_real] :
              ( aa_set_real_o(finite_finite_real,A6)
             => ( ! [X: real] :
                    ( aa_set_real_o(member_real2(X),A6)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,X),B3) )
               => ( aa_set_real_o(P,A6)
                 => aa_set_real_o(P,insert_real2(B3,A6)) ) ) )
         => aa_set_real_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_5394_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,A6: set_int] :
              ( aa_set_int_o(finite_finite_int,A6)
             => ( ! [X: int] :
                    ( aa_set_int_o(member_int2(X),A6)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,X),B3) )
               => ( aa_set_int_o(P,A6)
                 => aa_set_int_o(P,insert_int2(B3,A6)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_5395_finite__linorder__min__induct,axiom,
    ! [A2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [B3: $o,A6: set_o] :
              ( aa_set_o_o(finite_finite_o,A6)
             => ( ! [X: $o] :
                    ( aa_set_o_o(member_o2((X)),A6)
                   => aa_o_o(aa_o_fun_o_o(ord_less_o,(B3)),(X)) )
               => ( aa_set_o_o(P,A6)
                 => aa_set_o_o(P,insert_o2((B3),A6)) ) ) )
         => aa_set_o_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_5396_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,A6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A6)
             => ( ! [X: nat] :
                    ( aa_set_nat_o(member_nat2(X),A6)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B3),X) )
               => ( aa_set_nat_o(P,A6)
                 => aa_set_nat_o(P,insert_nat2(B3,A6)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_5397_finite__linorder__min__induct,axiom,
    ! [A2: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
       => ( ! [B3: extended_enat,A6: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A6)
             => ( ! [X: extended_enat] :
                    ( aa_set3337841544853333262enat_o(member_Extended_enat(X),A6)
                   => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B3),X) )
               => ( aa_set3337841544853333262enat_o(P,A6)
                 => aa_set3337841544853333262enat_o(P,insert_Extended_enat(B3,A6)) ) ) )
         => aa_set3337841544853333262enat_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_5398_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,A6: set_real] :
              ( aa_set_real_o(finite_finite_real,A6)
             => ( ! [X: real] :
                    ( aa_set_real_o(member_real2(X),A6)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,B3),X) )
               => ( aa_set_real_o(P,A6)
                 => aa_set_real_o(P,insert_real2(B3,A6)) ) ) )
         => aa_set_real_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_5399_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,A6: set_int] :
              ( aa_set_int_o(finite_finite_int,A6)
             => ( ! [X: int] :
                    ( aa_set_int_o(member_int2(X),A6)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,B3),X) )
               => ( aa_set_int_o(P,A6)
                 => aa_set_int_o(P,insert_int2(B3,A6)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_5400_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_nat2(A3),A2)
                 => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F4),A2)
                   => ( ~ aa_set_set_nat_o(member_set_nat2(A3),F4)
                     => ( aa_set_set_nat_o(P,F4)
                       => aa_set_set_nat_o(P,insert_set_nat2(A3,F4)) ) ) ) ) )
           => aa_set_set_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_5401_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_5402_finite__subset__induct_H,axiom,
    ! [F3: set_Extended_enat,A2: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F3)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(F3),A2)
       => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
         => ( ! [A3: extended_enat,F4: set_Extended_enat] :
                ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F4)
               => ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),A2)
                 => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(F4),A2)
                   => ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(A3),F4)
                     => ( aa_set3337841544853333262enat_o(P,F4)
                       => aa_set3337841544853333262enat_o(P,insert_Extended_enat(A3,F4)) ) ) ) ) )
           => aa_set3337841544853333262enat_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_5403_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_real2(A3),A2)
                 => ( aa_set_real_o(ord_less_eq_set_real(F4),A2)
                   => ( ~ aa_set_real_o(member_real2(A3),F4)
                     => ( aa_set_real_o(P,F4)
                       => aa_set_real_o(P,insert_real2(A3,F4)) ) ) ) ) )
           => aa_set_real_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_5404_finite__subset__induct_H,axiom,
    ! [F3: set_o,A2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F3)
     => ( aa_set_o_o(ord_less_eq_set_o(F3),A2)
       => ( aa_set_o_o(P,bot_bot_set_o)
         => ( ! [A3: $o,F4: set_o] :
                ( aa_set_o_o(finite_finite_o,F4)
               => ( aa_set_o_o(member_o2((A3)),A2)
                 => ( aa_set_o_o(ord_less_eq_set_o(F4),A2)
                   => ( ~ aa_set_o_o(member_o2((A3)),F4)
                     => ( aa_set_o_o(P,F4)
                       => aa_set_o_o(P,insert_o2((A3),F4)) ) ) ) ) )
           => aa_set_o_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_5405_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_int2(A3),A2)
                 => ( aa_set_int_o(ord_less_eq_set_int(F4),A2)
                   => ( ~ aa_set_int_o(member_int2(A3),F4)
                     => ( aa_set_int_o(P,F4)
                       => aa_set_int_o(P,insert_int2(A3,F4)) ) ) ) ) )
           => aa_set_int_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_5406_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_nat2(A3),A2)
                 => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F4),A2)
                   => ( ~ aa_set_nat_o(member_nat2(A3),F4)
                     => ( aa_set_nat_o(P,F4)
                       => aa_set_nat_o(P,insert_nat2(A3,F4)) ) ) ) ) )
           => aa_set_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_5407_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_nat2(A3),A2)
                 => ( ~ aa_set_set_nat_o(member_set_nat2(A3),F4)
                   => ( aa_set_set_nat_o(P,F4)
                     => aa_set_set_nat_o(P,insert_set_nat2(A3,F4)) ) ) ) )
           => aa_set_set_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_5408_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_5409_finite__subset__induct,axiom,
    ! [F3: set_Extended_enat,A2: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F3)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(F3),A2)
       => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
         => ( ! [A3: extended_enat,F4: set_Extended_enat] :
                ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F4)
               => ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),A2)
                 => ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(A3),F4)
                   => ( aa_set3337841544853333262enat_o(P,F4)
                     => aa_set3337841544853333262enat_o(P,insert_Extended_enat(A3,F4)) ) ) ) )
           => aa_set3337841544853333262enat_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_5410_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_real2(A3),A2)
                 => ( ~ aa_set_real_o(member_real2(A3),F4)
                   => ( aa_set_real_o(P,F4)
                     => aa_set_real_o(P,insert_real2(A3,F4)) ) ) ) )
           => aa_set_real_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_5411_finite__subset__induct,axiom,
    ! [F3: set_o,A2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F3)
     => ( aa_set_o_o(ord_less_eq_set_o(F3),A2)
       => ( aa_set_o_o(P,bot_bot_set_o)
         => ( ! [A3: $o,F4: set_o] :
                ( aa_set_o_o(finite_finite_o,F4)
               => ( aa_set_o_o(member_o2((A3)),A2)
                 => ( ~ aa_set_o_o(member_o2((A3)),F4)
                   => ( aa_set_o_o(P,F4)
                     => aa_set_o_o(P,insert_o2((A3),F4)) ) ) ) )
           => aa_set_o_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_5412_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_int2(A3),A2)
                 => ( ~ aa_set_int_o(member_int2(A3),F4)
                   => ( aa_set_int_o(P,F4)
                     => aa_set_int_o(P,insert_int2(A3,F4)) ) ) ) )
           => aa_set_int_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_5413_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_nat2(A3),A2)
                 => ( ~ aa_set_nat_o(member_nat2(A3),F4)
                   => ( aa_set_nat_o(P,F4)
                     => aa_set_nat_o(P,insert_nat2(A3,F4)) ) ) ) )
           => aa_set_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_5414_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_re6065469302365115155d_enat,Ia: real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real2(I2),S)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,I2)) )
       => ( ( groups2800946370649118462d_enat(F,S) = zero_z5237406670263579293d_enat )
         => ( aa_set_real_o(member_real2(Ia),S)
           => ( aa_rea1194391308913012860d_enat(F,Ia) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5415_sum__nonneg__0,axiom,
    ! [S: set_o,F: fun_o_Extended_enat,Ia: $o] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ! [I2: $o] :
            ( aa_set_o_o(member_o2((I2)),S)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_o_Extended_enat(F,(I2))) )
       => ( ( groups7198740251461348360d_enat(F,S) = zero_z5237406670263579293d_enat )
         => ( aa_set_o_o(member_o2((Ia)),S)
           => ( aa_o_Extended_enat(F,(Ia)) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5416_sum__nonneg__0,axiom,
    ! [S: set_nat,F: fun_na8797796813480665199d_enat,Ia: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat2(I2),S)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(F,I2)) )
       => ( ( groups7108830773950497114d_enat(F,S) = zero_z5237406670263579293d_enat )
         => ( aa_set_nat_o(member_nat2(Ia),S)
           => ( aa_nat_Extended_enat(F,Ia) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5417_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_co6496365337113880337d_enat,Ia: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(F,I2)) )
       => ( ( groups1752964319039525884d_enat(F,S) = zero_z5237406670263579293d_enat )
         => ( aa_set_complex_o(member_complex(Ia),S)
           => ( aa_com3683005661447327866d_enat(F,Ia) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5418_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_in5914218760682845459d_enat,Ia: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int2(I2),S)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(F,I2)) )
       => ( ( groups4225252721152677374d_enat(F,S) = zero_z5237406670263579293d_enat )
         => ( aa_set_int_o(member_int2(Ia),S)
           => ( aa_int_Extended_enat(F,Ia) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5419_sum__nonneg__0,axiom,
    ! [S: set_Extended_enat,F: fun_Ex5852502975177633731d_enat,Ia: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
     => ( ! [I2: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(I2),S)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(F,I2)) )
       => ( ( groups2433450451889696826d_enat(F,S) = zero_z5237406670263579293d_enat )
         => ( aa_set3337841544853333262enat_o(member_Extended_enat(Ia),S)
           => ( aa_Ext7969882492699152700d_enat(F,Ia) = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5420_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_real,Ia: real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real2(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)) )
       => ( ( groups8097168146408367636l_real(F,S) = zero_zero_real )
         => ( aa_set_real_o(member_real2(Ia),S)
           => ( aa_real_real(F,Ia) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5421_sum__nonneg__0,axiom,
    ! [S: set_o,F: fun_o_real,Ia: $o] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ! [I2: $o] :
            ( aa_set_o_o(member_o2((I2)),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(I2))) )
       => ( ( groups8691415230153176458o_real(F,S) = zero_zero_real )
         => ( aa_set_o_o(member_o2((Ia)),S)
           => ( aa_o_real(F,(Ia)) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5422_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_real,Ia: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)) )
       => ( ( groups5808333547571424918x_real(F,S) = zero_zero_real )
         => ( aa_set_complex_o(member_complex(Ia),S)
           => ( aa_complex_real(F,Ia) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5423_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_real,Ia: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int2(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)) )
       => ( ( groups8778361861064173332t_real(F,S) = zero_zero_real )
         => ( aa_set_int_o(member_int2(Ia),S)
           => ( aa_int_real(F,Ia) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5424_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_re6065469302365115155d_enat,B2: extended_enat,Ia: real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real2(I2),S)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,I2)) )
       => ( ( groups2800946370649118462d_enat(F,S) = B2 )
         => ( aa_set_real_o(member_real2(Ia),S)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_rea1194391308913012860d_enat(F,Ia)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5425_sum__nonneg__leq__bound,axiom,
    ! [S: set_o,F: fun_o_Extended_enat,B2: extended_enat,Ia: $o] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ! [I2: $o] :
            ( aa_set_o_o(member_o2((I2)),S)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_o_Extended_enat(F,(I2))) )
       => ( ( groups7198740251461348360d_enat(F,S) = B2 )
         => ( aa_set_o_o(member_o2((Ia)),S)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_o_Extended_enat(F,(Ia))),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5426_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: fun_na8797796813480665199d_enat,B2: extended_enat,Ia: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat2(I2),S)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(F,I2)) )
       => ( ( groups7108830773950497114d_enat(F,S) = B2 )
         => ( aa_set_nat_o(member_nat2(Ia),S)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,Ia)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5427_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_co6496365337113880337d_enat,B2: extended_enat,Ia: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(F,I2)) )
       => ( ( groups1752964319039525884d_enat(F,S) = B2 )
         => ( aa_set_complex_o(member_complex(Ia),S)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_com3683005661447327866d_enat(F,Ia)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5428_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_in5914218760682845459d_enat,B2: extended_enat,Ia: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int2(I2),S)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(F,I2)) )
       => ( ( groups4225252721152677374d_enat(F,S) = B2 )
         => ( aa_set_int_o(member_int2(Ia),S)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_int_Extended_enat(F,Ia)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5429_sum__nonneg__leq__bound,axiom,
    ! [S: set_Extended_enat,F: fun_Ex5852502975177633731d_enat,B2: extended_enat,Ia: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S)
     => ( ! [I2: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(I2),S)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(F,I2)) )
       => ( ( groups2433450451889696826d_enat(F,S) = B2 )
         => ( aa_set3337841544853333262enat_o(member_Extended_enat(Ia),S)
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(F,Ia)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5430_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_real,B2: real,Ia: real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real2(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)) )
       => ( ( groups8097168146408367636l_real(F,S) = B2 )
         => ( aa_set_real_o(member_real2(Ia),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Ia)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5431_sum__nonneg__leq__bound,axiom,
    ! [S: set_o,F: fun_o_real,B2: real,Ia: $o] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ! [I2: $o] :
            ( aa_set_o_o(member_o2((I2)),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(I2))) )
       => ( ( groups8691415230153176458o_real(F,S) = B2 )
         => ( aa_set_o_o(member_o2((Ia)),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(F,(Ia))),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5432_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_real,B2: real,Ia: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)) )
       => ( ( groups5808333547571424918x_real(F,S) = B2 )
         => ( aa_set_complex_o(member_complex(Ia),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,Ia)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5433_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_real,B2: real,Ia: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int2(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)) )
       => ( ( groups8778361861064173332t_real(F,S) = B2 )
         => ( aa_set_int_o(member_int2(Ia),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,Ia)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5434_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_5435_infinite__remove,axiom,
    ! [S3: set_Extended_enat,A: extended_enat] :
      ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
     => ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,minus_925952699566721837d_enat(S3,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) ) ).

% infinite_remove
tff(fact_5436_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_real2(A,bot_bot_set_real))) ) ).

% infinite_remove
tff(fact_5437_infinite__remove,axiom,
    ! [S3: set_o,A: $o] :
      ( ~ aa_set_o_o(finite_finite_o,S3)
     => ~ aa_set_o_o(finite_finite_o,minus_minus_set_o(S3,insert_o2((A),bot_bot_set_o))) ) ).

% infinite_remove
tff(fact_5438_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_int2(A,bot_bot_set_int))) ) ).

% infinite_remove
tff(fact_5439_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_nat2(A,bot_bot_set_nat))) ) ).

% infinite_remove
tff(fact_5440_infinite__coinduct,axiom,
    ! [X5: fun_set_complex_o,A2: set_complex] :
      ( aa_set_complex_o(X5,A2)
     => ( ! [A6: set_complex] :
            ( aa_set_complex_o(X5,A6)
           => ? [X: complex] :
                ( aa_set_complex_o(member_complex(X),A6)
                & ( aa_set_complex_o(X5,minus_811609699411566653omplex(A6,insert_complex(X,bot_bot_set_complex)))
                  | ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A6,insert_complex(X,bot_bot_set_complex))) ) ) )
       => ~ aa_set_complex_o(finite3207457112153483333omplex,A2) ) ) ).

% infinite_coinduct
tff(fact_5441_infinite__coinduct,axiom,
    ! [X5: fun_se8123728892507317397enat_o,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(X5,A2)
     => ( ! [A6: set_Extended_enat] :
            ( aa_set3337841544853333262enat_o(X5,A6)
           => ? [X: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(X),A6)
                & ( aa_set3337841544853333262enat_o(X5,minus_925952699566721837d_enat(A6,insert_Extended_enat(X,bot_bo7653980558646680370d_enat)))
                  | ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,minus_925952699566721837d_enat(A6,insert_Extended_enat(X,bot_bo7653980558646680370d_enat))) ) ) )
       => ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2) ) ) ).

% infinite_coinduct
tff(fact_5442_infinite__coinduct,axiom,
    ! [X5: fun_set_real_o,A2: set_real] :
      ( aa_set_real_o(X5,A2)
     => ( ! [A6: set_real] :
            ( aa_set_real_o(X5,A6)
           => ? [X: real] :
                ( aa_set_real_o(member_real2(X),A6)
                & ( aa_set_real_o(X5,minus_minus_set_real(A6,insert_real2(X,bot_bot_set_real)))
                  | ~ aa_set_real_o(finite_finite_real,minus_minus_set_real(A6,insert_real2(X,bot_bot_set_real))) ) ) )
       => ~ aa_set_real_o(finite_finite_real,A2) ) ) ).

% infinite_coinduct
tff(fact_5443_infinite__coinduct,axiom,
    ! [X5: fun_set_o_o,A2: set_o] :
      ( aa_set_o_o(X5,A2)
     => ( ! [A6: set_o] :
            ( aa_set_o_o(X5,A6)
           => ? [X: $o] :
                ( aa_set_o_o(member_o2((X)),A6)
                & ( aa_set_o_o(X5,minus_minus_set_o(A6,insert_o2((X),bot_bot_set_o)))
                  | ~ aa_set_o_o(finite_finite_o,minus_minus_set_o(A6,insert_o2((X),bot_bot_set_o))) ) ) )
       => ~ aa_set_o_o(finite_finite_o,A2) ) ) ).

% infinite_coinduct
tff(fact_5444_infinite__coinduct,axiom,
    ! [X5: fun_set_int_o,A2: set_int] :
      ( aa_set_int_o(X5,A2)
     => ( ! [A6: set_int] :
            ( aa_set_int_o(X5,A6)
           => ? [X: int] :
                ( aa_set_int_o(member_int2(X),A6)
                & ( aa_set_int_o(X5,minus_minus_set_int(A6,insert_int2(X,bot_bot_set_int)))
                  | ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(A6,insert_int2(X,bot_bot_set_int))) ) ) )
       => ~ aa_set_int_o(finite_finite_int,A2) ) ) ).

% infinite_coinduct
tff(fact_5445_infinite__coinduct,axiom,
    ! [X5: fun_set_nat_o,A2: set_nat] :
      ( aa_set_nat_o(X5,A2)
     => ( ! [A6: set_nat] :
            ( aa_set_nat_o(X5,A6)
           => ? [X: nat] :
                ( aa_set_nat_o(member_nat2(X),A6)
                & ( aa_set_nat_o(X5,minus_minus_set_nat(A6,insert_nat2(X,bot_bot_set_nat)))
                  | ~ aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A6,insert_nat2(X,bot_bot_set_nat))) ) ) )
       => ~ aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% infinite_coinduct
tff(fact_5446_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,A6: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A6)
             => ( aa_set_set_nat_o(member_set_nat2(A3),A6)
               => ( aa_set_set_nat_o(P,A6)
                 => aa_set_set_nat_o(P,minus_2163939370556025621et_nat(A6,insert_set_nat2(A3,bot_bot_set_set_nat))) ) ) )
         => aa_set_set_nat_o(P,bot_bot_set_set_nat) ) ) ) ).

% finite_empty_induct
tff(fact_5447_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,A6: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A6)
             => ( aa_set_complex_o(member_complex(A3),A6)
               => ( aa_set_complex_o(P,A6)
                 => aa_set_complex_o(P,minus_811609699411566653omplex(A6,insert_complex(A3,bot_bot_set_complex))) ) ) )
         => aa_set_complex_o(P,bot_bot_set_complex) ) ) ) ).

% finite_empty_induct
tff(fact_5448_finite__empty__induct,axiom,
    ! [A2: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(P,A2)
       => ( ! [A3: extended_enat,A6: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A6)
             => ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),A6)
               => ( aa_set3337841544853333262enat_o(P,A6)
                 => aa_set3337841544853333262enat_o(P,minus_925952699566721837d_enat(A6,insert_Extended_enat(A3,bot_bo7653980558646680370d_enat))) ) ) )
         => aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat) ) ) ) ).

% finite_empty_induct
tff(fact_5449_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,A6: set_real] :
              ( aa_set_real_o(finite_finite_real,A6)
             => ( aa_set_real_o(member_real2(A3),A6)
               => ( aa_set_real_o(P,A6)
                 => aa_set_real_o(P,minus_minus_set_real(A6,insert_real2(A3,bot_bot_set_real))) ) ) )
         => aa_set_real_o(P,bot_bot_set_real) ) ) ) ).

% finite_empty_induct
tff(fact_5450_finite__empty__induct,axiom,
    ! [A2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(P,A2)
       => ( ! [A3: $o,A6: set_o] :
              ( aa_set_o_o(finite_finite_o,A6)
             => ( aa_set_o_o(member_o2((A3)),A6)
               => ( aa_set_o_o(P,A6)
                 => aa_set_o_o(P,minus_minus_set_o(A6,insert_o2((A3),bot_bot_set_o))) ) ) )
         => aa_set_o_o(P,bot_bot_set_o) ) ) ) ).

% finite_empty_induct
tff(fact_5451_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,A6: set_int] :
              ( aa_set_int_o(finite_finite_int,A6)
             => ( aa_set_int_o(member_int2(A3),A6)
               => ( aa_set_int_o(P,A6)
                 => aa_set_int_o(P,minus_minus_set_int(A6,insert_int2(A3,bot_bot_set_int))) ) ) )
         => aa_set_int_o(P,bot_bot_set_int) ) ) ) ).

% finite_empty_induct
tff(fact_5452_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,A6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A6)
             => ( aa_set_nat_o(member_nat2(A3),A6)
               => ( aa_set_nat_o(P,A6)
                 => aa_set_nat_o(P,minus_minus_set_nat(A6,insert_nat2(A3,bot_bot_set_nat))) ) ) )
         => aa_set_nat_o(P,bot_bot_set_nat) ) ) ) ).

% finite_empty_induct
tff(fact_5453_Diff__single__insert,axiom,
    ! [A2: set_real,Xb: real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(minus_minus_set_real(A2,insert_real2(Xb,bot_bot_set_real))),B2)
     => aa_set_real_o(ord_less_eq_set_real(A2),insert_real2(Xb,B2)) ) ).

% Diff_single_insert
tff(fact_5454_Diff__single__insert,axiom,
    ! [A2: set_o,Xb: $o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(minus_minus_set_o(A2,insert_o2((Xb),bot_bot_set_o))),B2)
     => aa_set_o_o(ord_less_eq_set_o(A2),insert_o2((Xb),B2)) ) ).

% Diff_single_insert
tff(fact_5455_Diff__single__insert,axiom,
    ! [A2: set_int,Xb: int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A2,insert_int2(Xb,bot_bot_set_int))),B2)
     => aa_set_int_o(ord_less_eq_set_int(A2),insert_int2(Xb,B2)) ) ).

% Diff_single_insert
tff(fact_5456_Diff__single__insert,axiom,
    ! [A2: set_nat,Xb: nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,minus_minus_set_nat(A2,insert_nat2(Xb,bot_bot_set_nat))),B2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat2(Xb,B2)) ) ).

% Diff_single_insert
tff(fact_5457_subset__insert__iff,axiom,
    ! [A2: set_set_nat,Xb: set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),insert_set_nat2(Xb,B2))
    <=> $ite(aa_set_set_nat_o(member_set_nat2(Xb),A2),aa_set_set_nat_o(ord_le6893508408891458716et_nat(minus_2163939370556025621et_nat(A2,insert_set_nat2(Xb,bot_bot_set_set_nat))),B2),aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)) ) ).

% subset_insert_iff
tff(fact_5458_subset__insert__iff,axiom,
    ! [A2: set_real,Xb: real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),insert_real2(Xb,B2))
    <=> $ite(aa_set_real_o(member_real2(Xb),A2),aa_set_real_o(ord_less_eq_set_real(minus_minus_set_real(A2,insert_real2(Xb,bot_bot_set_real))),B2),aa_set_real_o(ord_less_eq_set_real(A2),B2)) ) ).

% subset_insert_iff
tff(fact_5459_subset__insert__iff,axiom,
    ! [A2: set_o,Xb: $o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),insert_o2((Xb),B2))
    <=> $ite(aa_set_o_o(member_o2((Xb)),A2),aa_set_o_o(ord_less_eq_set_o(minus_minus_set_o(A2,insert_o2((Xb),bot_bot_set_o))),B2),aa_set_o_o(ord_less_eq_set_o(A2),B2)) ) ).

% subset_insert_iff
tff(fact_5460_subset__insert__iff,axiom,
    ! [A2: set_int,Xb: int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),insert_int2(Xb,B2))
    <=> $ite(aa_set_int_o(member_int2(Xb),A2),aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A2,insert_int2(Xb,bot_bot_set_int))),B2),aa_set_int_o(ord_less_eq_set_int(A2),B2)) ) ).

% subset_insert_iff
tff(fact_5461_subset__insert__iff,axiom,
    ! [A2: set_nat,Xb: nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat2(Xb,B2))
    <=> $ite(aa_set_nat_o(member_nat2(Xb),A2),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,minus_minus_set_nat(A2,insert_nat2(Xb,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_5462_atLeast0__atMost__Suc,axiom,
    ! [Nb: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat2(aa_nat_nat(suc,Nb),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) ).

% atLeast0_atMost_Suc
tff(fact_5463_atLeastAtMost__insertL,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( insert_nat2(Mb,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),Nb)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb) ) ) ).

% atLeastAtMost_insertL
tff(fact_5464_atLeastAtMostSuc__conv,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(suc,Nb)) = insert_nat2(aa_nat_nat(suc,Nb),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_5465_Icc__eq__insert__lb__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb) = insert_nat2(Mb,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),Nb)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_5466_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_gs(G)))) = groups1935376822645274424al_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5467_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_gt(G)))) = groups5693394587270226106ex_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5468_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_gu(G)))) = groups4541462559716669496nt_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5469_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( groups2027974829824023292at_nat(G,minus_925952699566721837d_enat(A2,collec4429806609662206161d_enat(aTP_Lamm_gv(G)))) = groups2027974829824023292at_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5470_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_gw(G)))) = groups8097168146408367636l_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5471_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_gx(G)))) = groups5808333547571424918x_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5472_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_gy(G)))) = groups8778361861064173332t_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5473_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( groups4148127829035722712t_real(G,minus_925952699566721837d_enat(A2,collec4429806609662206161d_enat(aTP_Lamm_gz(G)))) = groups4148127829035722712t_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5474_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1932886352136224148al_int(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_ha(G)))) = groups1932886352136224148al_int(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5475_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_hb(G)))) = groups5690904116761175830ex_int(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5476_set__update__subset__insert,axiom,
    ! [Xs: list_real,Ia: nat,Xb: real] : aa_set_real_o(ord_less_eq_set_real(set_real2(list_update_real(Xs,Ia,Xb))),insert_real2(Xb,set_real2(Xs))) ).

% set_update_subset_insert
tff(fact_5477_set__update__subset__insert,axiom,
    ! [Xs: list_o,Ia: nat,Xb: $o] : aa_set_o_o(ord_less_eq_set_o(set_o2(list_update_o(Xs,Ia,(Xb)))),insert_o2((Xb),set_o2(Xs))) ).

% set_update_subset_insert
tff(fact_5478_set__update__subset__insert,axiom,
    ! [Xs: list_int,Ia: nat,Xb: int] : aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,Ia,Xb))),insert_int2(Xb,set_int2(Xs))) ).

% set_update_subset_insert
tff(fact_5479_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,Ia: nat,Xb: vEBT_VEBT] : ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,Ia,Xb)),insert_VEBT_VEBT2(Xb,set_VEBT_VEBT2(Xs))) ).

% set_update_subset_insert
tff(fact_5480_set__update__subset__insert,axiom,
    ! [Xs: list_nat,Ia: nat,Xb: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(list_update_nat(Xs,Ia,Xb))),insert_nat2(Xb,set_nat2(Xs))) ).

% set_update_subset_insert
tff(fact_5481_Compl__insert,axiom,
    ! [Xb: real,A2: set_real] : uminus612125837232591019t_real(insert_real2(Xb,A2)) = minus_minus_set_real(uminus612125837232591019t_real(A2),insert_real2(Xb,bot_bot_set_real)) ).

% Compl_insert
tff(fact_5482_Compl__insert,axiom,
    ! [Xb: $o,A2: set_o] : uminus_uminus_set_o(insert_o2((Xb),A2)) = minus_minus_set_o(uminus_uminus_set_o(A2),insert_o2((Xb),bot_bot_set_o)) ).

% Compl_insert
tff(fact_5483_Compl__insert,axiom,
    ! [Xb: int,A2: set_int] : uminus1532241313380277803et_int(insert_int2(Xb,A2)) = minus_minus_set_int(uminus1532241313380277803et_int(A2),insert_int2(Xb,bot_bot_set_int)) ).

% Compl_insert
tff(fact_5484_Compl__insert,axiom,
    ! [Xb: nat,A2: set_nat] : uminus5710092332889474511et_nat(insert_nat2(Xb,A2)) = minus_minus_set_nat(uminus5710092332889474511et_nat(A2),insert_nat2(Xb,bot_bot_set_nat)) ).

% Compl_insert
tff(fact_5485_sum__power__add,axiom,
    ! [Xb: int,Mb: nat,I5: set_nat] : aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_hc(Xb),Mb)),I5) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),Mb)),aa_set_nat_int(groups3539618377306564664at_int(power_power_int(Xb)),I5)) ).

% sum_power_add
tff(fact_5486_sum__power__add,axiom,
    ! [Xb: complex,Mb: nat,I5: set_nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_hd(Xb),Mb),I5) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),groups2073611262835488442omplex(power_power_complex(Xb),I5)) ).

% sum_power_add
tff(fact_5487_sum__power__add,axiom,
    ! [Xb: real,Mb: nat,I5: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_he(Xb),Mb)),I5) = aa_real_real(times_times_real(aa_nat_real(power_power_real(Xb),Mb)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(Xb)),I5)) ).

% sum_power_add
tff(fact_5488_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Nb),Mb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_hf(G),Nb),Mb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Nb),Mb)) ).

% sum.atLeastAtMost_rev
tff(fact_5489_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,Nb: nat,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Nb),Mb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_hg(G),Nb),Mb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Nb),Mb)) ).

% sum.atLeastAtMost_rev
tff(fact_5490_less__mask,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ) ).

% less_mask
tff(fact_5491_sum__pos2,axiom,
    ! [I5: set_real,Ia: real,F: fun_re6065469302365115155d_enat] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( aa_set_real_o(member_real2(Ia),I5)
       => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,Ia))
         => ( ! [I2: real] :
                ( aa_set_real_o(member_real2(I2),I5)
               => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,I2)) )
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups2800946370649118462d_enat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_5492_sum__pos2,axiom,
    ! [I5: set_o,Ia: $o,F: fun_o_Extended_enat] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( aa_set_o_o(member_o2((Ia)),I5)
       => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_o_Extended_enat(F,(Ia)))
         => ( ! [I2: $o] :
                ( aa_set_o_o(member_o2((I2)),I5)
               => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_o_Extended_enat(F,(I2))) )
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups7198740251461348360d_enat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_5493_sum__pos2,axiom,
    ! [I5: set_nat,Ia: nat,F: fun_na8797796813480665199d_enat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat2(Ia),I5)
       => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(F,Ia))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat2(I2),I5)
               => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(F,I2)) )
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups7108830773950497114d_enat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_5494_sum__pos2,axiom,
    ! [I5: set_complex,Ia: complex,F: fun_co6496365337113880337d_enat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(Ia),I5)
       => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(F,Ia))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I5)
               => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(F,I2)) )
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups1752964319039525884d_enat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_5495_sum__pos2,axiom,
    ! [I5: set_int,Ia: int,F: fun_in5914218760682845459d_enat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int2(Ia),I5)
       => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(F,Ia))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int2(I2),I5)
               => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(F,I2)) )
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups4225252721152677374d_enat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_5496_sum__pos2,axiom,
    ! [I5: set_Extended_enat,Ia: extended_enat,F: fun_Ex5852502975177633731d_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,I5)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Ia),I5)
       => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(F,Ia))
         => ( ! [I2: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(I2),I5)
               => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(F,I2)) )
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups2433450451889696826d_enat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_5497_sum__pos2,axiom,
    ! [I5: set_real,Ia: real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( aa_set_real_o(member_real2(Ia),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,Ia))
         => ( ! [I2: real] :
                ( aa_set_real_o(member_real2(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_5498_sum__pos2,axiom,
    ! [I5: set_o,Ia: $o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( aa_set_o_o(member_o2((Ia)),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_o_real(F,(Ia)))
         => ( ! [I2: $o] :
                ( aa_set_o_o(member_o2((I2)),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(I2))) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8691415230153176458o_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_5499_sum__pos2,axiom,
    ! [I5: set_complex,Ia: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(Ia),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,Ia))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_5500_sum__pos2,axiom,
    ! [I5: set_int,Ia: int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int2(Ia),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,Ia))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int2(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_5501_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_5502_sum__pos,axiom,
    ! [I5: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,I5)
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I2: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(I2),I5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Extended_enat_nat(F,I2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups2027974829824023292at_nat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_5503_sum__pos,axiom,
    ! [I5: set_real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real2(I2),I5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,I2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_5504_sum__pos,axiom,
    ! [I5: set_o,F: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( aa_set_o_o(member_o2((I2)),I5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(F,(I2))) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups8507830703676809646_o_nat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_5505_sum__pos,axiom,
    ! [I5: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int2(I2),I5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,I2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_5506_sum__pos,axiom,
    ! [I5: set_complex,F: fun_co6496365337113880337d_enat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I5)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(F,I2)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups1752964319039525884d_enat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_5507_sum__pos,axiom,
    ! [I5: set_Extended_enat,F: fun_Ex5852502975177633731d_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,I5)
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I2: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(I2),I5)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(F,I2)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups2433450451889696826d_enat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_5508_sum__pos,axiom,
    ! [I5: set_real,F: fun_re6065469302365115155d_enat] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real2(I2),I5)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,I2)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups2800946370649118462d_enat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_5509_sum__pos,axiom,
    ! [I5: set_o,F: fun_o_Extended_enat] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( aa_set_o_o(member_o2((I2)),I5)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_o_Extended_enat(F,(I2))) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups7198740251461348360d_enat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_5510_sum__pos,axiom,
    ! [I5: set_nat,F: fun_na8797796813480665199d_enat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat2(I2),I5)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(F,I2)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups7108830773950497114d_enat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_5511_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S3: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
       => ( ! [X4: real] :
              ( aa_set_real_o(member_real2(X4),minus_minus_set_real(T2,S3))
             => ( aa_real_nat(G,X4) = zero_zero_nat ) )
         => ( ! [X4: real] :
                ( aa_set_real_o(member_real2(X4),S3)
               => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
           => ( groups1935376822645274424al_nat(G,T2) = groups1935376822645274424al_nat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5512_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_o,S3: set_o,G: fun_o_nat,H: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(ord_less_eq_set_o(S3),T2)
       => ( ! [X4: $o] :
              ( aa_set_o_o(member_o2((X4)),minus_minus_set_o(T2,S3))
             => ( aa_o_nat(G,(X4)) = zero_zero_nat ) )
         => ( ! [X4: $o] :
                ( aa_set_o_o(member_o2((X4)),S3)
               => ( aa_o_nat(G,(X4)) = aa_o_nat(H,(X4)) ) )
           => ( groups8507830703676809646_o_nat(G,T2) = groups8507830703676809646_o_nat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5513_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( ! [X4: complex] :
                ( aa_set_complex_o(member_complex(X4),S3)
               => ( aa_complex_nat(G,X4) = aa_complex_nat(H,X4) ) )
           => ( groups5693394587270226106ex_nat(G,T2) = groups5693394587270226106ex_nat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5514_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
             => ( aa_int_nat(G,X4) = zero_zero_nat ) )
         => ( ! [X4: int] :
                ( aa_set_int_o(member_int2(X4),S3)
               => ( aa_int_nat(G,X4) = aa_int_nat(H,X4) ) )
           => ( groups4541462559716669496nt_nat(G,T2) = groups4541462559716669496nt_nat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5515_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,H: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(T2,S3))
             => ( aa_Extended_enat_nat(G,X4) = zero_zero_nat ) )
         => ( ! [X4: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),S3)
               => ( aa_Extended_enat_nat(G,X4) = aa_Extended_enat_nat(H,X4) ) )
           => ( groups2027974829824023292at_nat(G,T2) = groups2027974829824023292at_nat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5516_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S3: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
       => ( ! [X4: real] :
              ( aa_set_real_o(member_real2(X4),minus_minus_set_real(T2,S3))
             => ( aa_real_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: real] :
                ( aa_set_real_o(member_real2(X4),S3)
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups8097168146408367636l_real(G,T2) = groups8097168146408367636l_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5517_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_o,S3: set_o,G: fun_o_real,H: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(ord_less_eq_set_o(S3),T2)
       => ( ! [X4: $o] :
              ( aa_set_o_o(member_o2((X4)),minus_minus_set_o(T2,S3))
             => ( aa_o_real(G,(X4)) = zero_zero_real ) )
         => ( ! [X4: $o] :
                ( aa_set_o_o(member_o2((X4)),S3)
               => ( aa_o_real(G,(X4)) = aa_o_real(H,(X4)) ) )
           => ( groups8691415230153176458o_real(G,T2) = groups8691415230153176458o_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5518_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: complex] :
                ( aa_set_complex_o(member_complex(X4),S3)
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups5808333547571424918x_real(G,T2) = groups5808333547571424918x_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5519_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
             => ( aa_int_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: int] :
                ( aa_set_int_o(member_int2(X4),S3)
               => ( aa_int_real(G,X4) = aa_int_real(H,X4) ) )
           => ( groups8778361861064173332t_real(G,T2) = groups8778361861064173332t_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5520_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex1451620480498224165t_real,H: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(T2,S3))
             => ( aa_Ext2541572767299617110t_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),S3)
               => ( aa_Ext2541572767299617110t_real(G,X4) = aa_Ext2541572767299617110t_real(H,X4) ) )
           => ( groups4148127829035722712t_real(G,T2) = groups4148127829035722712t_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5521_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S3: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
       => ( ! [X4: real] :
              ( aa_set_real_o(member_real2(X4),minus_minus_set_real(T2,S3))
             => ( aa_real_nat(H,X4) = zero_zero_nat ) )
         => ( ! [X4: real] :
                ( aa_set_real_o(member_real2(X4),S3)
               => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
           => ( groups1935376822645274424al_nat(G,S3) = groups1935376822645274424al_nat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5522_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_o,S3: set_o,H: fun_o_nat,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(ord_less_eq_set_o(S3),T2)
       => ( ! [X4: $o] :
              ( aa_set_o_o(member_o2((X4)),minus_minus_set_o(T2,S3))
             => ( aa_o_nat(H,(X4)) = zero_zero_nat ) )
         => ( ! [X4: $o] :
                ( aa_set_o_o(member_o2((X4)),S3)
               => ( aa_o_nat(G,(X4)) = aa_o_nat(H,(X4)) ) )
           => ( groups8507830703676809646_o_nat(G,S3) = groups8507830703676809646_o_nat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5523_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S3: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_nat(H,X4) = zero_zero_nat ) )
         => ( ! [X4: complex] :
                ( aa_set_complex_o(member_complex(X4),S3)
               => ( aa_complex_nat(G,X4) = aa_complex_nat(H,X4) ) )
           => ( groups5693394587270226106ex_nat(G,S3) = groups5693394587270226106ex_nat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5524_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_int,S3: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
             => ( aa_int_nat(H,X4) = zero_zero_nat ) )
         => ( ! [X4: int] :
                ( aa_set_int_o(member_int2(X4),S3)
               => ( aa_int_nat(G,X4) = aa_int_nat(H,X4) ) )
           => ( groups4541462559716669496nt_nat(G,S3) = groups4541462559716669496nt_nat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5525_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_Extended_enat,S3: set_Extended_enat,H: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(T2,S3))
             => ( aa_Extended_enat_nat(H,X4) = zero_zero_nat ) )
         => ( ! [X4: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),S3)
               => ( aa_Extended_enat_nat(G,X4) = aa_Extended_enat_nat(H,X4) ) )
           => ( groups2027974829824023292at_nat(G,S3) = groups2027974829824023292at_nat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5526_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S3: set_real,H: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
       => ( ! [X4: real] :
              ( aa_set_real_o(member_real2(X4),minus_minus_set_real(T2,S3))
             => ( aa_real_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: real] :
                ( aa_set_real_o(member_real2(X4),S3)
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups8097168146408367636l_real(G,S3) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5527_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_o,S3: set_o,H: fun_o_real,G: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(ord_less_eq_set_o(S3),T2)
       => ( ! [X4: $o] :
              ( aa_set_o_o(member_o2((X4)),minus_minus_set_o(T2,S3))
             => ( aa_o_real(H,(X4)) = zero_zero_real ) )
         => ( ! [X4: $o] :
                ( aa_set_o_o(member_o2((X4)),S3)
               => ( aa_o_real(G,(X4)) = aa_o_real(H,(X4)) ) )
           => ( groups8691415230153176458o_real(G,S3) = groups8691415230153176458o_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5528_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: complex] :
                ( aa_set_complex_o(member_complex(X4),S3)
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups5808333547571424918x_real(G,S3) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5529_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_int,S3: set_int,H: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
             => ( aa_int_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: int] :
                ( aa_set_int_o(member_int2(X4),S3)
               => ( aa_int_real(G,X4) = aa_int_real(H,X4) ) )
           => ( groups8778361861064173332t_real(G,S3) = groups8778361861064173332t_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5530_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_Extended_enat,S3: set_Extended_enat,H: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(T2,S3))
             => ( aa_Ext2541572767299617110t_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),S3)
               => ( aa_Ext2541572767299617110t_real(G,X4) = aa_Ext2541572767299617110t_real(H,X4) ) )
           => ( groups4148127829035722712t_real(G,S3) = groups4148127829035722712t_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5531_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T2) = groups5693394587270226106ex_nat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5532_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
             => ( aa_int_nat(G,X4) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,T2) = groups4541462559716669496nt_nat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5533_sum_Omono__neutral__right,axiom,
    ! [T2: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(T2,S3))
             => ( aa_Extended_enat_nat(G,X4) = zero_zero_nat ) )
         => ( groups2027974829824023292at_nat(G,T2) = groups2027974829824023292at_nat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5534_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T2) = groups5808333547571424918x_real(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5535_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
             => ( aa_int_real(G,X4) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,T2) = groups8778361861064173332t_real(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5536_sum_Omono__neutral__right,axiom,
    ! [T2: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(T2,S3))
             => ( aa_Ext2541572767299617110t_real(G,X4) = zero_zero_real ) )
         => ( groups4148127829035722712t_real(G,T2) = groups4148127829035722712t_real(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5537_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T2) = groups5690904116761175830ex_int(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5538_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
             => ( aa_int_int(G,X4) = zero_zero_int ) )
         => ( groups4538972089207619220nt_int(G,T2) = groups4538972089207619220nt_int(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5539_sum_Omono__neutral__right,axiom,
    ! [T2: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(T2,S3))
             => ( aa_Extended_enat_int(G,X4) = zero_zero_int ) )
         => ( groups2025484359314973016at_int(G,T2) = groups2025484359314973016at_int(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5540_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
             => ( aa_int_complex(G,X4) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,T2) = groups3049146728041665814omplex(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5541_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S3) = groups5693394587270226106ex_nat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5542_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
             => ( aa_int_nat(G,X4) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,S3) = groups4541462559716669496nt_nat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5543_sum_Omono__neutral__left,axiom,
    ! [T2: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(T2,S3))
             => ( aa_Extended_enat_nat(G,X4) = zero_zero_nat ) )
         => ( groups2027974829824023292at_nat(G,S3) = groups2027974829824023292at_nat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5544_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S3) = groups5808333547571424918x_real(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5545_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
             => ( aa_int_real(G,X4) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,S3) = groups8778361861064173332t_real(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5546_sum_Omono__neutral__left,axiom,
    ! [T2: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(T2,S3))
             => ( aa_Ext2541572767299617110t_real(G,X4) = zero_zero_real ) )
         => ( groups4148127829035722712t_real(G,S3) = groups4148127829035722712t_real(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5547_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S3) = groups5690904116761175830ex_int(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5548_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
             => ( aa_int_int(G,X4) = zero_zero_int ) )
         => ( groups4538972089207619220nt_int(G,S3) = groups4538972089207619220nt_int(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5549_sum_Omono__neutral__left,axiom,
    ! [T2: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(T2,S3))
             => ( aa_Extended_enat_int(G,X4) = zero_zero_int ) )
         => ( groups2025484359314973016at_int(G,S3) = groups2025484359314973016at_int(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5550_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
             => ( aa_int_complex(G,X4) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,S3) = groups3049146728041665814omplex(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5551_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_real2(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real2(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_5552_sum_Osame__carrierI,axiom,
    ! [C4: set_o,A2: set_o,B2: set_o,G: fun_o_nat,H: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,C4)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),C4)
       => ( aa_set_o_o(ord_less_eq_set_o(B2),C4)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o2((A3)),minus_minus_set_o(C4,A2))
               => ( aa_o_nat(G,(A3)) = zero_zero_nat ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o2((B3)),minus_minus_set_o(C4,B2))
                 => ( aa_o_nat(H,(B3)) = zero_zero_nat ) )
             => ( ( groups8507830703676809646_o_nat(G,C4) = groups8507830703676809646_o_nat(H,C4) )
               => ( groups8507830703676809646_o_nat(G,A2) = groups8507830703676809646_o_nat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5553_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_5554_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,C4)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),C4)
       => ( aa_set_int_o(ord_less_eq_set_int(B2),C4)
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int2(A3),minus_minus_set_int(C4,A2))
               => ( aa_int_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int2(B3),minus_minus_set_int(C4,B2))
                 => ( aa_int_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups4541462559716669496nt_nat(G,C4) = groups4541462559716669496nt_nat(H,C4) )
               => ( groups4541462559716669496nt_nat(G,A2) = groups4541462559716669496nt_nat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5555_sum_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,H: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,C4)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),C4)
       => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B2),C4)
         => ( ! [A3: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2))
               => ( aa_Extended_enat_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: extended_enat] :
                  ( aa_set3337841544853333262enat_o(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B2))
                 => ( aa_Extended_enat_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups2027974829824023292at_nat(G,C4) = groups2027974829824023292at_nat(H,C4) )
               => ( groups2027974829824023292at_nat(G,A2) = groups2027974829824023292at_nat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5556_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_real2(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real2(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_5557_sum_Osame__carrierI,axiom,
    ! [C4: set_o,A2: set_o,B2: set_o,G: fun_o_real,H: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,C4)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),C4)
       => ( aa_set_o_o(ord_less_eq_set_o(B2),C4)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o2((A3)),minus_minus_set_o(C4,A2))
               => ( aa_o_real(G,(A3)) = zero_zero_real ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o2((B3)),minus_minus_set_o(C4,B2))
                 => ( aa_o_real(H,(B3)) = zero_zero_real ) )
             => ( ( groups8691415230153176458o_real(G,C4) = groups8691415230153176458o_real(H,C4) )
               => ( groups8691415230153176458o_real(G,A2) = groups8691415230153176458o_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5558_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_5559_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,C4)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),C4)
       => ( aa_set_int_o(ord_less_eq_set_int(B2),C4)
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int2(A3),minus_minus_set_int(C4,A2))
               => ( aa_int_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int2(B3),minus_minus_set_int(C4,B2))
                 => ( aa_int_real(H,B3) = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real(G,C4) = groups8778361861064173332t_real(H,C4) )
               => ( groups8778361861064173332t_real(G,A2) = groups8778361861064173332t_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5560_sum_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: fun_Ex1451620480498224165t_real,H: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,C4)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),C4)
       => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B2),C4)
         => ( ! [A3: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2))
               => ( aa_Ext2541572767299617110t_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: extended_enat] :
                  ( aa_set3337841544853333262enat_o(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B2))
                 => ( aa_Ext2541572767299617110t_real(H,B3) = zero_zero_real ) )
             => ( ( groups4148127829035722712t_real(G,C4) = groups4148127829035722712t_real(H,C4) )
               => ( groups4148127829035722712t_real(G,A2) = groups4148127829035722712t_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5561_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_real2(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real2(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_5562_sum_Osame__carrier,axiom,
    ! [C4: set_o,A2: set_o,B2: set_o,G: fun_o_nat,H: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,C4)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),C4)
       => ( aa_set_o_o(ord_less_eq_set_o(B2),C4)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o2((A3)),minus_minus_set_o(C4,A2))
               => ( aa_o_nat(G,(A3)) = zero_zero_nat ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o2((B3)),minus_minus_set_o(C4,B2))
                 => ( aa_o_nat(H,(B3)) = zero_zero_nat ) )
             => ( ( groups8507830703676809646_o_nat(G,A2) = groups8507830703676809646_o_nat(H,B2) )
              <=> ( groups8507830703676809646_o_nat(G,C4) = groups8507830703676809646_o_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5563_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_5564_sum_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,C4)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),C4)
       => ( aa_set_int_o(ord_less_eq_set_int(B2),C4)
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int2(A3),minus_minus_set_int(C4,A2))
               => ( aa_int_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int2(B3),minus_minus_set_int(C4,B2))
                 => ( aa_int_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups4541462559716669496nt_nat(G,A2) = groups4541462559716669496nt_nat(H,B2) )
              <=> ( groups4541462559716669496nt_nat(G,C4) = groups4541462559716669496nt_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5565_sum_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,H: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,C4)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),C4)
       => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B2),C4)
         => ( ! [A3: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2))
               => ( aa_Extended_enat_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: extended_enat] :
                  ( aa_set3337841544853333262enat_o(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B2))
                 => ( aa_Extended_enat_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups2027974829824023292at_nat(G,A2) = groups2027974829824023292at_nat(H,B2) )
              <=> ( groups2027974829824023292at_nat(G,C4) = groups2027974829824023292at_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5566_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_real2(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real2(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_5567_sum_Osame__carrier,axiom,
    ! [C4: set_o,A2: set_o,B2: set_o,G: fun_o_real,H: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,C4)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),C4)
       => ( aa_set_o_o(ord_less_eq_set_o(B2),C4)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o2((A3)),minus_minus_set_o(C4,A2))
               => ( aa_o_real(G,(A3)) = zero_zero_real ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o2((B3)),minus_minus_set_o(C4,B2))
                 => ( aa_o_real(H,(B3)) = zero_zero_real ) )
             => ( ( groups8691415230153176458o_real(G,A2) = groups8691415230153176458o_real(H,B2) )
              <=> ( groups8691415230153176458o_real(G,C4) = groups8691415230153176458o_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5568_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_5569_sum_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,C4)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),C4)
       => ( aa_set_int_o(ord_less_eq_set_int(B2),C4)
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int2(A3),minus_minus_set_int(C4,A2))
               => ( aa_int_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int2(B3),minus_minus_set_int(C4,B2))
                 => ( aa_int_real(H,B3) = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real(G,A2) = groups8778361861064173332t_real(H,B2) )
              <=> ( groups8778361861064173332t_real(G,C4) = groups8778361861064173332t_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5570_sum_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: fun_Ex1451620480498224165t_real,H: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,C4)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),C4)
       => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B2),C4)
         => ( ! [A3: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2))
               => ( aa_Ext2541572767299617110t_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: extended_enat] :
                  ( aa_set3337841544853333262enat_o(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B2))
                 => ( aa_Ext2541572767299617110t_real(H,B3) = zero_zero_real ) )
             => ( ( groups4148127829035722712t_real(G,A2) = groups4148127829035722712t_real(H,B2) )
              <=> ( groups4148127829035722712t_real(G,C4) = groups4148127829035722712t_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5571_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_5572_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_5573_sum_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B2),A2)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
       => ( groups2027974829824023292at_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups2027974829824023292at_nat(G,minus_925952699566721837d_enat(A2,B2))),groups2027974829824023292at_nat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_5574_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_5575_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_5576_sum_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B2),A2)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
       => ( groups2025484359314973016at_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups2025484359314973016at_int(G,minus_925952699566721837d_enat(A2,B2))),groups2025484359314973016at_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_5577_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(plus_plus_real(groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,B2))),groups5808333547571424918x_real(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_5578_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(plus_plus_real(groups8778361861064173332t_real(G,minus_minus_set_int(A2,B2))),groups8778361861064173332t_real(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_5579_sum_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B2),A2)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
       => ( groups4148127829035722712t_real(G,A2) = aa_real_real(plus_plus_real(groups4148127829035722712t_real(G,minus_925952699566721837d_enat(A2,B2))),groups4148127829035722712t_real(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_5580_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_co6496365337113880337d_enat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
       => ( groups1752964319039525884d_enat(G,A2) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups1752964319039525884d_enat(G,minus_811609699411566653omplex(A2,B2))),groups1752964319039525884d_enat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_5581_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_5582_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_5583_sum__diff,axiom,
    ! [A2: set_Extended_enat,B2: set_Extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B2),A2)
       => ( groups2025484359314973016at_int(F,minus_925952699566721837d_enat(A2,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups2025484359314973016at_int(F,A2)),groups2025484359314973016at_int(F,B2)) ) ) ) ).

% sum_diff
tff(fact_5584_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(minus_minus_real(groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B2)) ) ) ) ).

% sum_diff
tff(fact_5585_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(minus_minus_real(groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(F,B2)) ) ) ) ).

% sum_diff
tff(fact_5586_sum__diff,axiom,
    ! [A2: set_Extended_enat,B2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B2),A2)
       => ( groups4148127829035722712t_real(F,minus_925952699566721837d_enat(A2,B2)) = aa_real_real(minus_minus_real(groups4148127829035722712t_real(F,A2)),groups4148127829035722712t_real(F,B2)) ) ) ) ).

% sum_diff
tff(fact_5587_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)
       => ( aa_set_nat_int(groups3539618377306564664at_int(F),minus_minus_set_nat(A2,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups3539618377306564664at_int(F),A2)),aa_set_nat_int(groups3539618377306564664at_int(F),B2)) ) ) ) ).

% sum_diff
tff(fact_5588_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_5589_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)
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),minus_minus_set_nat(A2,B2)) = aa_real_real(minus_minus_real(aa_set_nat_real(groups6591440286371151544t_real(F),A2)),aa_set_nat_real(groups6591440286371151544t_real(F),B2)) ) ) ) ).

% sum_diff
tff(fact_5590_of__int__ceiling__le__add__one,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),aa_real_real(plus_plus_real(R2),one_one_real)) ).

% of_int_ceiling_le_add_one
tff(fact_5591_of__int__ceiling__diff__one__le,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),one_one_real)),R2) ).

% of_int_ceiling_diff_one_le
tff(fact_5592_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_5593_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_5594_sum__diff__nat,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B2),A2)
       => ( groups2027974829824023292at_nat(F,minus_925952699566721837d_enat(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups2027974829824023292at_nat(F,A2)),groups2027974829824023292at_nat(F,B2)) ) ) ) ).

% sum_diff_nat
tff(fact_5595_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_5596_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_int,K: nat] :
      ( ( aa_nat_int(F,zero_zero_nat) = zero_zero_int )
     => ( aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5597_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( groups2073611262835488442omplex(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),K)) = groups2073611262835488442omplex(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5598_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_na8797796813480665199d_enat,K: nat] :
      ( ( aa_nat_Extended_enat(F,zero_zero_nat) = zero_z5237406670263579293d_enat )
     => ( groups7108830773950497114d_enat(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),K)) = groups7108830773950497114d_enat(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5599_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),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5600_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5601_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5602_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_na8797796813480665199d_enat,Nb: nat] : groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))),aa_nat_Extended_enat(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5603_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5604_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5605_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,aa_nat_nat(suc,Nb))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5606_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_na8797796813480665199d_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(suc,Nb))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(G,aa_nat_nat(suc,Nb))),groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5607_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5608_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_nat_real(G,aa_nat_nat(suc,Nb))),aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5609_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Mb)),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5610_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_na8797796813480665199d_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(G,Mb)),groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5611_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5612_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_real_real(plus_plus_real(aa_nat_real(G,Mb)),aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5613_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,Mb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(bit_se1146084159140164899it_int(K),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(Mb),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_5614_bit__concat__bit__iff,axiom,
    ! [Mb: nat,K: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_concat_bit(Mb,K),L)),Nb)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
          & aa_nat_o(bit_se1146084159140164899it_int(K),Nb) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_nat_o(bit_se1146084159140164899it_int(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_5615_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)
       => ( ! [A6: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A6)
             => ( ( A6 != bot_bot_set_set_nat )
               => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A6),B2)
                 => ( ! [X: set_nat] :
                        ( aa_set_set_nat_o(member_set_nat2(X),A6)
                       => aa_set_set_nat_o(P,minus_2163939370556025621et_nat(A6,insert_set_nat2(X,bot_bot_set_set_nat))) )
                   => aa_set_set_nat_o(P,A6) ) ) ) )
         => aa_set_set_nat_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5616_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)
       => ( ! [A6: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A6)
             => ( ( A6 != bot_bot_set_complex )
               => ( aa_set_complex_o(ord_le211207098394363844omplex(A6),B2)
                 => ( ! [X: complex] :
                        ( aa_set_complex_o(member_complex(X),A6)
                       => aa_set_complex_o(P,minus_811609699411566653omplex(A6,insert_complex(X,bot_bot_set_complex))) )
                   => aa_set_complex_o(P,A6) ) ) ) )
         => aa_set_complex_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5617_finite__remove__induct,axiom,
    ! [B2: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
     => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
       => ( ! [A6: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A6)
             => ( ( A6 != bot_bo7653980558646680370d_enat )
               => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A6),B2)
                 => ( ! [X: extended_enat] :
                        ( aa_set3337841544853333262enat_o(member_Extended_enat(X),A6)
                       => aa_set3337841544853333262enat_o(P,minus_925952699566721837d_enat(A6,insert_Extended_enat(X,bot_bo7653980558646680370d_enat))) )
                   => aa_set3337841544853333262enat_o(P,A6) ) ) ) )
         => aa_set3337841544853333262enat_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5618_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)
       => ( ! [A6: set_real] :
              ( aa_set_real_o(finite_finite_real,A6)
             => ( ( A6 != bot_bot_set_real )
               => ( aa_set_real_o(ord_less_eq_set_real(A6),B2)
                 => ( ! [X: real] :
                        ( aa_set_real_o(member_real2(X),A6)
                       => aa_set_real_o(P,minus_minus_set_real(A6,insert_real2(X,bot_bot_set_real))) )
                   => aa_set_real_o(P,A6) ) ) ) )
         => aa_set_real_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5619_finite__remove__induct,axiom,
    ! [B2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,B2)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [A6: set_o] :
              ( aa_set_o_o(finite_finite_o,A6)
             => ( ( A6 != bot_bot_set_o )
               => ( aa_set_o_o(ord_less_eq_set_o(A6),B2)
                 => ( ! [X: $o] :
                        ( aa_set_o_o(member_o2((X)),A6)
                       => aa_set_o_o(P,minus_minus_set_o(A6,insert_o2((X),bot_bot_set_o))) )
                   => aa_set_o_o(P,A6) ) ) ) )
         => aa_set_o_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5620_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)
       => ( ! [A6: set_int] :
              ( aa_set_int_o(finite_finite_int,A6)
             => ( ( A6 != bot_bot_set_int )
               => ( aa_set_int_o(ord_less_eq_set_int(A6),B2)
                 => ( ! [X: int] :
                        ( aa_set_int_o(member_int2(X),A6)
                       => aa_set_int_o(P,minus_minus_set_int(A6,insert_int2(X,bot_bot_set_int))) )
                   => aa_set_int_o(P,A6) ) ) ) )
         => aa_set_int_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5621_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)
       => ( ! [A6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A6)
             => ( ( A6 != bot_bot_set_nat )
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A6),B2)
                 => ( ! [X: nat] :
                        ( aa_set_nat_o(member_nat2(X),A6)
                       => aa_set_nat_o(P,minus_minus_set_nat(A6,insert_nat2(X,bot_bot_set_nat))) )
                   => aa_set_nat_o(P,A6) ) ) ) )
         => aa_set_nat_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5622_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) )
       => ( ! [A6: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A6)
             => ( ( A6 != bot_bot_set_set_nat )
               => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A6),B2)
                 => ( ! [X: set_nat] :
                        ( aa_set_set_nat_o(member_set_nat2(X),A6)
                       => aa_set_set_nat_o(P,minus_2163939370556025621et_nat(A6,insert_set_nat2(X,bot_bot_set_set_nat))) )
                   => aa_set_set_nat_o(P,A6) ) ) ) )
         => aa_set_set_nat_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_5623_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) )
       => ( ! [A6: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A6)
             => ( ( A6 != bot_bot_set_complex )
               => ( aa_set_complex_o(ord_le211207098394363844omplex(A6),B2)
                 => ( ! [X: complex] :
                        ( aa_set_complex_o(member_complex(X),A6)
                       => aa_set_complex_o(P,minus_811609699411566653omplex(A6,insert_complex(X,bot_bot_set_complex))) )
                   => aa_set_complex_o(P,A6) ) ) ) )
         => aa_set_complex_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_5624_remove__induct,axiom,
    ! [P: fun_se8123728892507317397enat_o,B2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
     => ( ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
         => aa_set3337841544853333262enat_o(P,B2) )
       => ( ! [A6: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A6)
             => ( ( A6 != bot_bo7653980558646680370d_enat )
               => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A6),B2)
                 => ( ! [X: extended_enat] :
                        ( aa_set3337841544853333262enat_o(member_Extended_enat(X),A6)
                       => aa_set3337841544853333262enat_o(P,minus_925952699566721837d_enat(A6,insert_Extended_enat(X,bot_bo7653980558646680370d_enat))) )
                   => aa_set3337841544853333262enat_o(P,A6) ) ) ) )
         => aa_set3337841544853333262enat_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_5625_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) )
       => ( ! [A6: set_real] :
              ( aa_set_real_o(finite_finite_real,A6)
             => ( ( A6 != bot_bot_set_real )
               => ( aa_set_real_o(ord_less_eq_set_real(A6),B2)
                 => ( ! [X: real] :
                        ( aa_set_real_o(member_real2(X),A6)
                       => aa_set_real_o(P,minus_minus_set_real(A6,insert_real2(X,bot_bot_set_real))) )
                   => aa_set_real_o(P,A6) ) ) ) )
         => aa_set_real_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_5626_remove__induct,axiom,
    ! [P: fun_set_o_o,B2: set_o] :
      ( aa_set_o_o(P,bot_bot_set_o)
     => ( ( ~ aa_set_o_o(finite_finite_o,B2)
         => aa_set_o_o(P,B2) )
       => ( ! [A6: set_o] :
              ( aa_set_o_o(finite_finite_o,A6)
             => ( ( A6 != bot_bot_set_o )
               => ( aa_set_o_o(ord_less_eq_set_o(A6),B2)
                 => ( ! [X: $o] :
                        ( aa_set_o_o(member_o2((X)),A6)
                       => aa_set_o_o(P,minus_minus_set_o(A6,insert_o2((X),bot_bot_set_o))) )
                   => aa_set_o_o(P,A6) ) ) ) )
         => aa_set_o_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_5627_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) )
       => ( ! [A6: set_int] :
              ( aa_set_int_o(finite_finite_int,A6)
             => ( ( A6 != bot_bot_set_int )
               => ( aa_set_int_o(ord_less_eq_set_int(A6),B2)
                 => ( ! [X: int] :
                        ( aa_set_int_o(member_int2(X),A6)
                       => aa_set_int_o(P,minus_minus_set_int(A6,insert_int2(X,bot_bot_set_int))) )
                   => aa_set_int_o(P,A6) ) ) ) )
         => aa_set_int_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_5628_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) )
       => ( ! [A6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A6)
             => ( ( A6 != bot_bot_set_nat )
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A6),B2)
                 => ( ! [X: nat] :
                        ( aa_set_nat_o(member_nat2(X),A6)
                       => aa_set_nat_o(P,minus_minus_set_nat(A6,insert_nat2(X,bot_bot_set_nat))) )
                   => aa_set_nat_o(P,A6) ) ) ) )
         => aa_set_nat_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_5629_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)
               => ? [X: complex] :
                    ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(S3,T4))
                    & aa_set_complex_o(P,insert_complex(X,T4)) ) ) )
         => aa_set_complex_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_5630_finite__induct__select,axiom,
    ! [S3: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
     => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
       => ( ! [T4: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(ord_le2529575680413868914d_enat(T4),S3)
             => ( aa_set3337841544853333262enat_o(P,T4)
               => ? [X: extended_enat] :
                    ( aa_set3337841544853333262enat_o(member_Extended_enat(X),minus_925952699566721837d_enat(S3,T4))
                    & aa_set3337841544853333262enat_o(P,insert_Extended_enat(X,T4)) ) ) )
         => aa_set3337841544853333262enat_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_5631_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)
               => ? [X: real] :
                    ( aa_set_real_o(member_real2(X),minus_minus_set_real(S3,T4))
                    & aa_set_real_o(P,insert_real2(X,T4)) ) ) )
         => aa_set_real_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_5632_finite__induct__select,axiom,
    ! [S3: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,S3)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [T4: set_o] :
              ( aa_set_o_o(ord_less_set_o(T4),S3)
             => ( aa_set_o_o(P,T4)
               => ? [X: $o] :
                    ( aa_set_o_o(member_o2((X)),minus_minus_set_o(S3,T4))
                    & aa_set_o_o(P,insert_o2((X),T4)) ) ) )
         => aa_set_o_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_5633_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)
               => ? [X: int] :
                    ( aa_set_int_o(member_int2(X),minus_minus_set_int(S3,T4))
                    & aa_set_int_o(P,insert_int2(X,T4)) ) ) )
         => aa_set_int_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_5634_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)
               => ? [X: nat] :
                    ( aa_set_nat_o(member_nat2(X),minus_minus_set_nat(S3,T4))
                    & aa_set_nat_o(P,insert_nat2(X,T4)) ) ) )
         => aa_set_nat_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_5635_psubset__insert__iff,axiom,
    ! [A2: set_set_nat,Xb: set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A2),insert_set_nat2(Xb,B2))
    <=> $ite(
          aa_set_set_nat_o(member_set_nat2(Xb),B2),
          aa_set_set_nat_o(ord_less_set_set_nat(A2),B2),
          $ite(aa_set_set_nat_o(member_set_nat2(Xb),A2),aa_set_set_nat_o(ord_less_set_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat2(Xb,bot_bot_set_set_nat))),B2),aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_5636_psubset__insert__iff,axiom,
    ! [A2: set_real,Xb: real,B2: set_real] :
      ( aa_set_real_o(ord_less_set_real(A2),insert_real2(Xb,B2))
    <=> $ite(
          aa_set_real_o(member_real2(Xb),B2),
          aa_set_real_o(ord_less_set_real(A2),B2),
          $ite(aa_set_real_o(member_real2(Xb),A2),aa_set_real_o(ord_less_set_real(minus_minus_set_real(A2,insert_real2(Xb,bot_bot_set_real))),B2),aa_set_real_o(ord_less_eq_set_real(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_5637_psubset__insert__iff,axiom,
    ! [A2: set_o,Xb: $o,B2: set_o] :
      ( aa_set_o_o(ord_less_set_o(A2),insert_o2((Xb),B2))
    <=> $ite(
          aa_set_o_o(member_o2((Xb)),B2),
          aa_set_o_o(ord_less_set_o(A2),B2),
          $ite(aa_set_o_o(member_o2((Xb)),A2),aa_set_o_o(ord_less_set_o(minus_minus_set_o(A2,insert_o2((Xb),bot_bot_set_o))),B2),aa_set_o_o(ord_less_eq_set_o(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_5638_psubset__insert__iff,axiom,
    ! [A2: set_int,Xb: int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),insert_int2(Xb,B2))
    <=> $ite(
          aa_set_int_o(member_int2(Xb),B2),
          aa_set_int_o(ord_less_set_int(A2),B2),
          $ite(aa_set_int_o(member_int2(Xb),A2),aa_set_int_o(ord_less_set_int(minus_minus_set_int(A2,insert_int2(Xb,bot_bot_set_int))),B2),aa_set_int_o(ord_less_eq_set_int(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_5639_psubset__insert__iff,axiom,
    ! [A2: set_nat,Xb: nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),insert_nat2(Xb,B2))
    <=> $ite(
          aa_set_nat_o(member_nat2(Xb),B2),
          aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2),
          $ite(aa_set_nat_o(member_nat2(Xb),A2),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,minus_minus_set_nat(A2,insert_nat2(Xb,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_5640_set__replicate__Suc,axiom,
    ! [Nb: nat,Xb: vEBT_VEBT] : set_VEBT_VEBT2(replicate_VEBT_VEBT(aa_nat_nat(suc,Nb),Xb)) = insert_VEBT_VEBT2(Xb,bot_bo8194388402131092736T_VEBT) ).

% set_replicate_Suc
tff(fact_5641_set__replicate__Suc,axiom,
    ! [Nb: nat,Xb: real] : set_real2(replicate_real(aa_nat_nat(suc,Nb),Xb)) = insert_real2(Xb,bot_bot_set_real) ).

% set_replicate_Suc
tff(fact_5642_set__replicate__Suc,axiom,
    ! [Nb: nat,Xb: $o] : set_o2(replicate_o(aa_nat_nat(suc,Nb),(Xb))) = insert_o2((Xb),bot_bot_set_o) ).

% set_replicate_Suc
tff(fact_5643_set__replicate__Suc,axiom,
    ! [Nb: nat,Xb: nat] : set_nat2(replicate_nat(aa_nat_nat(suc,Nb),Xb)) = insert_nat2(Xb,bot_bot_set_nat) ).

% set_replicate_Suc
tff(fact_5644_set__replicate__Suc,axiom,
    ! [Nb: nat,Xb: int] : set_int2(replicate_int(aa_nat_nat(suc,Nb),Xb)) = insert_int2(Xb,bot_bot_set_int) ).

% set_replicate_Suc
tff(fact_5645_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xb: vEBT_VEBT] :
      set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Xb)) = $ite(Nb = zero_zero_nat,bot_bo8194388402131092736T_VEBT,insert_VEBT_VEBT2(Xb,bot_bo8194388402131092736T_VEBT)) ).

% set_replicate_conv_if
tff(fact_5646_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xb: real] :
      set_real2(replicate_real(Nb,Xb)) = $ite(Nb = zero_zero_nat,bot_bot_set_real,insert_real2(Xb,bot_bot_set_real)) ).

% set_replicate_conv_if
tff(fact_5647_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xb: $o] :
      set_o2(replicate_o(Nb,(Xb))) = $ite(Nb = zero_zero_nat,bot_bot_set_o,insert_o2((Xb),bot_bot_set_o)) ).

% set_replicate_conv_if
tff(fact_5648_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xb: nat] :
      set_nat2(replicate_nat(Nb,Xb)) = $ite(Nb = zero_zero_nat,bot_bot_set_nat,insert_nat2(Xb,bot_bot_set_nat)) ).

% set_replicate_conv_if
tff(fact_5649_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xb: int] :
      set_int2(replicate_int(Nb,Xb)) = $ite(Nb = zero_zero_nat,bot_bot_set_int,insert_int2(Xb,bot_bot_set_int)) ).

% set_replicate_conv_if
tff(fact_5650_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Mb)),aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_hh(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5651_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_na8797796813480665199d_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_nat_Extended_enat(G,aa_nat_nat(suc,Nb))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(G,Mb)),groups7108830773950497114d_enat(aTP_Lamm_hi(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5652_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_go(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5653_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_nat_real(G,Mb)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gp(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5654_sum__Suc__diff,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_hj(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,Nb))),aa_nat_int(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_5655_sum__Suc__diff,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hk(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_real_real(minus_minus_real(aa_nat_real(F,aa_nat_nat(suc,Nb))),aa_nat_real(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_5656_exp__eq__0__imp__not__bit,axiom,
    ! [Nb: nat,A: int] :
      ( ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb) = zero_zero_int )
     => ~ aa_nat_o(bit_se1146084159140164899it_int(A),Nb) ) ).

% exp_eq_0_imp_not_bit
tff(fact_5657_exp__eq__0__imp__not__bit,axiom,
    ! [Nb: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb) = zero_zero_nat )
     => ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb) ) ).

% exp_eq_0_imp_not_bit
tff(fact_5658_bit__Suc,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one)))),Nb) ) ).

% bit_Suc
tff(fact_5659_bit__Suc,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),Nb) ) ).

% bit_Suc
tff(fact_5660_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_re6065469302365115155d_enat] :
      ( 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_real2(B3),minus_minus_set_real(B2,A2))
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,B3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups2800946370649118462d_enat(F,A2)),groups2800946370649118462d_enat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_5661_sum__mono2,axiom,
    ! [B2: set_o,A2: set_o,F: fun_o_Extended_enat] :
      ( aa_set_o_o(finite_finite_o,B2)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
       => ( ! [B3: $o] :
              ( aa_set_o_o(member_o2((B3)),minus_minus_set_o(B2,A2))
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_o_Extended_enat(F,(B3))) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups7198740251461348360d_enat(F,A2)),groups7198740251461348360d_enat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_5662_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_co6496365337113880337d_enat] :
      ( 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_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(F,B3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups1752964319039525884d_enat(F,A2)),groups1752964319039525884d_enat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_5663_sum__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_in5914218760682845459d_enat] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
       => ( ! [B3: int] :
              ( aa_set_int_o(member_int2(B3),minus_minus_set_int(B2,A2))
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(F,B3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups4225252721152677374d_enat(F,A2)),groups4225252721152677374d_enat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_5664_sum__mono2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex5852502975177633731d_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B2)
       => ( ! [B3: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(B3),minus_925952699566721837d_enat(B2,A2))
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(F,B3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups2433450451889696826d_enat(F,A2)),groups2433450451889696826d_enat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_5665_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_real2(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_5666_sum__mono2,axiom,
    ! [B2: set_o,A2: set_o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,B2)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
       => ( ! [B3: $o] :
              ( aa_set_o_o(member_o2((B3)),minus_minus_set_o(B2,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(B3))) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8691415230153176458o_real(F,A2)),groups8691415230153176458o_real(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_5667_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_5668_sum__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
       => ( ! [B3: int] :
              ( aa_set_int_o(member_int2(B3),minus_minus_set_int(B2,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,B3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_5669_sum__mono2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B2)
       => ( ! [B3: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(B3),minus_925952699566721837d_enat(B2,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,B3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups4148127829035722712t_real(F,A2)),groups4148127829035722712t_real(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_5670_ceiling__correct,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb))),one_one_real)),Xb)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb))) ) ).

% ceiling_correct
tff(fact_5671_ceiling__unique,axiom,
    ! [Z: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,Z)),one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Z))
       => ( aa_real_int(archim7802044766580827645g_real,Xb) = Z ) ) ) ).

% ceiling_unique
tff(fact_5672_ceiling__eq__iff,axiom,
    ! [Xb: real,A: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,Xb) = A )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,A)),one_one_real)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,A)) ) ) ).

% ceiling_eq_iff
tff(fact_5673_ceiling__split,axiom,
    ! [P: fun_int_o,Ta: real] :
      ( aa_int_o(P,aa_real_int(archim7802044766580827645g_real,Ta))
    <=> ! [I4: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,I4)),one_one_real)),Ta)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),aa_int_real(ring_1_of_int_real,I4)) )
         => aa_int_o(P,I4) ) ) ).

% ceiling_split
tff(fact_5674_mult__ceiling__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(times_times_real(A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B))) ) ) ).

% mult_ceiling_le
tff(fact_5675_ceiling__less__iff,axiom,
    ! [Xb: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,Z)),one_one_real)) ) ).

% ceiling_less_iff
tff(fact_5676_le__ceiling__iff,axiom,
    ! [Z: int,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,Z)),one_one_real)),Xb) ) ).

% le_ceiling_iff
tff(fact_5677_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2)
             => ( aa_nat_o(bit_se1146084159140164899it_int(K),M2)
              <=> aa_nat_o(bit_se1146084159140164899it_int(K),N) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
              <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),N) ) ) ) ).

% int_bit_bound
tff(fact_5678_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_5679_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_na8797796813480665199d_enat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_5680_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_5681_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_5682_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_5683_sum__count__set,axiom,
    ! [Xs: list_Extended_enat,X5: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(set_Extended_enat2(Xs)),X5)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,X5)
       => ( groups2027974829824023292at_nat(count_101369445342291426d_enat(Xs),X5) = size_s3941691890525107288d_enat(Xs) ) ) ) ).

% sum_count_set
tff(fact_5684_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) = size_s6755466524823107622T_VEBT(Xs) ) ) ) ).

% sum_count_set
tff(fact_5685_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) = size_size_list_int(Xs) ) ) ) ).

% sum_count_set
tff(fact_5686_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) = size_size_list_nat(Xs) ) ) ) ).

% sum_count_set
tff(fact_5687_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_real2(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))
           => ( ! [X4: real] :
                  ( aa_set_real_o(member_real2(X4),B2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5688_sum__strict__mono2,axiom,
    ! [B2: set_o,A2: set_o,B: $o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,B2)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
       => ( aa_set_o_o(member_o2((B)),minus_minus_set_o(B2,A2))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_o_real(F,(B)))
           => ( ! [X4: $o] :
                  ( aa_set_o_o(member_o2((X4)),B2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(X4))) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8691415230153176458o_real(F,A2)),groups8691415230153176458o_real(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5689_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))
           => ( ! [X4: complex] :
                  ( aa_set_complex_o(member_complex(X4),B2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5690_sum__strict__mono2,axiom,
    ! [B2: set_int,A2: set_int,B: int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
       => ( aa_set_int_o(member_int2(B),minus_minus_set_int(B2,A2))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,B))
           => ( ! [X4: int] :
                  ( aa_set_int_o(member_int2(X4),B2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5691_sum__strict__mono2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,B: extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B2)
       => ( aa_set3337841544853333262enat_o(member_Extended_enat(B),minus_925952699566721837d_enat(B2,A2))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,B))
           => ( ! [X4: extended_enat] :
                  ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),B2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,X4)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups4148127829035722712t_real(F,A2)),groups4148127829035722712t_real(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5692_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_real2(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))
           => ( ! [X4: real] :
                  ( aa_set_real_o(member_real2(X4),B2)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5693_sum__strict__mono2,axiom,
    ! [B2: set_o,A2: set_o,B: $o,F: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,B2)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
       => ( aa_set_o_o(member_o2((B)),minus_minus_set_o(B2,A2))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(F,(B)))
           => ( ! [X4: $o] :
                  ( aa_set_o_o(member_o2((X4)),B2)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(F,(X4))) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups8507830703676809646_o_nat(F,A2)),groups8507830703676809646_o_nat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5694_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))
           => ( ! [X4: complex] :
                  ( aa_set_complex_o(member_complex(X4),B2)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X4)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5695_sum__strict__mono2,axiom,
    ! [B2: set_int,A2: set_int,B: int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
       => ( aa_set_int_o(member_int2(B),minus_minus_set_int(B2,A2))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,B))
           => ( ! [X4: int] :
                  ( aa_set_int_o(member_int2(X4),B2)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5696_sum__strict__mono2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,B: extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B2)
       => ( aa_set3337841544853333262enat_o(member_Extended_enat(B),minus_925952699566721837d_enat(B2,A2))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Extended_enat_nat(F,B))
           => ( ! [X4: extended_enat] :
                  ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),B2)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_Extended_enat_nat(F,X4)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups2027974829824023292at_nat(F,A2)),groups2027974829824023292at_nat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5697_ceiling__divide__upper,axiom,
    ! [Q2: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,P2),aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(P2),Q2)))),Q2)) ) ).

% ceiling_divide_upper
tff(fact_5698_Suc__mask__eq__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb) ).

% Suc_mask_eq_exp
tff(fact_5699_mask__nat__less__exp,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) ).

% mask_nat_less_exp
tff(fact_5700_sum__natinterval__diff,axiom,
    ! [F: fun_nat_complex,Mb: nat,Nb: nat] :
      groups2073611262835488442omplex(aTP_Lamm_hl(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),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,Nb),one_one_nat))),zero_zero_complex) ).

% sum_natinterval_diff
tff(fact_5701_sum__natinterval__diff,axiom,
    ! [F: fun_nat_int,Mb: nat,Nb: nat] :
      aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_hm(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Mb)),aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_int) ).

% sum_natinterval_diff
tff(fact_5702_sum__natinterval__diff,axiom,
    ! [F: fun_nat_real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hn(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_real_real(minus_minus_real(aa_nat_real(F,Mb)),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_real) ).

% sum_natinterval_diff
tff(fact_5703_sum__telescope_H_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_ho(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Nb)),aa_nat_int(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_5704_sum__telescope_H_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hp(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),Nb)) = aa_real_real(minus_minus_real(aa_nat_real(F,Nb)),aa_nat_real(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_5705_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_nat_int(bit_se2000444600071755411sk_int,Nb))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_5706_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_5707_even__bit__succ__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
     => ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),Nb)
      <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_5708_even__bit__succ__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),Nb)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_5709_odd__bit__iff__bit__pred,axiom,
    ! [A: int,Nb: nat] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
     => ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
      <=> ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),one_one_int)),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_5710_odd__bit__iff__bit__pred,axiom,
    ! [A: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),one_one_nat)),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_5711_set__decode__plus__power__2,axiom,
    ! [Nb: nat,Z: nat] :
      ( ~ aa_set_nat_o(member_nat2(Nb),aa_nat_set_nat(nat_set_decode,Z))
     => ( aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)),Z)) = insert_nat2(Nb,aa_nat_set_nat(nat_set_decode,Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_5712_ceiling__divide__lower,axiom,
    ! [Q2: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(P2),Q2)))),one_one_real)),Q2)),P2) ) ).

% ceiling_divide_lower
tff(fact_5713_ceiling__eq,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim7802044766580827645g_real,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_5714_mask__eq__sum__exp,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)),one_one_int) = aa_set_nat_int(groups3539618377306564664at_int(power_power_int(numeral_numeral_int(bit0(one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aj,Nb))) ).

% mask_eq_sum_exp
tff(fact_5715_mask__eq__sum__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)),one_one_nat) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aj,Nb))) ).

% mask_eq_sum_exp
tff(fact_5716_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),aa_set_nat_int(groups3539618377306564664at_int(power_power_int(Xb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),Mb)),aa_nat_int(power_power_int(Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_5717_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xb)),groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_5718_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_real_real(times_times_real(aa_real_real(minus_minus_real(one_one_real),Xb)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(Xb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) = aa_real_real(minus_minus_real(aa_nat_real(power_power_real(Xb),Mb)),aa_nat_real(power_power_real(Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_5719_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_hq(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) ).

% sum.in_pairs
tff(fact_5720_sum_Oin__pairs,axiom,
    ! [G: fun_na8797796813480665199d_enat,Mb: nat,Nb: nat] : groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))) = groups7108830773950497114d_enat(aTP_Lamm_hr(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) ).

% sum.in_pairs
tff(fact_5721_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hs(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) ).

% sum.in_pairs
tff(fact_5722_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ht(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) ).

% sum.in_pairs
tff(fact_5723_bit__sum__mult__2__cases,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(bit_se1146084159140164899it_int(A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B))),Nb)
      <=> $ite(Nb = zero_zero_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A),aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B)),Nb)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_5724_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B))),Nb)
      <=> $ite(Nb = zero_zero_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one))),B)),Nb)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_5725_bit__rec,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
    <=> $ite(Nb = zero_zero_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A),aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ).

% bit_rec
tff(fact_5726_bit__rec,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
    <=> $ite(Nb = zero_zero_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ).

% bit_rec
tff(fact_5727_mask__eq__sum__exp__nat,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aj,Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_5728_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% gauss_sum_nat
tff(fact_5729_take__bit__Suc__from__most,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),K) = aa_int_int(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(numeral_numeral_int(bit0(one))),Nb)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Nb)))),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ).

% take_bit_Suc_from_most
tff(fact_5730_sum__gp,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] :
      groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
        zero_zero_complex,
        $ite(Xb = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb)))),aa_complex_complex(minus_minus_complex(one_one_complex),Xb))) ) ).

% sum_gp
tff(fact_5731_sum__gp,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(power_power_real(Xb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
        zero_zero_real,
        $ite(Xb = one_one_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(power_power_real(Xb),Mb)),aa_nat_real(power_power_real(Xb),aa_nat_nat(suc,Nb)))),aa_real_real(minus_minus_real(one_one_real),Xb))) ) ).

% sum_gp
tff(fact_5732_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_set_nat_int(groups3539618377306564664at_int(semiri1314217659103216013at_int),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int))),numeral_numeral_int(bit0(one))) ).

% gauss_sum_from_Suc_0
tff(fact_5733_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% gauss_sum_from_Suc_0
tff(fact_5734_sum__gp__offset,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] :
      groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(Xb = one_one_complex,aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb))))),aa_complex_complex(minus_minus_complex(one_one_complex),Xb))) ).

% sum_gp_offset
tff(fact_5735_sum__gp__offset,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(power_power_real(Xb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(Xb = one_one_real,aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(power_power_real(Xb),Mb)),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(Xb),aa_nat_nat(suc,Nb))))),aa_real_real(minus_minus_real(one_one_real),Xb))) ).

% sum_gp_offset
tff(fact_5736_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),Nb))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)) ).

% double_gauss_sum_from_Suc_0
tff(fact_5737_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(bit0(one))),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),Nb))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb)),one_on7984719198319812577d_enat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_5738_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_set_nat_int(groups3539618377306564664at_int(semiri1314217659103216013at_int),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_5739_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_5740_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_set_nat_real(groups6591440286371151544t_real(semiri5074537144036343181t_real),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),Nb))) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)) ).

% double_gauss_sum_from_Suc_0
tff(fact_5741_arith__series,axiom,
    ! [A: int,D: int,Nb: nat] : aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_hu(A),D)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) = 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,Nb)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),D)))),numeral_numeral_int(bit0(one))) ).

% arith_series
tff(fact_5742_arith__series,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_hv(A),D)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),D)))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% arith_series
tff(fact_5743_gauss__sum,axiom,
    ! [Nb: nat] : aa_set_nat_int(groups3539618377306564664at_int(semiri1314217659103216013at_int),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int))),numeral_numeral_int(bit0(one))) ).

% gauss_sum
tff(fact_5744_gauss__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% gauss_sum
tff(fact_5745_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)) ).

% double_gauss_sum
tff(fact_5746_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(bit0(one))),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb)),one_on7984719198319812577d_enat)) ).

% double_gauss_sum
tff(fact_5747_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_set_nat_int(groups3539618377306564664at_int(semiri1314217659103216013at_int),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)) ).

% double_gauss_sum
tff(fact_5748_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)) ).

% double_gauss_sum
tff(fact_5749_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_set_nat_real(groups6591440286371151544t_real(semiri5074537144036343181t_real),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)) ).

% double_gauss_sum
tff(fact_5750_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_5751_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_5752_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Mb) = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_5753_numeral__le__real__of__nat__iff,axiom,
    ! [Nb: num,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Nb)),aa_nat_real(semiri5074537144036343181t_real,Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),Mb) ) ).

% numeral_le_real_of_nat_iff
tff(fact_5754_int__eq__iff__numeral,axiom,
    ! [Mb: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = numeral_numeral_int(V) )
    <=> ( Mb = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_5755_negative__eq__positive,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_int(semiri1314217659103216013at_int,Mb) )
    <=> ( ( Nb = zero_zero_nat )
        & ( Mb = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_5756_of__int__of__nat__eq,axiom,
    ! [Nb: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_real(semiri5074537144036343181t_real,Nb) ).

% of_int_of_nat_eq
tff(fact_5757_of__int__of__nat__eq,axiom,
    ! [Nb: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_int(semiri1314217659103216013at_int,Nb) ).

% of_int_of_nat_eq
tff(fact_5758_negative__zle,axiom,
    ! [Nb: 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,Nb))),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% negative_zle
tff(fact_5759_int__dvd__int__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ).

% int_dvd_int_iff
tff(fact_5760_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_5761_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_Extended_enat(semiri4216267220026989637d_enat,Mb) = zero_z5237406670263579293d_enat )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_5762_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_5763_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_5764_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_5765_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_5766_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_z5237406670263579293d_enat = aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_5767_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_5768_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_5769_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_5770_of__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% of_nat_0
tff(fact_5771_of__nat__0,axiom,
    aa_nat_Extended_enat(semiri4216267220026989637d_enat,zero_zero_nat) = zero_z5237406670263579293d_enat ).

% of_nat_0
tff(fact_5772_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_5773_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_5774_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_5775_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(semiri4216267220026989637d_enat,Mb)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_5776_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_5777_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_5778_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_5779_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_5780_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_5781_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_5782_real__of__nat__less__numeral__iff,axiom,
    ! [Nb: nat,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),numeral_numeral_real(W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,W2)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_5783_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(W2)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,W2)),Nb) ) ).

% numeral_less_real_of_nat_iff
tff(fact_5784_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Mb)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb)) ).

% of_nat_add
tff(fact_5785_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_add
tff(fact_5786_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_add
tff(fact_5787_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_add
tff(fact_5788_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ).

% of_nat_mult
tff(fact_5789_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Mb)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb)) ).

% of_nat_mult
tff(fact_5790_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_mult
tff(fact_5791_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mult
tff(fact_5792_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mult
tff(fact_5793_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_5794_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_5795_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_5796_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_5797_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_5798_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_5799_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_5800_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_5801_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Nb) = one_one_complex )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_5802_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Nb) = one_one_real )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_5803_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Nb) = one_one_int )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_5804_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Nb) = one_one_nat )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_5805_negative__zless,axiom,
    ! [Nb: nat,Mb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% negative_zless
tff(fact_5806_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_real(semiri5074537144036343181t_real,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_real(zero_n3304061248610475627l_real,(P)) ).

% of_nat_of_bool
tff(fact_5807_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_5808_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_5809_of__nat__sum,axiom,
    ! [F: fun_complex_nat,A2: set_complex] : aa_nat_complex(semiri8010041392384452111omplex,groups5693394587270226106ex_nat(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_hw(F),A2) ).

% of_nat_sum
tff(fact_5810_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_hx(F)),A2) ).

% of_nat_sum
tff(fact_5811_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_hy(F)),A2) ).

% of_nat_sum
tff(fact_5812_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hz(F)),A2) ).

% of_nat_sum
tff(fact_5813_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(semiri4216267220026989637d_enat,Mb)),zero_z5237406670263579293d_enat)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_5814_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_5815_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_5816_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_5817_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_5818_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(suc,Mb)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Mb)) ).

% of_nat_Suc
tff(fact_5819_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb)) = aa_real_real(plus_plus_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Mb)) ).

% of_nat_Suc
tff(fact_5820_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_5821_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_5822_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_5823_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_5824_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_5825_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_5826_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_5827_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_5828_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),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,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_5829_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W2)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_5830_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xb: 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,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_5831_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xb: 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,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_5832_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W2)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> 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)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_5833_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xb: 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,Xb))
    <=> 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)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_5834_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xb: 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,Xb))
    <=> 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)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_5835_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_5836_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),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,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_5837_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),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,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_5838_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_5839_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_5840_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_5841_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [Ia: num,Nb: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(numeral_numeral_real(Ia)),Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> 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,Ia)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5842_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [Ia: num,Nb: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(numeral_numeral_int(Ia)),Nb)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> 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,Ia)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5843_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [Ia: num,Nb: nat,Xb: 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,Ia)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> 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,Ia)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5844_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,Ia: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(power_power_real(numeral_numeral_real(Ia)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Ia)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5845_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,Ia: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(numeral_numeral_int(Ia)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Ia)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5846_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,Ia: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Ia)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Ia)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5847_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [Ia: num,Nb: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(numeral_numeral_real(Ia)),Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> 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,Ia)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5848_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [Ia: num,Nb: nat,Xb: 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,Ia)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> 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,Ia)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5849_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [Ia: num,Nb: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(numeral_numeral_int(Ia)),Nb)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> 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,Ia)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5850_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,Ia: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(power_power_real(numeral_numeral_real(Ia)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Ia)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5851_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,Ia: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Ia)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Ia)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5852_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,Ia: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(numeral_numeral_int(Ia)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Ia)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5853_int__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_hx(F)),A2) ).

% int_sum
tff(fact_5854_real__arch__simple,axiom,
    ! [Xb: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% real_arch_simple
tff(fact_5855_reals__Archimedean2,axiom,
    ! [Xb: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% reals_Archimedean2
tff(fact_5856_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Xb)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(semiri8010041392384452111omplex,Xb)) ).

% mult_of_nat_commute
tff(fact_5857_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Xb)),Y) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(Y),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Xb)) ).

% mult_of_nat_commute
tff(fact_5858_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: real] : aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Xb)),Y) = aa_real_real(times_times_real(Y),aa_nat_real(semiri5074537144036343181t_real,Xb)) ).

% mult_of_nat_commute
tff(fact_5859_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,Xb)) ).

% mult_of_nat_commute
tff(fact_5860_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,Xb)) ).

% mult_of_nat_commute
tff(fact_5861_nat__less__real__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ).

% nat_less_real_le
tff(fact_5862_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_5863_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_5864_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_5865_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(ring_1_of_int_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_5866_not__bit__Suc__0__Suc,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,Nb)) ).

% not_bit_Suc_0_Suc
tff(fact_5867_bit__Suc__0__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_5868_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb)) ).

% of_nat_0_le_iff
tff(fact_5869_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_0_le_iff
tff(fact_5870_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_0_le_iff
tff(fact_5871_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_0_le_iff
tff(fact_5872_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(semiri4216267220026989637d_enat,Mb)),zero_z5237406670263579293d_enat) ).

% of_nat_less_0_iff
tff(fact_5873_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_5874_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_5875_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_5876_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Nb)) != zero_zero_complex ).

% of_nat_neq_0
tff(fact_5877_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(suc,Nb)) != zero_z5237406670263579293d_enat ).

% of_nat_neq_0
tff(fact_5878_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_5879_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_5880_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_5881_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(semiri4216267220026989637d_enat,Mb)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_5882_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_5883_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_5884_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_5885_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(semiri4216267220026989637d_enat,Mb)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_5886_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_5887_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_5888_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_5889_of__nat__mono,axiom,
    ! [Ia: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Ia)),aa_nat_real(semiri5074537144036343181t_real,J)) ) ).

% of_nat_mono
tff(fact_5890_of__nat__mono,axiom,
    ! [Ia: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ia)),aa_nat_nat(semiri1316708129612266289at_nat,J)) ) ).

% of_nat_mono
tff(fact_5891_of__nat__mono,axiom,
    ! [Ia: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Ia)),aa_nat_int(semiri1314217659103216013at_int,J)) ) ).

% of_nat_mono
tff(fact_5892_nat__le__real__less,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_5893_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_5894_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_5895_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_5896_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_5897_zle__int,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% zle_int
tff(fact_5898_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_5899_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_5900_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_5901_zadd__int__left,axiom,
    ! [Mb: nat,Nb: nat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))),Z) ).

% zadd_int_left
tff(fact_5902_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_5903_int__plus,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% int_plus
tff(fact_5904_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_5905_not__int__zless__negative,axiom,
    ! [Nb: nat,Mb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Mb))) ).

% not_int_zless_negative
tff(fact_5906_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Xb)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Y)) ).

% of_nat_max
tff(fact_5907_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_real_real(ord_max_real(aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(semiri5074537144036343181t_real,Y)) ).

% of_nat_max
tff(fact_5908_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(semiri1314217659103216013at_int,Y)) ).

% of_nat_max
tff(fact_5909_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(semiri1316708129612266289at_nat,Y)) ).

% of_nat_max
tff(fact_5910_nat__less__as__int,axiom,
    ! [X: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_less_as_int
tff(fact_5911_nat__leq__as__int,axiom,
    ! [X: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_leq_as_int
tff(fact_5912_not__bit__Suc__0__numeral,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% not_bit_Suc_0_numeral
tff(fact_5913_ex__less__of__nat__mult,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),Xb)) ) ).

% ex_less_of_nat_mult
tff(fact_5914_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% of_nat_diff
tff(fact_5915_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ) ).

% of_nat_diff
tff(fact_5916_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ) ).

% of_nat_diff
tff(fact_5917_real__archimedian__rdiv__eq__0,axiom,
    ! [Xb: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( 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(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M)),Xb)),C) )
         => ( Xb = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_5918_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_5919_int__zle__neg,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)))
    <=> ( ( Nb = zero_zero_nat )
        & ( Mb = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_5920_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_5921_int__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ).

% int_Suc
tff(fact_5922_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_5923_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_5924_negative__zle__0,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),zero_zero_int) ).

% negative_zle_0
tff(fact_5925_simp__from__to,axiom,
    ! [Ia: int,J: int] :
      set_or1266510415728281911st_int(Ia,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),Ia),bot_bot_set_int,insert_int2(Ia,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ia),one_one_int),J))) ).

% simp_from_to
tff(fact_5926_sum__nth__roots,axiom,
    ! [Nb: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_bf,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_cb(Nb),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_5927_sum__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_bf,collect_complex(aTP_Lamm_cd(Nb))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_5928_mod__mult2__eq_H,axiom,
    ! [A: int,Mb: nat,Nb: 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,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_int(semiri1314217659103216013at_int,Nb)))),aa_int_int(modulo_modulo_int(A),aa_nat_int(semiri1314217659103216013at_int,Mb))) ).

% mod_mult2_eq'
tff(fact_5929_mod__mult2__eq_H,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Mb))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Mb))) ).

% mod_mult2_eq'
tff(fact_5930_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_5931_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_5932_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_5933_zmult__zless__mono2__lemma,axiom,
    ! [Ia: int,J: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ia),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)),Ia)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_5934_not__zle__0__negative,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))) ).

% not_zle_0_negative
tff(fact_5935_negD,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),zero_zero_int)
     => ? [N: nat] : Xb = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% negD
tff(fact_5936_negative__zless__0,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))),zero_zero_int) ).

% negative_zless_0
tff(fact_5937_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(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)))),E2) ) ).

% nat_approx_posE
tff(fact_5938_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(power_power_real(numeral_numeral_real(bit0(one))),Nb)) ).

% of_nat_less_two_power
tff(fact_5939_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)) ).

% of_nat_less_two_power
tff(fact_5940_inverse__of__nat__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( Nb != zero_zero_nat )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_5941_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_5942_zdiff__int__split,axiom,
    ! [P: fun_int_o,Xb: nat,Y: nat] :
      ( aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Y)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(semiri1314217659103216013at_int,Y))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_5943_double__arith__series,axiom,
    ! [A: complex,D: complex,Nb: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ia(A),D),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),A)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),D))) ).

% double_arith_series
tff(fact_5944_double__arith__series,axiom,
    ! [A: extended_enat,D: extended_enat,Nb: nat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(bit0(one))),groups7108830773950497114d_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_ib(A),D),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb)),one_on7984719198319812577d_enat)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(bit0(one))),A)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb)),D))) ).

% double_arith_series
tff(fact_5945_double__arith__series,axiom,
    ! [A: int,D: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_hu(A),D)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),D))) ).

% double_arith_series
tff(fact_5946_double__arith__series,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_hv(A),D)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),D))) ).

% double_arith_series
tff(fact_5947_double__arith__series,axiom,
    ! [A: real,D: real,Nb: nat] : aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ic(A),D)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),A)),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),D))) ).

% double_arith_series
tff(fact_5948_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_complex(semiri8010041392384452111omplex,Nb) = $ite(Nb = zero_zero_nat,zero_zero_complex,produc1917071388513777916omplex(aTP_Lamm_id,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% of_nat_code_if
tff(fact_5949_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb) = $ite(Nb = zero_zero_nat,zero_z5237406670263579293d_enat,produc2676513652042109336d_enat(aTP_Lamm_ie,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% of_nat_code_if
tff(fact_5950_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_real(semiri5074537144036343181t_real,Nb) = $ite(Nb = zero_zero_nat,zero_zero_real,aa_Pro6168199777849092593t_real(produc1703576794950452218t_real(aTP_Lamm_if),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% of_nat_code_if
tff(fact_5951_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_int(semiri1314217659103216013at_int,Nb) = $ite(Nb = zero_zero_nat,zero_zero_int,aa_Pro4240683299535347697at_int(produc6840382203811409530at_int(aTP_Lamm_ig),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% of_nat_code_if
tff(fact_5952_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_nat(semiri1316708129612266289at_nat,Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_ih),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% of_nat_code_if
tff(fact_5953_lemma__termdiff2,axiom,
    ! [H: complex,Z: complex,Nb: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),Nb)),aa_nat_complex(power_power_complex(Z),Nb))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) = aa_complex_complex(times_times_complex(H),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ij(H),Z),Nb),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_5954_lemma__termdiff2,axiom,
    ! [H: real,Z: real,Nb: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Z),H)),Nb)),aa_nat_real(power_power_real(Z),Nb))),H)),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(power_power_real(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) = aa_real_real(times_times_real(H),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_il(H),Z),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_5955_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K5: real,Nb: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(Z),H))),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Z),H)),Nb)),aa_nat_real(power_power_real(Z),Nb))),H)),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(power_power_real(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(power_power_real(K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_real_real(real_V7735802525324610683m_real,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_5956_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K5: real,Nb: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Z),H))),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),Nb)),aa_nat_complex(power_power_complex(Z),Nb))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(power_power_real(K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_complex_real(real_V1022390504157884413omplex,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_5957_pochhammer__double,axiom,
    ! [Z: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))),aa_nat_complex(comm_s2602460028002588243omplex(Z),Nb))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(bit0(one))))),Nb)) ).

% pochhammer_double
tff(fact_5958_pochhammer__double,axiom,
    ! [Z: real,Nb: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))),aa_nat_real(comm_s7457072308508201937r_real(Z),Nb))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(Z),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))),Nb)) ).

% pochhammer_double
tff(fact_5959_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,Nb) = aa_complex_complex(semiri2816024913162550771omplex(aTP_Lamm_im,Nb),zero_zero_complex) ).

% of_nat_code
tff(fact_5960_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb) = aa_Ext7969882492699152700d_enat(semiri8563196900006977889d_enat(aTP_Lamm_in,Nb),zero_z5237406670263579293d_enat) ).

% of_nat_code
tff(fact_5961_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_nat_fun_real_real(semiri7260567687927622513x_real(aTP_Lamm_io),Nb),zero_zero_real) ).

% of_nat_code
tff(fact_5962_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,Nb) = aa_int_int(semiri8420488043553186161ux_int(aTP_Lamm_ip,Nb),zero_zero_int) ).

% of_nat_code
tff(fact_5963_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_iq),Nb),zero_zero_nat) ).

% of_nat_code
tff(fact_5964_and__int_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(Xb),Xaa) = Y )
     => ( Y = $ite(
            ( aa_set_int_o(member_int2(Xb),insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(member_int2(Xaa),insert_int2(zero_zero_int,insert_int2(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,numeral_numeral_int(bit0(one))),Xb)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),Xaa) ))),
            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,numeral_numeral_int(bit0(one))),Xb)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),Xaa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(Xb),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(Xaa),numeral_numeral_int(bit0(one)))))) ) ) ) ).

% and_int.elims
tff(fact_5965_lessThan__iff,axiom,
    ! [Ia: $o,K: $o] :
      ( aa_set_o_o(member_o2((Ia)),set_ord_lessThan_o((K)))
    <=> aa_o_o(aa_o_fun_o_o(ord_less_o,(Ia)),(K)) ) ).

% lessThan_iff
tff(fact_5966_lessThan__iff,axiom,
    ! [Ia: set_nat,K: set_nat] :
      ( aa_set_set_nat_o(member_set_nat2(Ia),set_or890127255671739683et_nat(K))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Ia),K) ) ).

% lessThan_iff
tff(fact_5967_lessThan__iff,axiom,
    ! [Ia: extended_enat,K: extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(Ia),set_or8419480210114673929d_enat(K))
    <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Ia),K) ) ).

% lessThan_iff
tff(fact_5968_lessThan__iff,axiom,
    ! [Ia: real,K: real] :
      ( aa_set_real_o(member_real2(Ia),set_or5984915006950818249n_real(K))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ia),K) ) ).

% lessThan_iff
tff(fact_5969_lessThan__iff,axiom,
    ! [Ia: int,K: int] :
      ( aa_set_int_o(member_int2(Ia),set_ord_lessThan_int(K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Ia),K) ) ).

% lessThan_iff
tff(fact_5970_lessThan__iff,axiom,
    ! [Ia: nat,K: nat] :
      ( aa_set_nat_o(member_nat2(Ia),aa_nat_set_nat(set_ord_lessThan_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),K) ) ).

% lessThan_iff
tff(fact_5971_and__zero__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),zero_zero_int) = zero_zero_int ).

% and_zero_eq
tff(fact_5972_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_5973_zero__and__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),A) = zero_zero_int ).

% zero_and_eq
tff(fact_5974_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_5975_bit_Oconj__zero__left,axiom,
    ! [Xb: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),Xb) = zero_zero_int ).

% bit.conj_zero_left
tff(fact_5976_bit_Oconj__zero__right,axiom,
    ! [Xb: int] : aa_int_int(bit_se725231765392027082nd_int(Xb),zero_zero_int) = zero_zero_int ).

% bit.conj_zero_right
tff(fact_5977_lessThan__subset__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_or5984915006950818249n_real(Xb)),set_or5984915006950818249n_real(Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% lessThan_subset_iff
tff(fact_5978_lessThan__subset__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_ord_lessThan_int(Xb)),set_ord_lessThan_int(Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).

% lessThan_subset_iff
tff(fact_5979_lessThan__subset__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).

% lessThan_subset_iff
tff(fact_5980_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_5981_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_5982_pochhammer__0,axiom,
    ! [A: real] : aa_nat_real(comm_s7457072308508201937r_real(A),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_5983_pochhammer__0,axiom,
    ! [A: complex] : aa_nat_complex(comm_s2602460028002588243omplex(A),zero_zero_nat) = one_one_complex ).

% pochhammer_0
tff(fact_5984_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_5985_single__Diff__lessThan,axiom,
    ! [K: real] : minus_minus_set_real(insert_real2(K,bot_bot_set_real),set_or5984915006950818249n_real(K)) = insert_real2(K,bot_bot_set_real) ).

% single_Diff_lessThan
tff(fact_5986_single__Diff__lessThan,axiom,
    ! [K: $o] : minus_minus_set_o(insert_o2((K),bot_bot_set_o),set_ord_lessThan_o((K))) = insert_o2((K),bot_bot_set_o) ).

% single_Diff_lessThan
tff(fact_5987_single__Diff__lessThan,axiom,
    ! [K: int] : minus_minus_set_int(insert_int2(K,bot_bot_set_int),set_ord_lessThan_int(K)) = insert_int2(K,bot_bot_set_int) ).

% single_Diff_lessThan
tff(fact_5988_single__Diff__lessThan,axiom,
    ! [K: nat] : minus_minus_set_nat(insert_nat2(K,bot_bot_set_nat),aa_nat_set_nat(set_ord_lessThan_nat,K)) = insert_nat2(K,bot_bot_set_nat) ).

% single_Diff_lessThan
tff(fact_5989_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_int(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_5990_sum_OlessThan__Suc,axiom,
    ! [G: fun_na8797796813480665199d_enat,Nb: nat] : groups7108830773950497114d_enat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups7108830773950497114d_enat(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_Extended_enat(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_5991_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_nat(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_5992_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_real(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_5993_and__numerals_I5_J,axiom,
    ! [Xb: num] : aa_int_int(bit_se725231765392027082nd_int(numeral_numeral_int(bit0(Xb))),one_one_int) = zero_zero_int ).

% and_numerals(5)
tff(fact_5994_and__numerals_I5_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,bit0(Xb))),one_one_nat) = zero_zero_nat ).

% and_numerals(5)
tff(fact_5995_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),numeral_numeral_int(bit0(Y))) = zero_zero_int ).

% and_numerals(1)
tff(fact_5996_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,bit0(Y))) = zero_zero_nat ).

% and_numerals(1)
tff(fact_5997_and__numerals_I7_J,axiom,
    ! [Xb: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(numeral_numeral_int(aa_num_num(bit1,Xb))),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,numeral_numeral_int(bit0(one))),aa_int_int(bit_se725231765392027082nd_int(numeral_numeral_int(Xb)),numeral_numeral_int(Y)))) ).

% and_numerals(7)
tff(fact_5998_and__numerals_I7_J,axiom,
    ! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),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,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% and_numerals(7)
tff(fact_5999_int__int__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Mb = Nb ) ) ).

% int_int_eq
tff(fact_6000_lessThan__non__empty,axiom,
    ! [Xb: real] : set_or5984915006950818249n_real(Xb) != bot_bot_set_real ).

% lessThan_non_empty
tff(fact_6001_lessThan__non__empty,axiom,
    ! [Xb: int] : set_ord_lessThan_int(Xb) != bot_bot_set_int ).

% lessThan_non_empty
tff(fact_6002_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_ir,U)) ).

% lessThan_def
tff(fact_6003_lessThan__def,axiom,
    ! [U: extended_enat] : set_or8419480210114673929d_enat(U) = collec4429806609662206161d_enat(aa_Ext6368937152127170207enat_o(aTP_Lamm_is,U)) ).

% lessThan_def
tff(fact_6004_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_real_fun_real_o(aTP_Lamm_it,U)) ).

% lessThan_def
tff(fact_6005_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_iu,U)) ).

% lessThan_def
tff(fact_6006_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aj,U)) ).

% lessThan_def
tff(fact_6007_Iio__eq__empty__iff,axiom,
    ! [Nb: $o] :
      ( ( set_ord_lessThan_o((Nb)) = bot_bot_set_o )
    <=> ( (Nb)
      <=> bot_bot_o ) ) ).

% Iio_eq_empty_iff
tff(fact_6008_Iio__eq__empty__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,Nb) = bot_bot_set_nat )
    <=> ( Nb = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
tff(fact_6009_lessThan__strict__subset__iff,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le2529575680413868914d_enat(set_or8419480210114673929d_enat(Mb)),set_or8419480210114673929d_enat(Nb))
    <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_6010_lessThan__strict__subset__iff,axiom,
    ! [Mb: real,Nb: real] :
      ( aa_set_real_o(ord_less_set_real(set_or5984915006950818249n_real(Mb)),set_or5984915006950818249n_real(Nb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_6011_lessThan__strict__subset__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_set_int_o(ord_less_set_int(set_ord_lessThan_int(Mb)),set_ord_lessThan_int(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_6012_lessThan__strict__subset__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_6013_pochhammer__pos,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
     => 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,Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_6014_pochhammer__pos,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(comm_s7457072308508201937r_real(Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_6015_pochhammer__pos,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_6016_pochhammer__eq__0__mono,axiom,
    ! [A: real,Nb: nat,Mb: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),Nb) = zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),Mb) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_6017_pochhammer__eq__0__mono,axiom,
    ! [A: complex,Nb: nat,Mb: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),Mb) = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_6018_pochhammer__neq__0__mono,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),Mb) != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),Nb) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_6019_pochhammer__neq__0__mono,axiom,
    ! [A: complex,Mb: nat,Nb: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Mb) != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_6020_lessThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = insert_nat2(K,aa_nat_set_nat(set_ord_lessThan_nat,K)) ).

% lessThan_Suc
tff(fact_6021_lessThan__empty__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,Nb) = bot_bot_set_nat )
    <=> ( Nb = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_6022_pochhammer__nonneg,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(comm_s7457072308508201937r_real(Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_6023_pochhammer__nonneg,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
     => 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,Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_6024_pochhammer__nonneg,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_6025_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% pochhammer_0_left
tff(fact_6026_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_real(comm_s7457072308508201937r_real(zero_zero_real),Nb) = $ite(Nb = zero_zero_nat,one_one_real,zero_zero_real) ).

% pochhammer_0_left
tff(fact_6027_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),Nb) = $ite(Nb = zero_zero_nat,one_one_int,zero_zero_int) ).

% pochhammer_0_left
tff(fact_6028_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% pochhammer_0_left
tff(fact_6029_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_Extended_enat(comm_s3181272606743183617d_enat(zero_z5237406670263579293d_enat),Nb) = $ite(Nb = zero_zero_nat,one_on7984719198319812577d_enat,zero_z5237406670263579293d_enat) ).

% pochhammer_0_left
tff(fact_6030_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_iv(G),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nat_diff_reindex
tff(fact_6031_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_iw(G),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nat_diff_reindex
tff(fact_6032_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,P: fun_nat_nat,Nb: nat] :
      ( ! [X4: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Q,X4)),aa_nat_nat(P,X4))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(P),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_gn(Q),P)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ) ) ).

% sum_diff_distrib
tff(fact_6033_pochhammer__rec,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),Nb)) ).

% pochhammer_rec
tff(fact_6034_pochhammer__rec,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),Nb)) ).

% pochhammer_rec
tff(fact_6035_pochhammer__rec,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(A),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(A),one_one_real)),Nb)) ).

% pochhammer_rec
tff(fact_6036_pochhammer__rec,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),Nb)) ).

% pochhammer_rec
tff(fact_6037_pochhammer__rec,axiom,
    ! [A: extended_enat,Nb: nat] : aa_nat_Extended_enat(comm_s3181272606743183617d_enat(A),aa_nat_nat(suc,Nb)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),aa_nat_Extended_enat(comm_s3181272606743183617d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),one_on7984719198319812577d_enat)),Nb)) ).

% pochhammer_rec
tff(fact_6038_pochhammer__Suc,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(A),Nb)),aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,Nb))) ).

% pochhammer_Suc
tff(fact_6039_pochhammer__Suc,axiom,
    ! [A: extended_enat,Nb: nat] : aa_nat_Extended_enat(comm_s3181272606743183617d_enat(A),aa_nat_nat(suc,Nb)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(comm_s3181272606743183617d_enat(A),Nb)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb))) ).

% pochhammer_Suc
tff(fact_6040_pochhammer__Suc,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(aa_nat_real(comm_s7457072308508201937r_real(A),Nb)),aa_real_real(plus_plus_real(A),aa_nat_real(semiri5074537144036343181t_real,Nb))) ).

% pochhammer_Suc
tff(fact_6041_pochhammer__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(A),Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% pochhammer_Suc
tff(fact_6042_pochhammer__Suc,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) ).

% pochhammer_Suc
tff(fact_6043_pochhammer__rec_H,axiom,
    ! [Z: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,Nb))),aa_nat_complex(comm_s2602460028002588243omplex(Z),Nb)) ).

% pochhammer_rec'
tff(fact_6044_pochhammer__rec_H,axiom,
    ! [Z: extended_enat,Nb: nat] : aa_nat_Extended_enat(comm_s3181272606743183617d_enat(Z),aa_nat_nat(suc,Nb)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Z),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb))),aa_nat_Extended_enat(comm_s3181272606743183617d_enat(Z),Nb)) ).

% pochhammer_rec'
tff(fact_6045_pochhammer__rec_H,axiom,
    ! [Z: real,Nb: nat] : aa_nat_real(comm_s7457072308508201937r_real(Z),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(Z),aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(comm_s7457072308508201937r_real(Z),Nb)) ).

% pochhammer_rec'
tff(fact_6046_pochhammer__rec_H,axiom,
    ! [Z: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Nb))),aa_nat_int(comm_s4660882817536571857er_int(Z),Nb)) ).

% pochhammer_rec'
tff(fact_6047_pochhammer__rec_H,axiom,
    ! [Z: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb)) ).

% pochhammer_rec'
tff(fact_6048_pochhammer__eq__0__iff,axiom,
    ! [A: complex,Nb: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = zero_zero_complex )
    <=> ? [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),Nb)
          & ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K2)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_6049_pochhammer__eq__0__iff,axiom,
    ! [A: real,Nb: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),Nb) = zero_zero_real )
    <=> ? [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),Nb)
          & ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K2)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_6050_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K) = zero_zero_complex )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_6051_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K) = zero_zero_real )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_6052_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K) = zero_zero_int )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_6053_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K) = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_6054_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K) = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_6055_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_6056_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K) != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_6057_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K) != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_6058_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_6059_pochhammer__product_H,axiom,
    ! [Z: complex,Nb: nat,Mb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),Nb)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_6060_pochhammer__product_H,axiom,
    ! [Z: extended_enat,Nb: nat,Mb: nat] : aa_nat_Extended_enat(comm_s3181272606743183617d_enat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(comm_s3181272606743183617d_enat(Z),Nb)),aa_nat_Extended_enat(comm_s3181272606743183617d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Z),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_6061_pochhammer__product_H,axiom,
    ! [Z: real,Nb: nat,Mb: nat] : aa_nat_real(comm_s7457072308508201937r_real(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_real_real(times_times_real(aa_nat_real(comm_s7457072308508201937r_real(Z),Nb)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(Z),aa_nat_real(semiri5074537144036343181t_real,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_6062_pochhammer__product_H,axiom,
    ! [Z: int,Nb: nat,Mb: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),Nb)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_6063_pochhammer__product_H,axiom,
    ! [Z: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_6064_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_hh(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_6065_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_na8797796813480665199d_enat,Nb: nat] : groups7108830773950497114d_enat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(G,zero_zero_nat)),groups7108830773950497114d_enat(aTP_Lamm_hi(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_6066_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_go(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_6067_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gp(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_6068_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,Mb: nat] : aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_ix(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_6069_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_iy(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_real_real(minus_minus_real(aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_6070_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,Mb: nat] : aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_hj(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Mb)),aa_nat_int(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6071_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hk(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_real_real(minus_minus_real(aa_nat_real(F,Mb)),aa_nat_real(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6072_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_go(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.atLeast1_atMost_eq
tff(fact_6073_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gp(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.atLeast1_atMost_eq
tff(fact_6074_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(Z),Nb) = 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,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_6075_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_Extended_enat(comm_s3181272606743183617d_enat(Z),Nb) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(comm_s3181272606743183617d_enat(Z),Mb)),aa_nat_Extended_enat(comm_s3181272606743183617d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Z),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_6076_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_real(comm_s7457072308508201937r_real(Z),Nb) = aa_real_real(times_times_real(aa_nat_real(comm_s7457072308508201937r_real(Z),Mb)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(Z),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_6077_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_int(comm_s4660882817536571857er_int(Z),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),Mb)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_6078_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_6079_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: int,Nb: nat] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)) = zero_zero_int )
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(A),Nb) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_6080_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) = zero_zero_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_6081_lemma__termdiff1,axiom,
    ! [Z: int,H: int,Mb: nat] : aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_iz(Z),H),Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ja(Z),H),Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_6082_lemma__termdiff1,axiom,
    ! [Z: complex,H: complex,Mb: nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_jb(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_jc(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_6083_lemma__termdiff1,axiom,
    ! [Z: real,H: real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_jd(Z),H),Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_je(Z),H),Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_6084_power__diff__sumr2,axiom,
    ! [Xb: int,Nb: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),Nb)),aa_nat_int(power_power_int(Y),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_jf(Xb),Nb),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_6085_power__diff__sumr2,axiom,
    ! [Xb: complex,Nb: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(Y),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_jg(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_6086_power__diff__sumr2,axiom,
    ! [Xb: real,Nb: nat,Y: real] : aa_real_real(minus_minus_real(aa_nat_real(power_power_real(Xb),Nb)),aa_nat_real(power_power_real(Y),Nb)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(Xb),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_jh(Xb),Nb),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_6087_diff__power__eq__sum,axiom,
    ! [Xb: int,Nb: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),aa_nat_nat(suc,Nb))),aa_nat_int(power_power_int(Y),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_ji(Xb),Nb),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_6088_diff__power__eq__sum,axiom,
    ! [Xb: complex,Nb: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb))),aa_nat_complex(power_power_complex(Y),aa_nat_nat(suc,Nb))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_jj(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_6089_diff__power__eq__sum,axiom,
    ! [Xb: real,Nb: nat,Y: real] : aa_real_real(minus_minus_real(aa_nat_real(power_power_real(Xb),aa_nat_nat(suc,Nb))),aa_nat_real(power_power_real(Y),aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(Xb),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_jk(Xb),Nb),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_6090_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_6091_pochhammer__absorb__comp,axiom,
    ! [R2: real,K: nat] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(R2),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,R2)),K)) = aa_real_real(times_times_real(R2),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,R2)),one_one_real)),K)) ).

% pochhammer_absorb_comp
tff(fact_6092_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_6093_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_int,K5: int,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => 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,aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6094_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_nat,K5: nat,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => 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),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6095_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_real,K5: real,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => 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,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6096_one__diff__power__eq_H,axiom,
    ! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(Xb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_jl(Xb),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_6097_one__diff__power__eq_H,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xb)),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_jm(Xb),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_6098_one__diff__power__eq_H,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(Xb),Nb)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(one_one_real),Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_jn(Xb),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_6099_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_jo(F),G)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jp(F)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jq(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum_split_even_odd
tff(fact_6100_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_6101_pochhammer__minus_H,axiom,
    ! [B: real,K: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,B)),K)) ).

% pochhammer_minus'
tff(fact_6102_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_6103_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_6104_pochhammer__minus,axiom,
    ! [B: real,K: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,B)),K) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% pochhammer_minus
tff(fact_6105_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_6106_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K),L) = $ite(
        ( aa_set_int_o(member_int2(K),insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
        & aa_set_int_o(member_int2(L),insert_int2(zero_zero_int,insert_int2(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,numeral_numeral_int(bit0(one))),K)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(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,numeral_numeral_int(bit0(one))),K)
              & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(L),numeral_numeral_int(bit0(one)))))) ) ).

% and_int.simps
tff(fact_6107_norm__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Xb)),zero_zero_real)
    <=> ( Xb = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_6108_norm__le__zero__iff,axiom,
    ! [Xb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),zero_zero_real)
    <=> ( Xb = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_6109_zero__less__norm__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,Xb))
    <=> ( Xb != zero_zero_real ) ) ).

% zero_less_norm_iff
tff(fact_6110_zero__less__norm__iff,axiom,
    ! [Xb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,Xb))
    <=> ( Xb != zero_zero_complex ) ) ).

% zero_less_norm_iff
tff(fact_6111_norm__zero,axiom,
    aa_real_real(real_V7735802525324610683m_real,zero_zero_real) = zero_zero_real ).

% norm_zero
tff(fact_6112_norm__zero,axiom,
    aa_complex_real(real_V1022390504157884413omplex,zero_zero_complex) = zero_zero_real ).

% norm_zero
tff(fact_6113_norm__eq__zero,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(real_V7735802525324610683m_real,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_real ) ) ).

% norm_eq_zero
tff(fact_6114_norm__eq__zero,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_complex ) ) ).

% norm_eq_zero
tff(fact_6115_and__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,bit0(Xb))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_6116_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,bit0(Y))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_6117_and__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_6118_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_6119_Suc__0__and__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% Suc_0_and_eq
tff(fact_6120_and__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% and_Suc_0_eq
tff(fact_6121_and__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = $ite(
        ( ( Mb = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ).

% and_nat_unfold
tff(fact_6122_and__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)
            & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% and_nat_rec
tff(fact_6123_nonzero__norm__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ) ) ).

% nonzero_norm_divide
tff(fact_6124_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(divide_divide_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ) ) ).

% nonzero_norm_divide
tff(fact_6125_norm__diff__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(A),B))) ).

% norm_diff_ineq
tff(fact_6126_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))) ).

% norm_diff_ineq
tff(fact_6127_norm__uminus__minus,axiom,
    ! [Xb: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,Xb)),Y)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(Xb),Y)) ).

% norm_uminus_minus
tff(fact_6128_norm__uminus__minus,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),Y)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Y)) ).

% norm_uminus_minus
tff(fact_6129_power__eq__imp__eq__norm,axiom,
    ! [W2: real,Nb: nat,Z: real] :
      ( ( aa_nat_real(power_power_real(W2),Nb) = aa_nat_real(power_power_real(Z),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_real_real(real_V7735802525324610683m_real,W2) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_6130_power__eq__imp__eq__norm,axiom,
    ! [W2: complex,Nb: nat,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W2),Nb) = aa_nat_complex(power_power_complex(Z),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_complex_real(real_V1022390504157884413omplex,W2) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_6131_norm__triangle__lt,axiom,
    ! [Xb: real,Y: real,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,Xb)),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(plus_plus_real(Xb),Y))),E2) ) ).

% norm_triangle_lt
tff(fact_6132_norm__triangle__lt,axiom,
    ! [Xb: complex,Y: complex,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Xb)),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(Xb),Y))),E2) ) ).

% norm_triangle_lt
tff(fact_6133_norm__add__less,axiom,
    ! [Xb: real,R2: real,Y: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),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(plus_plus_real(Xb),Y))),aa_real_real(plus_plus_real(R2),S)) ) ) ).

% norm_add_less
tff(fact_6134_norm__add__less,axiom,
    ! [Xb: complex,R2: real,Y: complex,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),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(Xb),Y))),aa_real_real(plus_plus_real(R2),S)) ) ) ).

% norm_add_less
tff(fact_6135_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(A),B))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,A)),C)) ) ).

% norm_add_leD
tff(fact_6136_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,A)),C)) ) ).

% norm_add_leD
tff(fact_6137_norm__triangle__le,axiom,
    ! [Xb: real,Y: real,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,Xb)),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(plus_plus_real(Xb),Y))),E2) ) ).

% norm_triangle_le
tff(fact_6138_norm__triangle__le,axiom,
    ! [Xb: complex,Y: complex,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Xb)),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(Xb),Y))),E2) ) ).

% norm_triangle_le
tff(fact_6139_norm__triangle__ineq,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(Xb),Y))),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))) ).

% norm_triangle_ineq
tff(fact_6140_norm__triangle__ineq,axiom,
    ! [Xb: 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(Xb),Y))),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))) ).

% norm_triangle_ineq
tff(fact_6141_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(plus_plus_real(A),B))),aa_real_real(plus_plus_real(R2),S)) ) ) ).

% norm_triangle_mono
tff(fact_6142_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(plus_plus_real(R2),S)) ) ) ).

% norm_triangle_mono
tff(fact_6143_power__eq__1__iff,axiom,
    ! [W2: real,Nb: nat] :
      ( ( aa_nat_real(power_power_real(W2),Nb) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W2) = one_one_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_6144_power__eq__1__iff,axiom,
    ! [W2: complex,Nb: nat] :
      ( ( aa_nat_complex(power_power_complex(W2),Nb) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W2) = one_one_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_6145_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),aa_real_real(plus_plus_real(C),D)))),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(A),C))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_6146_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B)),aa_complex_complex(plus_plus_complex(C),D)))),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),C))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_6147_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_go(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),Mm)) ).

% sum_bounds_lt_plus1
tff(fact_6148_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_real,Mm: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gp(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),Mm)) ).

% sum_bounds_lt_plus1
tff(fact_6149_sumr__cos__zero__one,axiom,
    ! [Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jr),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_6150_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,Nb))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(bit0(one))))),aa_nat_nat(suc,Nb))) = groups6464643781859351333omplex(aTP_Lamm_js(Z),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_6151_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,Nb: nat] : aa_real_real(times_times_real(aa_nat_real(comm_s7457072308508201937r_real(Z),aa_nat_nat(suc,Nb))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(Z),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))),aa_nat_nat(suc,Nb))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_jt(Z)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_6152_pochhammer__code,axiom,
    ! [A: complex,Nb: nat] :
      aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_ju(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_complex)) ).

% pochhammer_code
tff(fact_6153_pochhammer__code,axiom,
    ! [A: extended_enat,Nb: nat] :
      aa_nat_Extended_enat(comm_s3181272606743183617d_enat(A),Nb) = $ite(Nb = zero_zero_nat,one_on7984719198319812577d_enat,aa_Ext7969882492699152700d_enat(set_fo2538466533108834004d_enat(aTP_Lamm_jv(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_on7984719198319812577d_enat)) ).

% pochhammer_code
tff(fact_6154_pochhammer__code,axiom,
    ! [A: real,Nb: nat] :
      aa_nat_real(comm_s7457072308508201937r_real(A),Nb) = $ite(Nb = zero_zero_nat,one_one_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_jw(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_real)) ).

% pochhammer_code
tff(fact_6155_pochhammer__code,axiom,
    ! [A: int,Nb: nat] :
      aa_nat_int(comm_s4660882817536571857er_int(A),Nb) = $ite(Nb = zero_zero_nat,one_one_int,aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_jx(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_int)) ).

% pochhammer_code
tff(fact_6156_pochhammer__code,axiom,
    ! [A: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_jy(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_nat)) ).

% pochhammer_code
tff(fact_6157_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,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(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,Nb)),one_one_int) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),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,Nb),one_one_nat))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_6158_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
     => aa_real_o(sums_real(aTP_Lamm_jz(Z)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(one_one_real),Z)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% geometric_deriv_sums
tff(fact_6159_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_ka(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,bit0(one))))) ) ).

% geometric_deriv_sums
tff(fact_6160_of__nat__prod,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_kb(F),A2) ).

% of_nat_prod
tff(fact_6161_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_hz(F)),A2) ).

% of_nat_prod
tff(fact_6162_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_hx(F)),A2) ).

% of_nat_prod
tff(fact_6163_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_hy(F)),A2) ).

% of_nat_prod
tff(fact_6164_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_real(ring_1_of_int_real,aa_set_nat_int(groups705719431365010083at_int(F),A2)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_dz(F)),A2) ).

% of_int_prod
tff(fact_6165_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_int(ring_1_of_int_int,aa_set_nat_int(groups705719431365010083at_int(F),A2)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_kc(F)),A2) ).

% of_int_prod
tff(fact_6166_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_kd(F),A2) ).

% of_int_prod
tff(fact_6167_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_ke(F),A2) ).

% of_int_prod
tff(fact_6168_prod__zero__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( groups861055069439313189ex_nat(F,A2) = zero_zero_nat )
      <=> ? [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
            & ( aa_complex_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_6169_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 )
      <=> ? [X3: int] :
            ( aa_set_int_o(member_int2(X3),A2)
            & ( aa_int_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_6170_prod__zero__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( groups2880970938130013265at_nat(F,A2) = zero_zero_nat )
      <=> ? [X3: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X3),A2)
            & ( aa_Extended_enat_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_6171_prod__zero__iff,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( aa_set_nat_real(groups129246275422532515t_real(F),A2) = zero_zero_real )
      <=> ? [X3: nat] :
            ( aa_set_nat_o(member_nat2(X3),A2)
            & ( aa_nat_real(F,X3) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_6172_prod__zero__iff,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( groups766887009212190081x_real(F,A2) = zero_zero_real )
      <=> ? [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
            & ( aa_complex_real(F,X3) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_6173_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 )
      <=> ? [X3: int] :
            ( aa_set_int_o(member_int2(X3),A2)
            & ( aa_int_real(F,X3) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_6174_prod__zero__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( groups97031904164794029t_real(F,A2) = zero_zero_real )
      <=> ? [X3: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X3),A2)
            & ( aa_Ext2541572767299617110t_real(F,X3) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_6175_prod__zero__iff,axiom,
    ! [A2: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( groups858564598930262913ex_int(F,A2) = zero_zero_int )
      <=> ? [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
            & ( aa_complex_int(F,X3) = zero_zero_int ) ) ) ) ).

% prod_zero_iff
tff(fact_6176_prod__zero__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( groups2878480467620962989at_int(F,A2) = zero_zero_int )
      <=> ? [X3: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X3),A2)
            & ( aa_Extended_enat_int(F,X3) = zero_zero_int ) ) ) ) ).

% prod_zero_iff
tff(fact_6177_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 )
      <=> ? [X3: nat] :
            ( aa_set_nat_o(member_nat2(X3),A2)
            & ( aa_nat_complex(F,X3) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_6178_prod_Oempty,axiom,
    ! [G: fun_real_nat] : groups4696554848551431203al_nat(G,bot_bot_set_real) = one_one_nat ).

% prod.empty
tff(fact_6179_prod_Oempty,axiom,
    ! [G: fun_real_int] : groups4694064378042380927al_int(G,bot_bot_set_real) = one_one_int ).

% prod.empty
tff(fact_6180_prod_Oempty,axiom,
    ! [G: fun_real_real] : groups1681761925125756287l_real(G,bot_bot_set_real) = one_one_real ).

% prod.empty
tff(fact_6181_prod_Oempty,axiom,
    ! [G: fun_real_complex] : groups713298508707869441omplex(G,bot_bot_set_real) = one_one_complex ).

% prod.empty
tff(fact_6182_prod_Oempty,axiom,
    ! [G: fun_o_nat] : groups3504817904513533571_o_nat(G,bot_bot_set_o) = one_one_nat ).

% prod.empty
tff(fact_6183_prod_Oempty,axiom,
    ! [G: fun_o_int] : groups3502327434004483295_o_int(G,bot_bot_set_o) = one_one_int ).

% prod.empty
tff(fact_6184_prod_Oempty,axiom,
    ! [G: fun_o_real] : groups234877984723959775o_real(G,bot_bot_set_o) = one_one_real ).

% prod.empty
tff(fact_6185_prod_Oempty,axiom,
    ! [G: fun_o_complex] : groups4859619685533338977omplex(G,bot_bot_set_o) = one_one_complex ).

% prod.empty
tff(fact_6186_prod_Oempty,axiom,
    ! [G: fun_nat_real] : aa_set_nat_real(groups129246275422532515t_real(G),bot_bot_set_nat) = one_one_real ).

% prod.empty
tff(fact_6187_prod_Oempty,axiom,
    ! [G: fun_nat_complex] : groups6464643781859351333omplex(G,bot_bot_set_nat) = one_one_complex ).

% prod.empty
tff(fact_6188_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_6189_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_real(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_6190_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_complex,Nb: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_complex(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_6191_prod_OlessThan__Suc,axiom,
    ! [G: fun_na8797796813480665199d_enat,Nb: nat] : groups7961826882256487087d_enat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(groups7961826882256487087d_enat(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_Extended_enat(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_6192_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_int(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_6193_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_nat(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_6194_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_real,Xb: real] :
      ( aa_real_o(sums_real(aTP_Lamm_ev(A)),Xb)
    <=> ( aa_nat_real(A,zero_zero_nat) = Xb ) ) ).

% powser_sums_zero_iff
tff(fact_6195_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_complex,Xb: complex] :
      ( aa_complex_o(sums_complex(aTP_Lamm_eu(A)),Xb)
    <=> ( aa_nat_complex(A,zero_zero_nat) = Xb ) ) ).

% powser_sums_zero_iff
tff(fact_6196_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_real,aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_6197_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_complex,Mb: nat,Nb: nat] :
      groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_complex,aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_6198_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_na8797796813480665199d_enat,Mb: nat,Nb: nat] :
      groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_on7984719198319812577d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_nat_Extended_enat(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_6199_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] :
      aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_6200_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] :
      aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_6201_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real2(I2),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I2)),aa_real_real(G,I2)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A2)),groups1681761925125756287l_real(G,A2)) ) ).

% prod_mono
tff(fact_6202_prod__mono,axiom,
    ! [A2: set_o,F: fun_o_real,G: fun_o_real] :
      ( ! [I2: $o] :
          ( aa_set_o_o(member_o2((I2)),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(I2)))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(F,(I2))),aa_o_real(G,(I2))) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups234877984723959775o_real(F,A2)),groups234877984723959775o_real(G,A2)) ) ).

% prod_mono
tff(fact_6203_prod__mono,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat2(I2),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,I2)),aa_nat_real(G,I2)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F),A2)),aa_set_nat_real(groups129246275422532515t_real(G),A2)) ) ).

% prod_mono
tff(fact_6204_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int2(I2),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I2)),aa_int_real(G,I2)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A2)),groups2316167850115554303t_real(G,A2)) ) ).

% prod_mono
tff(fact_6205_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real2(I2),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2)) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4696554848551431203al_nat(F,A2)),groups4696554848551431203al_nat(G,A2)) ) ).

% prod_mono
tff(fact_6206_prod__mono,axiom,
    ! [A2: set_o,F: fun_o_nat,G: fun_o_nat] :
      ( ! [I2: $o] :
          ( aa_set_o_o(member_o2((I2)),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(F,(I2)))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(I2))),aa_o_nat(G,(I2))) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups3504817904513533571_o_nat(F,A2)),groups3504817904513533571_o_nat(G,A2)) ) ).

% prod_mono
tff(fact_6207_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int2(I2),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2)) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1707563613775114915nt_nat(F,A2)),groups1707563613775114915nt_nat(G,A2)) ) ).

% prod_mono
tff(fact_6208_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real2(I2),A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,I2))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I2)),aa_real_int(G,I2)) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4694064378042380927al_int(F,A2)),groups4694064378042380927al_int(G,A2)) ) ).

% prod_mono
tff(fact_6209_prod__mono,axiom,
    ! [A2: set_o,F: fun_o_int,G: fun_o_int] :
      ( ! [I2: $o] :
          ( aa_set_o_o(member_o2((I2)),A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(F,(I2)))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,(I2))),aa_o_int(G,(I2))) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3502327434004483295_o_int(F,A2)),groups3502327434004483295_o_int(G,A2)) ) ).

% prod_mono
tff(fact_6210_prod__mono,axiom,
    ! [A2: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat2(I2),A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,I2))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I2)),aa_nat_int(G,I2)) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_nat_int(groups705719431365010083at_int(F),A2)),aa_set_nat_int(groups705719431365010083at_int(G),A2)) ) ).

% prod_mono
tff(fact_6211_prod__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat2(X4),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_set_nat_int(groups705719431365010083at_int(F),A2)) ) ).

% prod_nonneg
tff(fact_6212_prod__nonneg,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int2(X4),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,X4)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F,A2)) ) ).

% prod_nonneg
tff(fact_6213_prod__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat2(X4),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) ) ).

% prod_nonneg
tff(fact_6214_prod__pos,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat2(X4),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,X4)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_set_nat_int(groups705719431365010083at_int(F),A2)) ) ).

% prod_pos
tff(fact_6215_prod__pos,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int2(X4),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(F,X4)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(F,A2)) ) ).

% prod_pos
tff(fact_6216_prod__pos,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat2(X4),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) ) ).

% prod_pos
tff(fact_6217_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( aa_set_real_o(member_real2(X4),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,X4)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups1681761925125756287l_real(F,A2)) ) ).

% prod_ge_1
tff(fact_6218_prod__ge__1,axiom,
    ! [A2: set_o,F: fun_o_real] :
      ( ! [X4: $o] :
          ( aa_set_o_o(member_o2((X4)),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_o_real(F,(X4))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups234877984723959775o_real(F,A2)) ) ).

% prod_ge_1
tff(fact_6219_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat2(X4),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(F,X4)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),A2)) ) ).

% prod_ge_1
tff(fact_6220_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int2(X4),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,X4)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups2316167850115554303t_real(F,A2)) ) ).

% prod_ge_1
tff(fact_6221_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( aa_set_real_o(member_real2(X4),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_real_nat(F,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups4696554848551431203al_nat(F,A2)) ) ).

% prod_ge_1
tff(fact_6222_prod__ge__1,axiom,
    ! [A2: set_o,F: fun_o_nat] :
      ( ! [X4: $o] :
          ( aa_set_o_o(member_o2((X4)),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_o_nat(F,(X4))) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups3504817904513533571_o_nat(F,A2)) ) ).

% prod_ge_1
tff(fact_6223_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int2(X4),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_int_nat(F,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups1707563613775114915nt_nat(F,A2)) ) ).

% prod_ge_1
tff(fact_6224_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_int] :
      ( ! [X4: real] :
          ( aa_set_real_o(member_real2(X4),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(F,X4)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),groups4694064378042380927al_int(F,A2)) ) ).

% prod_ge_1
tff(fact_6225_prod__ge__1,axiom,
    ! [A2: set_o,F: fun_o_int] :
      ( ! [X4: $o] :
          ( aa_set_o_o(member_o2((X4)),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_o_int(F,(X4))) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),groups3502327434004483295_o_int(F,A2)) ) ).

% prod_ge_1
tff(fact_6226_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat2(X4),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(F,X4)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_set_nat_int(groups705719431365010083at_int(F),A2)) ) ).

% prod_ge_1
tff(fact_6227_prod__zero,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ? [X: complex] :
            ( aa_set_complex_o(member_complex(X),A2)
            & ( aa_complex_nat(F,X) = zero_zero_nat ) )
       => ( groups861055069439313189ex_nat(F,A2) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_6228_prod__zero,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ? [X: int] :
            ( aa_set_int_o(member_int2(X),A2)
            & ( aa_int_nat(F,X) = zero_zero_nat ) )
       => ( groups1707563613775114915nt_nat(F,A2) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_6229_prod__zero,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ? [X: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X),A2)
            & ( aa_Extended_enat_nat(F,X) = zero_zero_nat ) )
       => ( groups2880970938130013265at_nat(F,A2) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_6230_prod__zero,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ? [X: nat] :
            ( aa_set_nat_o(member_nat2(X),A2)
            & ( aa_nat_real(F,X) = zero_zero_real ) )
       => ( aa_set_nat_real(groups129246275422532515t_real(F),A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_6231_prod__zero,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ? [X: complex] :
            ( aa_set_complex_o(member_complex(X),A2)
            & ( aa_complex_real(F,X) = zero_zero_real ) )
       => ( groups766887009212190081x_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_6232_prod__zero,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ? [X: int] :
            ( aa_set_int_o(member_int2(X),A2)
            & ( aa_int_real(F,X) = zero_zero_real ) )
       => ( groups2316167850115554303t_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_6233_prod__zero,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ? [X: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X),A2)
            & ( aa_Ext2541572767299617110t_real(F,X) = zero_zero_real ) )
       => ( groups97031904164794029t_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_6234_prod__zero,axiom,
    ! [A2: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ? [X: complex] :
            ( aa_set_complex_o(member_complex(X),A2)
            & ( aa_complex_int(F,X) = zero_zero_int ) )
       => ( groups858564598930262913ex_int(F,A2) = zero_zero_int ) ) ) ).

% prod_zero
tff(fact_6235_prod__zero,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ? [X: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X),A2)
            & ( aa_Extended_enat_int(F,X) = zero_zero_int ) )
       => ( groups2878480467620962989at_int(F,A2) = zero_zero_int ) ) ) ).

% prod_zero
tff(fact_6236_prod__zero,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ? [X: nat] :
            ( aa_set_nat_o(member_nat2(X),A2)
            & ( aa_nat_complex(F,X) = zero_zero_complex ) )
       => ( groups6464643781859351333omplex(F,A2) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_6237_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_hh(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_6238_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_go(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_6239_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_int,Mb: nat,K: nat,Nb: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(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,Nb),K))) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_kf(G),K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_6240_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,K: nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(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,Nb),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gq(G),K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_6241_prod__le__1,axiom,
    ! [A2: set_real,F: fun_re6065469302365115155d_enat] :
      ( ! [X4: real] :
          ( aa_set_real_o(member_real2(X4),A2)
         => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,X4))
            & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_rea1194391308913012860d_enat(F,X4)),one_on7984719198319812577d_enat) ) )
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups7973222482632965587d_enat(F,A2)),one_on7984719198319812577d_enat) ) ).

% prod_le_1
tff(fact_6242_prod__le__1,axiom,
    ! [A2: set_o,F: fun_o_Extended_enat] :
      ( ! [X4: $o] :
          ( aa_set_o_o(member_o2((X4)),A2)
         => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_o_Extended_enat(F,(X4)))
            & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_o_Extended_enat(F,(X4))),one_on7984719198319812577d_enat) ) )
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups783334030178737011d_enat(F,A2)),one_on7984719198319812577d_enat) ) ).

% prod_le_1
tff(fact_6243_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_na8797796813480665199d_enat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat2(X4),A2)
         => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(F,X4))
            & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,X4)),one_on7984719198319812577d_enat) ) )
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups7961826882256487087d_enat(F,A2)),one_on7984719198319812577d_enat) ) ).

% prod_le_1
tff(fact_6244_prod__le__1,axiom,
    ! [A2: set_int,F: fun_in5914218760682845459d_enat] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int2(X4),A2)
         => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(F,X4))
            & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_int_Extended_enat(F,X4)),one_on7984719198319812577d_enat) ) )
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,groups5078248829458667347d_enat(F,A2)),one_on7984719198319812577d_enat) ) ).

% prod_le_1
tff(fact_6245_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( aa_set_real_o(member_real2(X4),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X4)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A2)),one_one_real) ) ).

% prod_le_1
tff(fact_6246_prod__le__1,axiom,
    ! [A2: set_o,F: fun_o_real] :
      ( ! [X4: $o] :
          ( aa_set_o_o(member_o2((X4)),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(X4)))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(F,(X4))),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups234877984723959775o_real(F,A2)),one_one_real) ) ).

% prod_le_1
tff(fact_6247_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat2(X4),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X4))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,X4)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F),A2)),one_one_real) ) ).

% prod_le_1
tff(fact_6248_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int2(X4),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X4)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A2)),one_one_real) ) ).

% prod_le_1
tff(fact_6249_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( aa_set_real_o(member_real2(X4),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X4)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4696554848551431203al_nat(F,A2)),one_one_nat) ) ).

% prod_le_1
tff(fact_6250_prod__le__1,axiom,
    ! [A2: set_o,F: fun_o_nat] :
      ( ! [X4: $o] :
          ( aa_set_o_o(member_o2((X4)),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(F,(X4)))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(X4))),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups3504817904513533571_o_nat(F,A2)),one_one_nat) ) ).

% prod_le_1
tff(fact_6251_prod__dvd__prod__subset2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real2(A3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_real_nat(F,A3)),aa_real_nat(G,A3)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups4696554848551431203al_nat(F,A2)),groups4696554848551431203al_nat(G,B2)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_6252_prod__dvd__prod__subset2,axiom,
    ! [B2: set_o,A2: set_o,F: fun_o_nat,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,B2)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o2((A3)),A2)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_o_nat(F,(A3))),aa_o_nat(G,(A3))) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups3504817904513533571_o_nat(F,A2)),groups3504817904513533571_o_nat(G,B2)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_6253_prod__dvd__prod__subset2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
       => ( ! [A3: complex] :
              ( aa_set_complex_o(member_complex(A3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_complex_nat(F,A3)),aa_complex_nat(G,A3)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups861055069439313189ex_nat(F,A2)),groups861055069439313189ex_nat(G,B2)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_6254_prod__dvd__prod__subset2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int2(A3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(F,A3)),aa_int_nat(G,A3)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups1707563613775114915nt_nat(F,A2)),groups1707563613775114915nt_nat(G,B2)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_6255_prod__dvd__prod__subset2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B2)
       => ( ! [A3: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_Extended_enat_nat(F,A3)),aa_Extended_enat_nat(G,A3)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups2880970938130013265at_nat(F,A2)),groups2880970938130013265at_nat(G,B2)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_6256_prod__dvd__prod__subset2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_int,G: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real2(A3),A2)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_real_int(F,A3)),aa_real_int(G,A3)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups4694064378042380927al_int(F,A2)),groups4694064378042380927al_int(G,B2)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_6257_prod__dvd__prod__subset2,axiom,
    ! [B2: set_o,A2: set_o,F: fun_o_int,G: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,B2)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o2((A3)),A2)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_o_int(F,(A3))),aa_o_int(G,(A3))) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups3502327434004483295_o_int(F,A2)),groups3502327434004483295_o_int(G,B2)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_6258_prod__dvd__prod__subset2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
       => ( ! [A3: complex] :
              ( aa_set_complex_o(member_complex(A3),A2)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_complex_int(F,A3)),aa_complex_int(G,A3)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups858564598930262913ex_int(F,A2)),groups858564598930262913ex_int(G,B2)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_6259_prod__dvd__prod__subset2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex3482447378695272101at_int,G: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B2)
       => ( ! [A3: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),A2)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_Extended_enat_int(F,A3)),aa_Extended_enat_int(G,A3)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups2878480467620962989at_int(F,A2)),groups2878480467620962989at_int(G,B2)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_6260_prod__dvd__prod__subset2,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_int,G: 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)
       => ( ! [A3: nat] :
              ( aa_set_nat_o(member_nat2(A3),A2)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(F,A3)),aa_nat_int(G,A3)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_set_nat_int(groups705719431365010083at_int(F),A2)),aa_set_nat_int(groups705719431365010083at_int(G),B2)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_6261_prod__dvd__prod__subset,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)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups861055069439313189ex_nat(F,A2)),groups861055069439313189ex_nat(F,B2)) ) ) ).

% prod_dvd_prod_subset
tff(fact_6262_prod__dvd__prod__subset,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(A2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups1707563613775114915nt_nat(F,A2)),groups1707563613775114915nt_nat(F,B2)) ) ) ).

% prod_dvd_prod_subset
tff(fact_6263_prod__dvd__prod__subset,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups2880970938130013265at_nat(F,A2)),groups2880970938130013265at_nat(F,B2)) ) ) ).

% prod_dvd_prod_subset
tff(fact_6264_prod__dvd__prod__subset,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)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups858564598930262913ex_int(F,A2)),groups858564598930262913ex_int(F,B2)) ) ) ).

% prod_dvd_prod_subset
tff(fact_6265_prod__dvd__prod__subset,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups2878480467620962989at_int(F,A2)),groups2878480467620962989at_int(F,B2)) ) ) ).

% prod_dvd_prod_subset
tff(fact_6266_prod__dvd__prod__subset,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)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_set_nat_int(groups705719431365010083at_int(F),A2)),aa_set_nat_int(groups705719431365010083at_int(F),B2)) ) ) ).

% prod_dvd_prod_subset
tff(fact_6267_prod__dvd__prod__subset,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups1705073143266064639nt_int(F,A2)),groups1705073143266064639nt_int(F,B2)) ) ) ).

% prod_dvd_prod_subset
tff(fact_6268_prod__dvd__prod__subset,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,A2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)),aa_set_nat_nat(groups708209901874060359at_nat(F),B2)) ) ) ).

% prod_dvd_prod_subset
tff(fact_6269_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_int,Nb: nat] : aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_kg(G),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.nat_diff_reindex
tff(fact_6270_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_iv(G),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.nat_diff_reindex
tff(fact_6271_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_int,Nb: nat,Mb: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Nb),Mb)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_kh(G),Nb),Mb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Nb),Mb)) ).

% prod.atLeastAtMost_rev
tff(fact_6272_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Nb: nat,Mb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Nb),Mb)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_hf(G),Nb),Mb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Nb),Mb)) ).

% prod.atLeastAtMost_rev
tff(fact_6273_less__1__prod2,axiom,
    ! [I5: set_real,Ia: real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( aa_set_real_o(member_real2(Ia),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(F,Ia))
         => ( ! [I2: real] :
                ( aa_set_real_o(member_real2(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_6274_less__1__prod2,axiom,
    ! [I5: set_o,Ia: $o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( aa_set_o_o(member_o2((Ia)),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_o_real(F,(Ia)))
         => ( ! [I2: $o] :
                ( aa_set_o_o(member_o2((I2)),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_o_real(F,(I2))) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups234877984723959775o_real(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_6275_less__1__prod2,axiom,
    ! [I5: set_nat,Ia: nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat2(Ia),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(F,Ia))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat2(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_6276_less__1__prod2,axiom,
    ! [I5: set_complex,Ia: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(Ia),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_complex_real(F,Ia))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_complex_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups766887009212190081x_real(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_6277_less__1__prod2,axiom,
    ! [I5: set_int,Ia: int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int2(Ia),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(F,Ia))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int2(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_6278_less__1__prod2,axiom,
    ! [I5: set_Extended_enat,Ia: extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,I5)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Ia),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_Ext2541572767299617110t_real(F,Ia))
         => ( ! [I2: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_Ext2541572767299617110t_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups97031904164794029t_real(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_6279_less__1__prod2,axiom,
    ! [I5: set_real,Ia: real,F: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( aa_set_real_o(member_real2(Ia),I5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(F,Ia))
         => ( ! [I2: real] :
                ( aa_set_real_o(member_real2(I2),I5)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(F,I2)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups4694064378042380927al_int(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_6280_less__1__prod2,axiom,
    ! [I5: set_o,Ia: $o,F: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( aa_set_o_o(member_o2((Ia)),I5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_o_int(F,(Ia)))
         => ( ! [I2: $o] :
                ( aa_set_o_o(member_o2((I2)),I5)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_o_int(F,(I2))) )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups3502327434004483295_o_int(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_6281_less__1__prod2,axiom,
    ! [I5: set_complex,Ia: complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(Ia),I5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_complex_int(F,Ia))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I5)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_complex_int(F,I2)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups858564598930262913ex_int(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_6282_less__1__prod2,axiom,
    ! [I5: set_Extended_enat,Ia: extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,I5)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Ia),I5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_Extended_enat_int(F,Ia))
         => ( ! [I2: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(I2),I5)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_Extended_enat_int(F,I2)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups2878480467620962989at_int(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_6283_less__1__prod,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_complex_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups766887009212190081x_real(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_6284_less__1__prod,axiom,
    ! [I5: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,I5)
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I2: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_Ext2541572767299617110t_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups97031904164794029t_real(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_6285_less__1__prod,axiom,
    ! [I5: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real2(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_6286_less__1__prod,axiom,
    ! [I5: set_o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( aa_set_o_o(member_o2((I2)),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_o_real(F,(I2))) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups234877984723959775o_real(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_6287_less__1__prod,axiom,
    ! [I5: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat2(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),I5)) ) ) ) ).

% less_1_prod
tff(fact_6288_less__1__prod,axiom,
    ! [I5: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int2(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_6289_less__1__prod,axiom,
    ! [I5: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I5)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_complex_int(F,I2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups858564598930262913ex_int(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_6290_less__1__prod,axiom,
    ! [I5: set_Extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,I5)
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I2: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(I2),I5)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_Extended_enat_int(F,I2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups2878480467620962989at_int(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_6291_less__1__prod,axiom,
    ! [I5: set_real,F: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real2(I2),I5)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(F,I2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups4694064378042380927al_int(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_6292_less__1__prod,axiom,
    ! [I5: set_o,F: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( aa_set_o_o(member_o2((I2)),I5)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_o_int(F,(I2))) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups3502327434004483295_o_int(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_6293_prod_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)
       => ( groups861055069439313189ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A2,B2))),groups861055069439313189ex_nat(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_6294_prod_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)
       => ( groups1707563613775114915nt_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G,minus_minus_set_int(A2,B2))),groups1707563613775114915nt_nat(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_6295_prod_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B2),A2)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
       => ( groups2880970938130013265at_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups2880970938130013265at_nat(G,minus_925952699566721837d_enat(A2,B2))),groups2880970938130013265at_nat(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_6296_prod_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)
       => ( groups858564598930262913ex_int(G,A2) = aa_int_int(aa_int_fun_int_int(times_times_int,groups858564598930262913ex_int(G,minus_811609699411566653omplex(A2,B2))),groups858564598930262913ex_int(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_6297_prod_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B2),A2)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
       => ( groups2878480467620962989at_int(G,A2) = aa_int_int(aa_int_fun_int_int(times_times_int,groups2878480467620962989at_int(G,minus_925952699566721837d_enat(A2,B2))),groups2878480467620962989at_int(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_6298_prod_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)
       => ( groups766887009212190081x_real(G,A2) = aa_real_real(times_times_real(groups766887009212190081x_real(G,minus_811609699411566653omplex(A2,B2))),groups766887009212190081x_real(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_6299_prod_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)
       => ( groups2316167850115554303t_real(G,A2) = aa_real_real(times_times_real(groups2316167850115554303t_real(G,minus_minus_set_int(A2,B2))),groups2316167850115554303t_real(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_6300_prod_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B2),A2)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
       => ( groups97031904164794029t_real(G,A2) = aa_real_real(times_times_real(groups97031904164794029t_real(G,minus_925952699566721837d_enat(A2,B2))),groups97031904164794029t_real(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_6301_prod_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
       => ( groups3708469109370488835omplex(G,A2) = aa_complex_complex(times_times_complex(groups3708469109370488835omplex(G,minus_811609699411566653omplex(A2,B2))),groups3708469109370488835omplex(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_6302_prod_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_complex] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A2)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ( groups7440179247065528705omplex(G,A2) = aa_complex_complex(times_times_complex(groups7440179247065528705omplex(G,minus_minus_set_int(A2,B2))),groups7440179247065528705omplex(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_6303_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S3: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
       => ( ! [X4: real] :
              ( aa_set_real_o(member_real2(X4),minus_minus_set_real(T2,S3))
             => ( aa_real_nat(G,X4) = one_one_nat ) )
         => ( ! [X4: real] :
                ( aa_set_real_o(member_real2(X4),S3)
               => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
           => ( groups4696554848551431203al_nat(G,T2) = groups4696554848551431203al_nat(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6304_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_o,S3: set_o,G: fun_o_nat,H: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(ord_less_eq_set_o(S3),T2)
       => ( ! [X4: $o] :
              ( aa_set_o_o(member_o2((X4)),minus_minus_set_o(T2,S3))
             => ( aa_o_nat(G,(X4)) = one_one_nat ) )
         => ( ! [X4: $o] :
                ( aa_set_o_o(member_o2((X4)),S3)
               => ( aa_o_nat(G,(X4)) = aa_o_nat(H,(X4)) ) )
           => ( groups3504817904513533571_o_nat(G,T2) = groups3504817904513533571_o_nat(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6305_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_nat(G,X4) = one_one_nat ) )
         => ( ! [X4: complex] :
                ( aa_set_complex_o(member_complex(X4),S3)
               => ( aa_complex_nat(G,X4) = aa_complex_nat(H,X4) ) )
           => ( groups861055069439313189ex_nat(G,T2) = groups861055069439313189ex_nat(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6306_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
             => ( aa_int_nat(G,X4) = one_one_nat ) )
         => ( ! [X4: int] :
                ( aa_set_int_o(member_int2(X4),S3)
               => ( aa_int_nat(G,X4) = aa_int_nat(H,X4) ) )
           => ( groups1707563613775114915nt_nat(G,T2) = groups1707563613775114915nt_nat(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6307_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,H: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(T2,S3))
             => ( aa_Extended_enat_nat(G,X4) = one_one_nat ) )
         => ( ! [X4: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),S3)
               => ( aa_Extended_enat_nat(G,X4) = aa_Extended_enat_nat(H,X4) ) )
           => ( groups2880970938130013265at_nat(G,T2) = groups2880970938130013265at_nat(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6308_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S3: set_real,G: fun_real_int,H: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
       => ( ! [X4: real] :
              ( aa_set_real_o(member_real2(X4),minus_minus_set_real(T2,S3))
             => ( aa_real_int(G,X4) = one_one_int ) )
         => ( ! [X4: real] :
                ( aa_set_real_o(member_real2(X4),S3)
               => ( aa_real_int(G,X4) = aa_real_int(H,X4) ) )
           => ( groups4694064378042380927al_int(G,T2) = groups4694064378042380927al_int(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6309_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_o,S3: set_o,G: fun_o_int,H: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(ord_less_eq_set_o(S3),T2)
       => ( ! [X4: $o] :
              ( aa_set_o_o(member_o2((X4)),minus_minus_set_o(T2,S3))
             => ( aa_o_int(G,(X4)) = one_one_int ) )
         => ( ! [X4: $o] :
                ( aa_set_o_o(member_o2((X4)),S3)
               => ( aa_o_int(G,(X4)) = aa_o_int(H,(X4)) ) )
           => ( groups3502327434004483295_o_int(G,T2) = groups3502327434004483295_o_int(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6310_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_int(G,X4) = one_one_int ) )
         => ( ! [X4: complex] :
                ( aa_set_complex_o(member_complex(X4),S3)
               => ( aa_complex_int(G,X4) = aa_complex_int(H,X4) ) )
           => ( groups858564598930262913ex_int(G,T2) = groups858564598930262913ex_int(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6311_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex3482447378695272101at_int,H: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(T2,S3))
             => ( aa_Extended_enat_int(G,X4) = one_one_int ) )
         => ( ! [X4: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),S3)
               => ( aa_Extended_enat_int(G,X4) = aa_Extended_enat_int(H,X4) ) )
           => ( groups2878480467620962989at_int(G,T2) = groups2878480467620962989at_int(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6312_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S3: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
       => ( ! [X4: real] :
              ( aa_set_real_o(member_real2(X4),minus_minus_set_real(T2,S3))
             => ( aa_real_real(G,X4) = one_one_real ) )
         => ( ! [X4: real] :
                ( aa_set_real_o(member_real2(X4),S3)
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups1681761925125756287l_real(G,T2) = groups1681761925125756287l_real(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_6313_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S3: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
       => ( ! [X4: real] :
              ( aa_set_real_o(member_real2(X4),minus_minus_set_real(T2,S3))
             => ( aa_real_nat(H,X4) = one_one_nat ) )
         => ( ! [X4: real] :
                ( aa_set_real_o(member_real2(X4),S3)
               => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
           => ( groups4696554848551431203al_nat(G,S3) = groups4696554848551431203al_nat(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6314_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_o,S3: set_o,H: fun_o_nat,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(ord_less_eq_set_o(S3),T2)
       => ( ! [X4: $o] :
              ( aa_set_o_o(member_o2((X4)),minus_minus_set_o(T2,S3))
             => ( aa_o_nat(H,(X4)) = one_one_nat ) )
         => ( ! [X4: $o] :
                ( aa_set_o_o(member_o2((X4)),S3)
               => ( aa_o_nat(G,(X4)) = aa_o_nat(H,(X4)) ) )
           => ( groups3504817904513533571_o_nat(G,S3) = groups3504817904513533571_o_nat(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6315_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S3: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_nat(H,X4) = one_one_nat ) )
         => ( ! [X4: complex] :
                ( aa_set_complex_o(member_complex(X4),S3)
               => ( aa_complex_nat(G,X4) = aa_complex_nat(H,X4) ) )
           => ( groups861055069439313189ex_nat(G,S3) = groups861055069439313189ex_nat(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6316_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_int,S3: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
             => ( aa_int_nat(H,X4) = one_one_nat ) )
         => ( ! [X4: int] :
                ( aa_set_int_o(member_int2(X4),S3)
               => ( aa_int_nat(G,X4) = aa_int_nat(H,X4) ) )
           => ( groups1707563613775114915nt_nat(G,S3) = groups1707563613775114915nt_nat(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6317_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_Extended_enat,S3: set_Extended_enat,H: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(T2,S3))
             => ( aa_Extended_enat_nat(H,X4) = one_one_nat ) )
         => ( ! [X4: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),S3)
               => ( aa_Extended_enat_nat(G,X4) = aa_Extended_enat_nat(H,X4) ) )
           => ( groups2880970938130013265at_nat(G,S3) = groups2880970938130013265at_nat(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6318_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S3: set_real,H: fun_real_int,G: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
       => ( ! [X4: real] :
              ( aa_set_real_o(member_real2(X4),minus_minus_set_real(T2,S3))
             => ( aa_real_int(H,X4) = one_one_int ) )
         => ( ! [X4: real] :
                ( aa_set_real_o(member_real2(X4),S3)
               => ( aa_real_int(G,X4) = aa_real_int(H,X4) ) )
           => ( groups4694064378042380927al_int(G,S3) = groups4694064378042380927al_int(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6319_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_o,S3: set_o,H: fun_o_int,G: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(ord_less_eq_set_o(S3),T2)
       => ( ! [X4: $o] :
              ( aa_set_o_o(member_o2((X4)),minus_minus_set_o(T2,S3))
             => ( aa_o_int(H,(X4)) = one_one_int ) )
         => ( ! [X4: $o] :
                ( aa_set_o_o(member_o2((X4)),S3)
               => ( aa_o_int(G,(X4)) = aa_o_int(H,(X4)) ) )
           => ( groups3502327434004483295_o_int(G,S3) = groups3502327434004483295_o_int(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6320_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S3: set_complex,H: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_int(H,X4) = one_one_int ) )
         => ( ! [X4: complex] :
                ( aa_set_complex_o(member_complex(X4),S3)
               => ( aa_complex_int(G,X4) = aa_complex_int(H,X4) ) )
           => ( groups858564598930262913ex_int(G,S3) = groups858564598930262913ex_int(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6321_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_Extended_enat,S3: set_Extended_enat,H: fun_Ex3482447378695272101at_int,G: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(T2,S3))
             => ( aa_Extended_enat_int(H,X4) = one_one_int ) )
         => ( ! [X4: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),S3)
               => ( aa_Extended_enat_int(G,X4) = aa_Extended_enat_int(H,X4) ) )
           => ( groups2878480467620962989at_int(G,S3) = groups2878480467620962989at_int(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6322_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S3: set_real,H: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
       => ( ! [X4: real] :
              ( aa_set_real_o(member_real2(X4),minus_minus_set_real(T2,S3))
             => ( aa_real_real(H,X4) = one_one_real ) )
         => ( ! [X4: real] :
                ( aa_set_real_o(member_real2(X4),S3)
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups1681761925125756287l_real(G,S3) = groups1681761925125756287l_real(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_6323_prod_Omono__neutral__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_nat(G,X4) = one_one_nat ) )
         => ( groups861055069439313189ex_nat(G,T2) = groups861055069439313189ex_nat(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6324_prod_Omono__neutral__right,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
             => ( aa_int_nat(G,X4) = one_one_nat ) )
         => ( groups1707563613775114915nt_nat(G,T2) = groups1707563613775114915nt_nat(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6325_prod_Omono__neutral__right,axiom,
    ! [T2: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(T2,S3))
             => ( aa_Extended_enat_nat(G,X4) = one_one_nat ) )
         => ( groups2880970938130013265at_nat(G,T2) = groups2880970938130013265at_nat(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6326_prod_Omono__neutral__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_int(G,X4) = one_one_int ) )
         => ( groups858564598930262913ex_int(G,T2) = groups858564598930262913ex_int(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6327_prod_Omono__neutral__right,axiom,
    ! [T2: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(T2,S3))
             => ( aa_Extended_enat_int(G,X4) = one_one_int ) )
         => ( groups2878480467620962989at_int(G,T2) = groups2878480467620962989at_int(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6328_prod_Omono__neutral__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_real(G,X4) = one_one_real ) )
         => ( groups766887009212190081x_real(G,T2) = groups766887009212190081x_real(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6329_prod_Omono__neutral__right,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
             => ( aa_int_real(G,X4) = one_one_real ) )
         => ( groups2316167850115554303t_real(G,T2) = groups2316167850115554303t_real(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6330_prod_Omono__neutral__right,axiom,
    ! [T2: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(T2,S3))
             => ( aa_Ext2541572767299617110t_real(G,X4) = one_one_real ) )
         => ( groups97031904164794029t_real(G,T2) = groups97031904164794029t_real(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6331_prod_Omono__neutral__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_complex(G,X4) = one_one_complex ) )
         => ( groups3708469109370488835omplex(G,T2) = groups3708469109370488835omplex(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6332_prod_Omono__neutral__right,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
             => ( aa_int_complex(G,X4) = one_one_complex ) )
         => ( groups7440179247065528705omplex(G,T2) = groups7440179247065528705omplex(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_6333_prod_Omono__neutral__left,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_nat(G,X4) = one_one_nat ) )
         => ( groups861055069439313189ex_nat(G,S3) = groups861055069439313189ex_nat(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6334_prod_Omono__neutral__left,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
             => ( aa_int_nat(G,X4) = one_one_nat ) )
         => ( groups1707563613775114915nt_nat(G,S3) = groups1707563613775114915nt_nat(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6335_prod_Omono__neutral__left,axiom,
    ! [T2: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(T2,S3))
             => ( aa_Extended_enat_nat(G,X4) = one_one_nat ) )
         => ( groups2880970938130013265at_nat(G,S3) = groups2880970938130013265at_nat(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6336_prod_Omono__neutral__left,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_int(G,X4) = one_one_int ) )
         => ( groups858564598930262913ex_int(G,S3) = groups858564598930262913ex_int(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6337_prod_Omono__neutral__left,axiom,
    ! [T2: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(T2,S3))
             => ( aa_Extended_enat_int(G,X4) = one_one_int ) )
         => ( groups2878480467620962989at_int(G,S3) = groups2878480467620962989at_int(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6338_prod_Omono__neutral__left,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_real(G,X4) = one_one_real ) )
         => ( groups766887009212190081x_real(G,S3) = groups766887009212190081x_real(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6339_prod_Omono__neutral__left,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
             => ( aa_int_real(G,X4) = one_one_real ) )
         => ( groups2316167850115554303t_real(G,S3) = groups2316167850115554303t_real(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6340_prod_Omono__neutral__left,axiom,
    ! [T2: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(S3),T2)
       => ( ! [X4: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(X4),minus_925952699566721837d_enat(T2,S3))
             => ( aa_Ext2541572767299617110t_real(G,X4) = one_one_real ) )
         => ( groups97031904164794029t_real(G,S3) = groups97031904164794029t_real(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6341_prod_Omono__neutral__left,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T2)
       => ( ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_complex(G,X4) = one_one_complex ) )
         => ( groups3708469109370488835omplex(G,S3) = groups3708469109370488835omplex(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6342_prod_Omono__neutral__left,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
             => ( aa_int_complex(G,X4) = one_one_complex ) )
         => ( groups7440179247065528705omplex(G,S3) = groups7440179247065528705omplex(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_6343_prod_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_real2(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real2(B3),minus_minus_set_real(C4,B2))
                 => ( aa_real_nat(H,B3) = one_one_nat ) )
             => ( ( groups4696554848551431203al_nat(G,C4) = groups4696554848551431203al_nat(H,C4) )
               => ( groups4696554848551431203al_nat(G,A2) = groups4696554848551431203al_nat(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6344_prod_Osame__carrierI,axiom,
    ! [C4: set_o,A2: set_o,B2: set_o,G: fun_o_nat,H: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,C4)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),C4)
       => ( aa_set_o_o(ord_less_eq_set_o(B2),C4)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o2((A3)),minus_minus_set_o(C4,A2))
               => ( aa_o_nat(G,(A3)) = one_one_nat ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o2((B3)),minus_minus_set_o(C4,B2))
                 => ( aa_o_nat(H,(B3)) = one_one_nat ) )
             => ( ( groups3504817904513533571_o_nat(G,C4) = groups3504817904513533571_o_nat(H,C4) )
               => ( groups3504817904513533571_o_nat(G,A2) = groups3504817904513533571_o_nat(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6345_prod_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) = one_one_nat ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B2))
                 => ( aa_complex_nat(H,B3) = one_one_nat ) )
             => ( ( groups861055069439313189ex_nat(G,C4) = groups861055069439313189ex_nat(H,C4) )
               => ( groups861055069439313189ex_nat(G,A2) = groups861055069439313189ex_nat(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6346_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,C4)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),C4)
       => ( aa_set_int_o(ord_less_eq_set_int(B2),C4)
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int2(A3),minus_minus_set_int(C4,A2))
               => ( aa_int_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int2(B3),minus_minus_set_int(C4,B2))
                 => ( aa_int_nat(H,B3) = one_one_nat ) )
             => ( ( groups1707563613775114915nt_nat(G,C4) = groups1707563613775114915nt_nat(H,C4) )
               => ( groups1707563613775114915nt_nat(G,A2) = groups1707563613775114915nt_nat(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6347_prod_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,H: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,C4)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),C4)
       => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B2),C4)
         => ( ! [A3: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2))
               => ( aa_Extended_enat_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: extended_enat] :
                  ( aa_set3337841544853333262enat_o(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B2))
                 => ( aa_Extended_enat_nat(H,B3) = one_one_nat ) )
             => ( ( groups2880970938130013265at_nat(G,C4) = groups2880970938130013265at_nat(H,C4) )
               => ( groups2880970938130013265at_nat(G,A2) = groups2880970938130013265at_nat(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6348_prod_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_real2(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_int(G,A3) = one_one_int ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real2(B3),minus_minus_set_real(C4,B2))
                 => ( aa_real_int(H,B3) = one_one_int ) )
             => ( ( groups4694064378042380927al_int(G,C4) = groups4694064378042380927al_int(H,C4) )
               => ( groups4694064378042380927al_int(G,A2) = groups4694064378042380927al_int(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6349_prod_Osame__carrierI,axiom,
    ! [C4: set_o,A2: set_o,B2: set_o,G: fun_o_int,H: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,C4)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),C4)
       => ( aa_set_o_o(ord_less_eq_set_o(B2),C4)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o2((A3)),minus_minus_set_o(C4,A2))
               => ( aa_o_int(G,(A3)) = one_one_int ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o2((B3)),minus_minus_set_o(C4,B2))
                 => ( aa_o_int(H,(B3)) = one_one_int ) )
             => ( ( groups3502327434004483295_o_int(G,C4) = groups3502327434004483295_o_int(H,C4) )
               => ( groups3502327434004483295_o_int(G,A2) = groups3502327434004483295_o_int(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6350_prod_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) = one_one_int ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B2))
                 => ( aa_complex_int(H,B3) = one_one_int ) )
             => ( ( groups858564598930262913ex_int(G,C4) = groups858564598930262913ex_int(H,C4) )
               => ( groups858564598930262913ex_int(G,A2) = groups858564598930262913ex_int(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6351_prod_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: fun_Ex3482447378695272101at_int,H: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,C4)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),C4)
       => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B2),C4)
         => ( ! [A3: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2))
               => ( aa_Extended_enat_int(G,A3) = one_one_int ) )
           => ( ! [B3: extended_enat] :
                  ( aa_set3337841544853333262enat_o(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B2))
                 => ( aa_Extended_enat_int(H,B3) = one_one_int ) )
             => ( ( groups2878480467620962989at_int(G,C4) = groups2878480467620962989at_int(H,C4) )
               => ( groups2878480467620962989at_int(G,A2) = groups2878480467620962989at_int(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6352_prod_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_real2(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_real(G,A3) = one_one_real ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real2(B3),minus_minus_set_real(C4,B2))
                 => ( aa_real_real(H,B3) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,C4) = groups1681761925125756287l_real(H,C4) )
               => ( groups1681761925125756287l_real(G,A2) = groups1681761925125756287l_real(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_6353_prod_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_real2(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real2(B3),minus_minus_set_real(C4,B2))
                 => ( aa_real_nat(H,B3) = one_one_nat ) )
             => ( ( groups4696554848551431203al_nat(G,A2) = groups4696554848551431203al_nat(H,B2) )
              <=> ( groups4696554848551431203al_nat(G,C4) = groups4696554848551431203al_nat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6354_prod_Osame__carrier,axiom,
    ! [C4: set_o,A2: set_o,B2: set_o,G: fun_o_nat,H: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,C4)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),C4)
       => ( aa_set_o_o(ord_less_eq_set_o(B2),C4)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o2((A3)),minus_minus_set_o(C4,A2))
               => ( aa_o_nat(G,(A3)) = one_one_nat ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o2((B3)),minus_minus_set_o(C4,B2))
                 => ( aa_o_nat(H,(B3)) = one_one_nat ) )
             => ( ( groups3504817904513533571_o_nat(G,A2) = groups3504817904513533571_o_nat(H,B2) )
              <=> ( groups3504817904513533571_o_nat(G,C4) = groups3504817904513533571_o_nat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6355_prod_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) = one_one_nat ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B2))
                 => ( aa_complex_nat(H,B3) = one_one_nat ) )
             => ( ( groups861055069439313189ex_nat(G,A2) = groups861055069439313189ex_nat(H,B2) )
              <=> ( groups861055069439313189ex_nat(G,C4) = groups861055069439313189ex_nat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6356_prod_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,C4)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),C4)
       => ( aa_set_int_o(ord_less_eq_set_int(B2),C4)
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int2(A3),minus_minus_set_int(C4,A2))
               => ( aa_int_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int2(B3),minus_minus_set_int(C4,B2))
                 => ( aa_int_nat(H,B3) = one_one_nat ) )
             => ( ( groups1707563613775114915nt_nat(G,A2) = groups1707563613775114915nt_nat(H,B2) )
              <=> ( groups1707563613775114915nt_nat(G,C4) = groups1707563613775114915nt_nat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6357_prod_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,H: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,C4)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),C4)
       => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B2),C4)
         => ( ! [A3: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2))
               => ( aa_Extended_enat_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: extended_enat] :
                  ( aa_set3337841544853333262enat_o(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B2))
                 => ( aa_Extended_enat_nat(H,B3) = one_one_nat ) )
             => ( ( groups2880970938130013265at_nat(G,A2) = groups2880970938130013265at_nat(H,B2) )
              <=> ( groups2880970938130013265at_nat(G,C4) = groups2880970938130013265at_nat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6358_prod_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_real2(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_int(G,A3) = one_one_int ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real2(B3),minus_minus_set_real(C4,B2))
                 => ( aa_real_int(H,B3) = one_one_int ) )
             => ( ( groups4694064378042380927al_int(G,A2) = groups4694064378042380927al_int(H,B2) )
              <=> ( groups4694064378042380927al_int(G,C4) = groups4694064378042380927al_int(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6359_prod_Osame__carrier,axiom,
    ! [C4: set_o,A2: set_o,B2: set_o,G: fun_o_int,H: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,C4)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),C4)
       => ( aa_set_o_o(ord_less_eq_set_o(B2),C4)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o2((A3)),minus_minus_set_o(C4,A2))
               => ( aa_o_int(G,(A3)) = one_one_int ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o2((B3)),minus_minus_set_o(C4,B2))
                 => ( aa_o_int(H,(B3)) = one_one_int ) )
             => ( ( groups3502327434004483295_o_int(G,A2) = groups3502327434004483295_o_int(H,B2) )
              <=> ( groups3502327434004483295_o_int(G,C4) = groups3502327434004483295_o_int(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6360_prod_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) = one_one_int ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C4,B2))
                 => ( aa_complex_int(H,B3) = one_one_int ) )
             => ( ( groups858564598930262913ex_int(G,A2) = groups858564598930262913ex_int(H,B2) )
              <=> ( groups858564598930262913ex_int(G,C4) = groups858564598930262913ex_int(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6361_prod_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: fun_Ex3482447378695272101at_int,H: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,C4)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),C4)
       => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B2),C4)
         => ( ! [A3: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2))
               => ( aa_Extended_enat_int(G,A3) = one_one_int ) )
           => ( ! [B3: extended_enat] :
                  ( aa_set3337841544853333262enat_o(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B2))
                 => ( aa_Extended_enat_int(H,B3) = one_one_int ) )
             => ( ( groups2878480467620962989at_int(G,A2) = groups2878480467620962989at_int(H,B2) )
              <=> ( groups2878480467620962989at_int(G,C4) = groups2878480467620962989at_int(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6362_prod_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_real2(A3),minus_minus_set_real(C4,A2))
               => ( aa_real_real(G,A3) = one_one_real ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real2(B3),minus_minus_set_real(C4,B2))
                 => ( aa_real_real(H,B3) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,A2) = groups1681761925125756287l_real(H,B2) )
              <=> ( groups1681761925125756287l_real(G,C4) = groups1681761925125756287l_real(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_6363_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6364_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_complex,Nb: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6365_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_na8797796813480665199d_enat,Nb: nat] : groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))),aa_nat_Extended_enat(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6366_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6367_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6368_powser__sums__zero,axiom,
    ! [A: fun_nat_real] : aa_real_o(sums_real(aTP_Lamm_ev(A)),aa_nat_real(A,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_6369_powser__sums__zero,axiom,
    ! [A: fun_nat_complex] : aa_complex_o(sums_complex(aTP_Lamm_eu(A)),aa_nat_complex(A,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_6370_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_real_real(times_times_real(aa_nat_real(G,Mb)),aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6371_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(G,Mb)),groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6372_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_na8797796813480665199d_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(G,Mb)),groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6373_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,Mb)),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6374_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6375_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_nat_real(G,aa_nat_nat(suc,Nb))),aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6376_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(suc,Nb))) = aa_complex_complex(times_times_complex(aa_nat_complex(G,aa_nat_nat(suc,Nb))),groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6377_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_na8797796813480665199d_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(suc,Nb))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(G,aa_nat_nat(suc,Nb))),groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6378_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,aa_nat_nat(suc,Nb))),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6379_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,aa_nat_nat(suc,Nb))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6380_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_gp(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_6381_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_complex,Nb: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_complex_complex(times_times_complex(aa_nat_complex(G,zero_zero_nat)),groups6464643781859351333omplex(aTP_Lamm_ki(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_6382_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_na8797796813480665199d_enat,Nb: nat] : groups7961826882256487087d_enat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(G,zero_zero_nat)),groups7961826882256487087d_enat(aTP_Lamm_hi(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_6383_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_hh(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_6384_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_go(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_6385_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_nat_real(G,Mb)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_gp(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6386_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb))) = aa_complex_complex(times_times_complex(aa_nat_complex(G,Mb)),groups6464643781859351333omplex(aTP_Lamm_ki(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6387_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_na8797796813480665199d_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_nat_Extended_enat(G,aa_nat_nat(suc,Nb))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(G,Mb)),groups7961826882256487087d_enat(aTP_Lamm_hi(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6388_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,Mb)),aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_hh(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6389_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_go(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6390_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_int,Nb: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_hh(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.atLeast1_atMost_eq
tff(fact_6391_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_go(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.atLeast1_atMost_eq
tff(fact_6392_prod__mono__strict,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),A2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,I2)),aa_complex_real(G,I2)) ) )
       => ( ( A2 != bot_bot_set_complex )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups766887009212190081x_real(F,A2)),groups766887009212190081x_real(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_6393_prod__mono__strict,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ! [I2: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(I2),A2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I2))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Ext2541572767299617110t_real(F,I2)),aa_Ext2541572767299617110t_real(G,I2)) ) )
       => ( ( A2 != bot_bo7653980558646680370d_enat )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups97031904164794029t_real(F,A2)),groups97031904164794029t_real(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_6394_prod__mono__strict,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real2(I2),A2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,I2)),aa_real_real(G,I2)) ) )
       => ( ( A2 != bot_bot_set_real )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups1681761925125756287l_real(F,A2)),groups1681761925125756287l_real(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_6395_prod__mono__strict,axiom,
    ! [A2: set_o,F: fun_o_real,G: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ! [I2: $o] :
            ( aa_set_o_o(member_o2((I2)),A2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(I2)))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_o_real(F,(I2))),aa_o_real(G,(I2))) ) )
       => ( ( A2 != bot_bot_set_o )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups234877984723959775o_real(F,A2)),groups234877984723959775o_real(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_6396_prod__mono__strict,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat2(I2),A2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,I2)),aa_nat_real(G,I2)) ) )
       => ( ( A2 != bot_bot_set_nat )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_set_nat_real(groups129246275422532515t_real(F),A2)),aa_set_nat_real(groups129246275422532515t_real(G),A2)) ) ) ) ).

% prod_mono_strict
tff(fact_6397_prod__mono__strict,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int2(I2),A2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,I2)),aa_int_real(G,I2)) ) )
       => ( ( A2 != bot_bot_set_int )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups2316167850115554303t_real(F,A2)),groups2316167850115554303t_real(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_6398_prod__mono__strict,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),A2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2))
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2)) ) )
       => ( ( A2 != bot_bot_set_complex )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups861055069439313189ex_nat(F,A2)),groups861055069439313189ex_nat(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_6399_prod__mono__strict,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ! [I2: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(I2),A2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_Extended_enat_nat(F,I2))
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Extended_enat_nat(F,I2)),aa_Extended_enat_nat(G,I2)) ) )
       => ( ( A2 != bot_bo7653980558646680370d_enat )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups2880970938130013265at_nat(F,A2)),groups2880970938130013265at_nat(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_6400_prod__mono__strict,axiom,
    ! [A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real2(I2),A2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2))
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2)) ) )
       => ( ( A2 != bot_bot_set_real )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4696554848551431203al_nat(F,A2)),groups4696554848551431203al_nat(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_6401_prod__mono__strict,axiom,
    ! [A2: set_o,F: fun_o_nat,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ! [I2: $o] :
            ( aa_set_o_o(member_o2((I2)),A2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(F,(I2)))
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(F,(I2))),aa_o_nat(G,(I2))) ) )
       => ( ( A2 != bot_bot_set_o )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups3504817904513533571_o_nat(F,A2)),groups3504817904513533571_o_nat(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_6402_prod_Oremove,axiom,
    ! [A2: set_complex,Xb: complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(Xb),A2)
       => ( groups861055069439313189ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_complex_nat(G,Xb)),groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(Xb,bot_bot_set_complex)))) ) ) ) ).

% prod.remove
tff(fact_6403_prod_Oremove,axiom,
    ! [A2: set_Extended_enat,Xb: extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Xb),A2)
       => ( groups2880970938130013265at_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_Extended_enat_nat(G,Xb)),groups2880970938130013265at_nat(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(Xb,bot_bo7653980558646680370d_enat)))) ) ) ) ).

% prod.remove
tff(fact_6404_prod_Oremove,axiom,
    ! [A2: set_real,Xb: real,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real2(Xb),A2)
       => ( groups4696554848551431203al_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_real_nat(G,Xb)),groups4696554848551431203al_nat(G,minus_minus_set_real(A2,insert_real2(Xb,bot_bot_set_real)))) ) ) ) ).

% prod.remove
tff(fact_6405_prod_Oremove,axiom,
    ! [A2: set_o,Xb: $o,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o2((Xb)),A2)
       => ( groups3504817904513533571_o_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(G,(Xb))),groups3504817904513533571_o_nat(G,minus_minus_set_o(A2,insert_o2((Xb),bot_bot_set_o)))) ) ) ) ).

% prod.remove
tff(fact_6406_prod_Oremove,axiom,
    ! [A2: set_int,Xb: int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int2(Xb),A2)
       => ( groups1707563613775114915nt_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(G,Xb)),groups1707563613775114915nt_nat(G,minus_minus_set_int(A2,insert_int2(Xb,bot_bot_set_int)))) ) ) ) ).

% prod.remove
tff(fact_6407_prod_Oremove,axiom,
    ! [A2: set_complex,Xb: complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(Xb),A2)
       => ( groups858564598930262913ex_int(G,A2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_complex_int(G,Xb)),groups858564598930262913ex_int(G,minus_811609699411566653omplex(A2,insert_complex(Xb,bot_bot_set_complex)))) ) ) ) ).

% prod.remove
tff(fact_6408_prod_Oremove,axiom,
    ! [A2: set_Extended_enat,Xb: extended_enat,G: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(member_Extended_enat(Xb),A2)
       => ( groups2878480467620962989at_int(G,A2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Extended_enat_int(G,Xb)),groups2878480467620962989at_int(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(Xb,bot_bo7653980558646680370d_enat)))) ) ) ) ).

% prod.remove
tff(fact_6409_prod_Oremove,axiom,
    ! [A2: set_real,Xb: real,G: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real2(Xb),A2)
       => ( groups4694064378042380927al_int(G,A2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(G,Xb)),groups4694064378042380927al_int(G,minus_minus_set_real(A2,insert_real2(Xb,bot_bot_set_real)))) ) ) ) ).

% prod.remove
tff(fact_6410_prod_Oremove,axiom,
    ! [A2: set_o,Xb: $o,G: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o2((Xb)),A2)
       => ( groups3502327434004483295_o_int(G,A2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(G,(Xb))),groups3502327434004483295_o_int(G,minus_minus_set_o(A2,insert_o2((Xb),bot_bot_set_o)))) ) ) ) ).

% prod.remove
tff(fact_6411_prod_Oremove,axiom,
    ! [A2: set_complex,Xb: complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(Xb),A2)
       => ( groups766887009212190081x_real(G,A2) = aa_real_real(times_times_real(aa_complex_real(G,Xb)),groups766887009212190081x_real(G,minus_811609699411566653omplex(A2,insert_complex(Xb,bot_bot_set_complex)))) ) ) ) ).

% prod.remove
tff(fact_6412_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_nat,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups861055069439313189ex_nat(G,insert_complex(Xb,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_complex_nat(G,Xb)),groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(Xb,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_6413_prod_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,Xb: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( groups2880970938130013265at_nat(G,insert_Extended_enat(Xb,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_Extended_enat_nat(G,Xb)),groups2880970938130013265at_nat(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(Xb,bot_bo7653980558646680370d_enat)))) ) ) ).

% prod.insert_remove
tff(fact_6414_prod_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_nat,Xb: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups4696554848551431203al_nat(G,insert_real2(Xb,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_real_nat(G,Xb)),groups4696554848551431203al_nat(G,minus_minus_set_real(A2,insert_real2(Xb,bot_bot_set_real)))) ) ) ).

% prod.insert_remove
tff(fact_6415_prod_Oinsert__remove,axiom,
    ! [A2: set_o,G: fun_o_nat,Xb: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups3504817904513533571_o_nat(G,insert_o2((Xb),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(G,(Xb))),groups3504817904513533571_o_nat(G,minus_minus_set_o(A2,insert_o2((Xb),bot_bot_set_o)))) ) ) ).

% prod.insert_remove
tff(fact_6416_prod_Oinsert__remove,axiom,
    ! [A2: set_int,G: fun_int_nat,Xb: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1707563613775114915nt_nat(G,insert_int2(Xb,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(G,Xb)),groups1707563613775114915nt_nat(G,minus_minus_set_int(A2,insert_int2(Xb,bot_bot_set_int)))) ) ) ).

% prod.insert_remove
tff(fact_6417_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_int,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups858564598930262913ex_int(G,insert_complex(Xb,A2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_complex_int(G,Xb)),groups858564598930262913ex_int(G,minus_811609699411566653omplex(A2,insert_complex(Xb,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_6418_prod_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex3482447378695272101at_int,Xb: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( groups2878480467620962989at_int(G,insert_Extended_enat(Xb,A2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Extended_enat_int(G,Xb)),groups2878480467620962989at_int(G,minus_925952699566721837d_enat(A2,insert_Extended_enat(Xb,bot_bo7653980558646680370d_enat)))) ) ) ).

% prod.insert_remove
tff(fact_6419_prod_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_int,Xb: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups4694064378042380927al_int(G,insert_real2(Xb,A2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(G,Xb)),groups4694064378042380927al_int(G,minus_minus_set_real(A2,insert_real2(Xb,bot_bot_set_real)))) ) ) ).

% prod.insert_remove
tff(fact_6420_prod_Oinsert__remove,axiom,
    ! [A2: set_o,G: fun_o_int,Xb: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups3502327434004483295_o_int(G,insert_o2((Xb),A2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(G,(Xb))),groups3502327434004483295_o_int(G,minus_minus_set_o(A2,insert_o2((Xb),bot_bot_set_o)))) ) ) ).

% prod.insert_remove
tff(fact_6421_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_real,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups766887009212190081x_real(G,insert_complex(Xb,A2)) = aa_real_real(times_times_real(aa_complex_real(G,Xb)),groups766887009212190081x_real(G,minus_811609699411566653omplex(A2,insert_complex(Xb,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_6422_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_6423_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_complex,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_6424_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_na8797796813480665199d_enat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_6425_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_6426_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_6427_prod_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_nat,C: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups861055069439313189ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_fn(A),B),C),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_complex_nat(B,A)),groups861055069439313189ex_nat(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))),groups861055069439313189ex_nat(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% prod.delta_remove
tff(fact_6428_prod_Odelta__remove,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: fun_Ex7660298398204468809at_nat,C: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
     => ( groups2880970938130013265at_nat(aa_fun1810565495048290044at_nat(aa_fun5812849712608808208at_nat(aTP_Lamm_fo(A),B),C),S3) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(A),S3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_Extended_enat_nat(B,A)),groups2880970938130013265at_nat(C,minus_925952699566721837d_enat(S3,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))),groups2880970938130013265at_nat(C,minus_925952699566721837d_enat(S3,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))) ) ) ).

% prod.delta_remove
tff(fact_6429_prod_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: fun_real_nat,C: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups4696554848551431203al_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_fv(A),B),C),S3) = $ite(aa_set_real_o(member_real2(A),S3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_real_nat(B,A)),groups4696554848551431203al_nat(C,minus_minus_set_real(S3,insert_real2(A,bot_bot_set_real)))),groups4696554848551431203al_nat(C,minus_minus_set_real(S3,insert_real2(A,bot_bot_set_real)))) ) ) ).

% prod.delta_remove
tff(fact_6430_prod_Odelta__remove,axiom,
    ! [S3: set_o,A: $o,B: fun_o_nat,C: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S3)
     => ( groups3504817904513533571_o_nat(aa_fun6176012476734143100_o_nat(aa_fun6251207968994965954_o_nat(aTP_Lamm_kj((A)),B),C),S3) = $ite(aa_set_o_o(member_o2((A)),S3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(B,(A))),groups3504817904513533571_o_nat(C,minus_minus_set_o(S3,insert_o2((A),bot_bot_set_o)))),groups3504817904513533571_o_nat(C,minus_minus_set_o(S3,insert_o2((A),bot_bot_set_o)))) ) ) ).

% prod.delta_remove
tff(fact_6431_prod_Odelta__remove,axiom,
    ! [S3: set_int,A: int,B: fun_int_nat,C: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_kk(A),B),C),S3) = $ite(aa_set_int_o(member_int2(A),S3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(B,A)),groups1707563613775114915nt_nat(C,minus_minus_set_int(S3,insert_int2(A,bot_bot_set_int)))),groups1707563613775114915nt_nat(C,minus_minus_set_int(S3,insert_int2(A,bot_bot_set_int)))) ) ) ).

% prod.delta_remove
tff(fact_6432_prod_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_int,C: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups858564598930262913ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_fp(A),B),C),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_int_int(aa_int_fun_int_int(times_times_int,aa_complex_int(B,A)),groups858564598930262913ex_int(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))),groups858564598930262913ex_int(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% prod.delta_remove
tff(fact_6433_prod_Odelta__remove,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: fun_Ex3482447378695272101at_int,C: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S3)
     => ( groups2878480467620962989at_int(aa_fun1695976788104708604at_int(aa_fun8556764835550362732at_int(aTP_Lamm_fq(A),B),C),S3) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(A),S3),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Extended_enat_int(B,A)),groups2878480467620962989at_int(C,minus_925952699566721837d_enat(S3,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))),groups2878480467620962989at_int(C,minus_925952699566721837d_enat(S3,insert_Extended_enat(A,bot_bo7653980558646680370d_enat)))) ) ) ).

% prod.delta_remove
tff(fact_6434_prod_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: fun_real_int,C: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups4694064378042380927al_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_fw(A),B),C),S3) = $ite(aa_set_real_o(member_real2(A),S3),aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(B,A)),groups4694064378042380927al_int(C,minus_minus_set_real(S3,insert_real2(A,bot_bot_set_real)))),groups4694064378042380927al_int(C,minus_minus_set_real(S3,insert_real2(A,bot_bot_set_real)))) ) ) ).

% prod.delta_remove
tff(fact_6435_prod_Odelta__remove,axiom,
    ! [S3: set_o,A: $o,B: fun_o_int,C: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,S3)
     => ( groups3502327434004483295_o_int(aa_fun4727766026823762812_o_int(aa_fun5648456818326528798_o_int(aTP_Lamm_kl((A)),B),C),S3) = $ite(aa_set_o_o(member_o2((A)),S3),aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(B,(A))),groups3502327434004483295_o_int(C,minus_minus_set_o(S3,insert_o2((A),bot_bot_set_o)))),groups3502327434004483295_o_int(C,minus_minus_set_o(S3,insert_o2((A),bot_bot_set_o)))) ) ) ).

% prod.delta_remove
tff(fact_6436_prod_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real,C: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_fr(A),B),C),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_real_real(times_times_real(aa_complex_real(B,A)),groups766887009212190081x_real(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))),groups766887009212190081x_real(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% prod.delta_remove
tff(fact_6437_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat,Acc2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A),Acc2,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc2))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_6438_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: nat,Xba: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,Xaa),Xba),Xc) = Y )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xba),Xaa),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xaa),one_one_nat)),Xba),aa_nat_nat(aa_nat_fun_nat_nat(Xb,Xaa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_6439_prod__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_real2(B3),minus_minus_set_real(B2,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,B3)) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real2(A3),A2)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,A3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A2)),groups1681761925125756287l_real(F,B2)) ) ) ) ) ).

% prod_mono2
tff(fact_6440_prod__mono2,axiom,
    ! [B2: set_o,A2: set_o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,B2)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
       => ( ! [B3: $o] :
              ( aa_set_o_o(member_o2((B3)),minus_minus_set_o(B2,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_o_real(F,(B3))) )
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o2((A3)),A2)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(A3))) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups234877984723959775o_real(F,A2)),groups234877984723959775o_real(F,B2)) ) ) ) ) ).

% prod_mono2
tff(fact_6441_prod__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,one_one_real),aa_complex_real(F,B3)) )
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),A2)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,A3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A2)),groups766887009212190081x_real(F,B2)) ) ) ) ) ).

% prod_mono2
tff(fact_6442_prod__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
       => ( ! [B3: int] :
              ( aa_set_int_o(member_int2(B3),minus_minus_set_int(B2,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,B3)) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int2(A3),A2)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,A3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A2)),groups2316167850115554303t_real(F,B2)) ) ) ) ) ).

% prod_mono2
tff(fact_6443_prod__mono2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B2)
       => ( ! [B3: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(B3),minus_925952699566721837d_enat(B2,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_Ext2541572767299617110t_real(F,B3)) )
         => ( ! [A3: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),A2)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,A3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups97031904164794029t_real(F,A2)),groups97031904164794029t_real(F,B2)) ) ) ) ) ).

% prod_mono2
tff(fact_6444_prod__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_real2(B3),minus_minus_set_real(B2,A2))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(F,B3)) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real2(A3),A2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,A3)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4694064378042380927al_int(F,A2)),groups4694064378042380927al_int(F,B2)) ) ) ) ) ).

% prod_mono2
tff(fact_6445_prod__mono2,axiom,
    ! [B2: set_o,A2: set_o,F: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,B2)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
       => ( ! [B3: $o] :
              ( aa_set_o_o(member_o2((B3)),minus_minus_set_o(B2,A2))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_o_int(F,(B3))) )
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o2((A3)),A2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(F,(A3))) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3502327434004483295_o_int(F,A2)),groups3502327434004483295_o_int(F,B2)) ) ) ) ) ).

% prod_mono2
tff(fact_6446_prod__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,one_one_int),aa_complex_int(F,B3)) )
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),A2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,A3)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups858564598930262913ex_int(F,A2)),groups858564598930262913ex_int(F,B2)) ) ) ) ) ).

% prod_mono2
tff(fact_6447_prod__mono2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B2)
       => ( ! [B3: extended_enat] :
              ( aa_set3337841544853333262enat_o(member_Extended_enat(B3),minus_925952699566721837d_enat(B2,A2))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_Extended_enat_int(F,B3)) )
         => ( ! [A3: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(A3),A2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_Extended_enat_int(F,A3)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups2878480467620962989at_int(F,A2)),groups2878480467620962989at_int(F,B2)) ) ) ) ) ).

% prod_mono2
tff(fact_6448_prod__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_real] :
      ( 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_nat2(B3),minus_minus_set_nat(B2,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(F,B3)) )
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat2(A3),A2)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,A3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F),A2)),aa_set_nat_real(groups129246275422532515t_real(F),B2)) ) ) ) ) ).

% prod_mono2
tff(fact_6449_prod__diff1,axiom,
    ! [A2: set_complex,F: fun_complex_complex,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( aa_complex_complex(F,A) != zero_zero_complex )
       => ( groups3708469109370488835omplex(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A2),aa_complex_complex(divide1717551699836669952omplex(groups3708469109370488835omplex(F,A2)),aa_complex_complex(F,A)),groups3708469109370488835omplex(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_6450_prod__diff1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex3741802288643809319omplex,A: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( aa_Ext8748583412541189208omplex(F,A) != zero_zero_complex )
       => ( groups4622424608036095791omplex(F,minus_925952699566721837d_enat(A2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(A),A2),aa_complex_complex(divide1717551699836669952omplex(groups4622424608036095791omplex(F,A2)),aa_Ext8748583412541189208omplex(F,A)),groups4622424608036095791omplex(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_6451_prod__diff1,axiom,
    ! [A2: set_real,F: fun_real_complex,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( aa_real_complex(F,A) != zero_zero_complex )
       => ( groups713298508707869441omplex(F,minus_minus_set_real(A2,insert_real2(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real2(A),A2),aa_complex_complex(divide1717551699836669952omplex(groups713298508707869441omplex(F,A2)),aa_real_complex(F,A)),groups713298508707869441omplex(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_6452_prod__diff1,axiom,
    ! [A2: set_o,F: fun_o_complex,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( aa_o_complex(F,(A)) != zero_zero_complex )
       => ( groups4859619685533338977omplex(F,minus_minus_set_o(A2,insert_o2((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o2((A)),A2),aa_complex_complex(divide1717551699836669952omplex(groups4859619685533338977omplex(F,A2)),aa_o_complex(F,(A))),groups4859619685533338977omplex(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_6453_prod__diff1,axiom,
    ! [A2: set_int,F: fun_int_complex,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( aa_int_complex(F,A) != zero_zero_complex )
       => ( groups7440179247065528705omplex(F,minus_minus_set_int(A2,insert_int2(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int2(A),A2),aa_complex_complex(divide1717551699836669952omplex(groups7440179247065528705omplex(F,A2)),aa_int_complex(F,A)),groups7440179247065528705omplex(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_6454_prod__diff1,axiom,
    ! [A2: set_nat,F: fun_nat_complex,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( aa_nat_complex(F,A) != zero_zero_complex )
       => ( groups6464643781859351333omplex(F,minus_minus_set_nat(A2,insert_nat2(A,bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat2(A),A2),aa_complex_complex(divide1717551699836669952omplex(groups6464643781859351333omplex(F,A2)),aa_nat_complex(F,A)),groups6464643781859351333omplex(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_6455_prod__diff1,axiom,
    ! [A2: set_complex,F: fun_complex_nat,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( aa_complex_nat(F,A) != zero_zero_nat )
       => ( groups861055069439313189ex_nat(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups861055069439313189ex_nat(F,A2)),aa_complex_nat(F,A)),groups861055069439313189ex_nat(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_6456_prod__diff1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat,A: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( aa_Extended_enat_nat(F,A) != zero_zero_nat )
       => ( groups2880970938130013265at_nat(F,minus_925952699566721837d_enat(A2,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups2880970938130013265at_nat(F,A2)),aa_Extended_enat_nat(F,A)),groups2880970938130013265at_nat(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_6457_prod__diff1,axiom,
    ! [A2: set_real,F: fun_real_nat,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( aa_real_nat(F,A) != zero_zero_nat )
       => ( groups4696554848551431203al_nat(F,minus_minus_set_real(A2,insert_real2(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real2(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups4696554848551431203al_nat(F,A2)),aa_real_nat(F,A)),groups4696554848551431203al_nat(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_6458_prod__diff1,axiom,
    ! [A2: set_o,F: fun_o_nat,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( aa_o_nat(F,(A)) != zero_zero_nat )
       => ( groups3504817904513533571_o_nat(F,minus_minus_set_o(A2,insert_o2((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o2((A)),A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups3504817904513533571_o_nat(F,A2)),aa_o_nat(F,(A))),groups3504817904513533571_o_nat(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_6459_log__of__power__less,axiom,
    ! [Mb: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(power_power_real(B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_less
tff(fact_6460_pochhammer__Suc__prod,axiom,
    ! [A: extended_enat,Nb: nat] : aa_nat_Extended_enat(comm_s3181272606743183617d_enat(A),aa_nat_nat(suc,Nb)) = groups7961826882256487087d_enat(aTP_Lamm_km(A),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_6461_pochhammer__Suc__prod,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,Nb)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_kn(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_6462_pochhammer__Suc__prod,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_ko(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_6463_pochhammer__Suc__prod,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kp,A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_6464_pochhammer__prod__rev,axiom,
    ! [A: extended_enat,Nb: nat] : aa_nat_Extended_enat(comm_s3181272606743183617d_enat(A),Nb) = groups7961826882256487087d_enat(aa_nat2025370463774033377d_enat(aTP_Lamm_kq(A),Nb),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_6465_pochhammer__prod__rev,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),Nb) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_kr(A),Nb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_6466_pochhammer__prod__rev,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),Nb) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ks(A),Nb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_6467_pochhammer__prod__rev,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kt(A),Nb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_6468_log__of__power__le,axiom,
    ! [Mb: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(power_power_real(B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_le
tff(fact_6469_prod_Oin__pairs,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_ku(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) ).

% prod.in_pairs
tff(fact_6470_prod_Oin__pairs,axiom,
    ! [G: fun_nat_complex,Mb: nat,Nb: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))) = groups6464643781859351333omplex(aTP_Lamm_kv(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) ).

% prod.in_pairs
tff(fact_6471_prod_Oin__pairs,axiom,
    ! [G: fun_na8797796813480665199d_enat,Mb: nat,Nb: nat] : groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))) = groups7961826882256487087d_enat(aTP_Lamm_kw(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) ).

% prod.in_pairs
tff(fact_6472_prod_Oin__pairs,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_kx(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) ).

% prod.in_pairs
tff(fact_6473_prod_Oin__pairs,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ky(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) ).

% prod.in_pairs
tff(fact_6474_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_kz(F),A,B),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_6475_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,A: nat,B: nat] : groups2073611262835488442omplex(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_la(F),A,B),zero_zero_complex) ).

% sum_atLeastAtMost_code
tff(fact_6476_sum__atLeastAtMost__code,axiom,
    ! [F: fun_na8797796813480665199d_enat,A: nat,B: nat] : groups7108830773950497114d_enat(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = aa_Ext7969882492699152700d_enat(set_fo2538466533108834004d_enat(aTP_Lamm_lb(F),A,B),zero_z5237406670263579293d_enat) ).

% sum_atLeastAtMost_code
tff(fact_6477_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_lc(F),A),B),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_6478_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B: nat] : aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_ld(F),A),B),zero_zero_real) ).

% sum_atLeastAtMost_code
tff(fact_6479_pochhammer__Suc__prod__rev,axiom,
    ! [A: extended_enat,Nb: nat] : aa_nat_Extended_enat(comm_s3181272606743183617d_enat(A),aa_nat_nat(suc,Nb)) = groups7961826882256487087d_enat(aa_nat2025370463774033377d_enat(aTP_Lamm_kq(A),Nb),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_6480_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,Nb)) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_kr(A),Nb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_6481_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ks(A),Nb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_6482_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kt(A),Nb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_6483_less__log2__of__power,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)),Mb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ).

% less_log2_of_power
tff(fact_6484_le__log2__of__power,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)),Mb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ).

% le_log2_of_power
tff(fact_6485_log2__of__power__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_less
tff(fact_6486_log2__of__power__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_le
tff(fact_6487_ceiling__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_6488_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B)
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_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,Nb)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_6489_power__half__series,axiom,
    aa_real_o(sums_real(aTP_Lamm_le),one_one_real) ).

% power_half_series
tff(fact_6490_sums__zero,axiom,
    aa_nat_o(sums_nat(aTP_Lamm_ax),zero_zero_nat) ).

% sums_zero
tff(fact_6491_sums__zero,axiom,
    aa_real_o(sums_real(aTP_Lamm_dr),zero_zero_real) ).

% sums_zero
tff(fact_6492_sums__zero,axiom,
    aa_int_o(sums_int(aTP_Lamm_lf),zero_zero_int) ).

% sums_zero
tff(fact_6493_sums__zero,axiom,
    aa_complex_o(sums_complex(aTP_Lamm_lg),zero_zero_complex) ).

% sums_zero
tff(fact_6494_sums__If__finite__set_H,axiom,
    ! [G: fun_nat_real,S3: real,A2: set_nat,S5: real,F: fun_nat_real] :
      ( aa_real_o(sums_real(G),S3)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( ( S5 = aa_real_real(plus_plus_real(S3),aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_lh(G),F)),A2)) )
         => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_li(G),A2),F)),S5) ) ) ) ).

% sums_If_finite_set'
tff(fact_6495_sums__iff__shift_H,axiom,
    ! [F: fun_nat_real,Nb: nat,S: real] :
      ( aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_gr(F),Nb)),aa_real_real(minus_minus_real(S),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))))
    <=> aa_real_o(sums_real(F),S) ) ).

% sums_iff_shift'
tff(fact_6496_sums__split__initial__segment,axiom,
    ! [F: fun_nat_real,S: real,Nb: nat] :
      ( aa_real_o(sums_real(F),S)
     => aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_gr(F),Nb)),aa_real_real(minus_minus_real(S),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))) ) ).

% sums_split_initial_segment
tff(fact_6497_sums__iff__shift,axiom,
    ! [F: fun_nat_real,Nb: nat,S: real] :
      ( aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_gr(F),Nb)),S)
    <=> aa_real_o(sums_real(F),aa_real_real(plus_plus_real(S),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))) ) ).

% sums_iff_shift
tff(fact_6498_prod__pos__nat__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups861055069439313189ex_nat(F,A2))
      <=> ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,X3)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_6499_prod__pos__nat__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups1707563613775114915nt_nat(F,A2))
      <=> ! [X3: int] :
            ( aa_set_int_o(member_int2(X3),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,X3)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_6500_prod__pos__nat__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups2880970938130013265at_nat(F,A2))
      <=> ! [X3: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X3),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Extended_enat_nat(F,X3)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_6501_prod__pos__nat__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A2))
      <=> ! [X3: nat] :
            ( aa_set_nat_o(member_nat2(X3),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X3)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_6502_int__prod,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_kb(F),A2) ).

% int_prod
tff(fact_6503_int__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_hx(F)),A2) ).

% int_prod
tff(fact_6504_prod__int__plus__eq,axiom,
    ! [Ia: nat,J: nat] : aa_set_nat_int(groups705719431365010083at_int(semiri1314217659103216013at_int),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Ia),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),J))) = groups1705073143266064639nt_int(aTP_Lamm_bd,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,Ia),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),J)))) ).

% prod_int_plus_eq
tff(fact_6505_sums__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,S: real,Ta: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
     => ( aa_real_o(sums_real(F),S)
       => ( aa_real_o(sums_real(G),Ta)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,S),Ta) ) ) ) ).

% sums_le
tff(fact_6506_sums__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,S: nat,Ta: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(G,N))
     => ( aa_nat_o(sums_nat(F),S)
       => ( aa_nat_o(sums_nat(G),Ta)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,S),Ta) ) ) ) ).

% sums_le
tff(fact_6507_sums__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,S: int,Ta: int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(G,N))
     => ( aa_int_o(sums_int(F),S)
       => ( aa_int_o(sums_int(G),Ta)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,S),Ta) ) ) ) ).

% sums_le
tff(fact_6508_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_6509_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_6510_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_6511_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_6512_sums__single,axiom,
    ! [Ia: nat,F: fun_nat_nat] : aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_lj(Ia),F)),aa_nat_nat(F,Ia)) ).

% sums_single
tff(fact_6513_sums__single,axiom,
    ! [Ia: nat,F: fun_nat_real] : aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_lk(Ia),F)),aa_nat_real(F,Ia)) ).

% sums_single
tff(fact_6514_sums__single,axiom,
    ! [Ia: nat,F: fun_nat_int] : aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_ll(Ia),F)),aa_nat_int(F,Ia)) ).

% sums_single
tff(fact_6515_sums__single,axiom,
    ! [Ia: nat,F: fun_nat_complex] : aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lm(Ia),F)),aa_nat_complex(F,Ia)) ).

% sums_single
tff(fact_6516_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_ey(F),G)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% sums_add
tff(fact_6517_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_ln(F),G)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% sums_add
tff(fact_6518_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_fa(F),G)),aa_real_real(plus_plus_real(A),B)) ) ) ).

% sums_add
tff(fact_6519_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_lo(C),F)),aa_real_real(times_times_real(D),C))
      <=> aa_real_o(sums_real(F),D) ) ) ).

% sums_mult2_iff
tff(fact_6520_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_lp(C),F)),aa_complex_complex(times_times_complex(D),C))
      <=> aa_complex_o(sums_complex(F),D) ) ) ).

% sums_mult2_iff
tff(fact_6521_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_lq(C),F)),aa_real_real(times_times_real(C),D))
      <=> aa_real_o(sums_real(F),D) ) ) ).

% sums_mult_iff
tff(fact_6522_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_lr(C),F)),aa_complex_complex(times_times_complex(C),D))
      <=> aa_complex_o(sums_complex(F),D) ) ) ).

% sums_mult_iff
tff(fact_6523_sums__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex,A: complex] :
      ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lr(C),F)),A)
     => ( ( C != zero_zero_complex )
       => aa_complex_o(sums_complex(F),aa_complex_complex(divide1717551699836669952omplex(A),C)) ) ) ).

% sums_mult_D
tff(fact_6524_sums__mult__D,axiom,
    ! [C: real,F: fun_nat_real,A: real] :
      ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_lq(C),F)),A)
     => ( ( C != zero_zero_real )
       => aa_real_o(sums_real(F),aa_real_real(divide_divide_real(A),C)) ) ) ).

% sums_mult_D
tff(fact_6525_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_gp(F)),S)
       => aa_real_o(sums_real(F),S) ) ) ).

% sums_Suc_imp
tff(fact_6526_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_ki(F)),S)
       => aa_complex_o(sums_complex(F),S) ) ) ).

% sums_Suc_imp
tff(fact_6527_sums__Suc__iff,axiom,
    ! [F: fun_nat_real,S: real] :
      ( aa_real_o(sums_real(aTP_Lamm_gp(F)),S)
    <=> aa_real_o(sums_real(F),aa_real_real(plus_plus_real(S),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc_iff
tff(fact_6528_sums__Suc,axiom,
    ! [F: fun_nat_nat,L: nat] :
      ( aa_nat_o(sums_nat(aTP_Lamm_go(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_6529_sums__Suc,axiom,
    ! [F: fun_nat_int,L: int] :
      ( aa_int_o(sums_int(aTP_Lamm_hh(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_6530_sums__Suc,axiom,
    ! [F: fun_nat_real,L: real] :
      ( aa_real_o(sums_real(aTP_Lamm_gp(F)),L)
     => aa_real_o(sums_real(F),aa_real_real(plus_plus_real(L),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_6531_sums__zero__iff__shift,axiom,
    ! [Nb: nat,F: fun_nat_real,S: real] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
         => ( aa_nat_real(F,I2) = zero_zero_real ) )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ls(Nb),F)),S)
      <=> aa_real_o(sums_real(F),S) ) ) ).

% sums_zero_iff_shift
tff(fact_6532_sums__zero__iff__shift,axiom,
    ! [Nb: nat,F: fun_nat_complex,S: complex] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
         => ( aa_nat_complex(F,I2) = zero_zero_complex ) )
     => ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lt(Nb),F)),S)
      <=> aa_complex_o(sums_complex(F),S) ) ) ).

% sums_zero_iff_shift
tff(fact_6533_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_nat2(N),N4)
           => ( aa_nat_int(F,N) = zero_zero_int ) )
       => aa_int_o(sums_int(F),aa_set_nat_int(groups3539618377306564664at_int(F),N4)) ) ) ).

% sums_finite
tff(fact_6534_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_nat2(N),N4)
           => ( aa_nat_complex(F,N) = zero_zero_complex ) )
       => aa_complex_o(sums_complex(F),groups2073611262835488442omplex(F,N4)) ) ) ).

% sums_finite
tff(fact_6535_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_nat2(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_6536_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_nat2(N),N4)
           => ( aa_nat_real(F,N) = zero_zero_real ) )
       => aa_real_o(sums_real(F),aa_set_nat_real(groups6591440286371151544t_real(F),N4)) ) ) ).

% sums_finite
tff(fact_6537_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_lu(P),F)),aa_set_nat_int(groups3539618377306564664at_int(F),collect_nat(P))) ) ).

% sums_If_finite
tff(fact_6538_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_lv(P),F)),groups2073611262835488442omplex(F,collect_nat(P))) ) ).

% sums_If_finite
tff(fact_6539_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_lw(P),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),collect_nat(P))) ) ).

% sums_If_finite
tff(fact_6540_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_lx(P),F)),aa_set_nat_real(groups6591440286371151544t_real(F),collect_nat(P))) ) ).

% sums_If_finite
tff(fact_6541_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_ly(A2),F)),aa_set_nat_int(groups3539618377306564664at_int(F),A2)) ) ).

% sums_If_finite_set
tff(fact_6542_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_lz(A2),F)),groups2073611262835488442omplex(F,A2)) ) ).

% sums_If_finite_set
tff(fact_6543_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_ma(A2),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) ) ).

% sums_If_finite_set
tff(fact_6544_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_mb(A2),F)),aa_set_nat_real(groups6591440286371151544t_real(F),A2)) ) ).

% sums_If_finite_set
tff(fact_6545_powser__sums__if,axiom,
    ! [Mb: nat,Z: int] : aa_int_o(sums_int(aa_int_fun_nat_int(aTP_Lamm_mc(Mb),Z)),aa_nat_int(power_power_int(Z),Mb)) ).

% powser_sums_if
tff(fact_6546_powser__sums__if,axiom,
    ! [Mb: nat,Z: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_md(Mb),Z)),aa_nat_real(power_power_real(Z),Mb)) ).

% powser_sums_if
tff(fact_6547_powser__sums__if,axiom,
    ! [Mb: nat,Z: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_me(Mb),Z)),aa_nat_complex(power_power_complex(Z),Mb)) ).

% powser_sums_if
tff(fact_6548_ln__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),numeral_numeral_real(bit0(one)))
       => ( aa_real_real(ln_ln_real,Xb) = suminf_real(aTP_Lamm_mf(Xb)) ) ) ) ).

% ln_series
tff(fact_6549_ceiling__log__eq__powr__iff,axiom,
    ! [Xb: real,B: real,K: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(B),Xb)) = 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(powr_real(B),aa_nat_real(semiri5074537144036343181t_real,K))),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_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_6550_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,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(log(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_6551_and__int_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(Xb),Xaa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa))
       => ~ ( ( Y = $ite(
                  ( aa_set_int_o(member_int2(Xb),insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & aa_set_int_o(member_int2(Xaa),insert_int2(zero_zero_int,insert_int2(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,numeral_numeral_int(bit0(one))),Xb)
                      & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),Xaa) ))),
                  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,numeral_numeral_int(bit0(one))),Xb)
                        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),Xaa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(Xb),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(Xaa),numeral_numeral_int(bit0(one)))))) ) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_6552_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_int2(K),insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(member_int2(L),insert_int2(zero_zero_int,insert_int2(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,numeral_numeral_int(bit0(one))),K)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(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,numeral_numeral_int(bit0(one))),K)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(L),numeral_numeral_int(bit0(one)))))) ) ) ) ).

% and_int.psimps
tff(fact_6553_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,Mb: nat] : groups2073611262835488442omplex(aTP_Lamm_mg(R2),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Mb)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Mb))),numera6690914467698888265omplex(bit0(one)))),aa_nat_complex(gbinomial_complex(R2),aa_nat_nat(suc,Mb))) ).

% gchoose_row_sum_weighted
tff(fact_6554_gchoose__row__sum__weighted,axiom,
    ! [R2: real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_mh(R2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Mb)) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb))),numeral_numeral_real(bit0(one)))),aa_nat_real(gbinomial_real(R2),aa_nat_nat(suc,Mb))) ).

% gchoose_row_sum_weighted
tff(fact_6555_powr__0,axiom,
    ! [Z: real] : aa_real_real(powr_real(zero_zero_real),Z) = zero_zero_real ).

% powr_0
tff(fact_6556_powr__eq__0__iff,axiom,
    ! [W2: real,Z: real] :
      ( ( aa_real_real(powr_real(W2),Z) = zero_zero_real )
    <=> ( W2 = zero_zero_real ) ) ).

% powr_eq_0_iff
tff(fact_6557_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_ax) = zero_zero_nat ).

% suminf_zero
tff(fact_6558_suminf__zero,axiom,
    suminf_real(aTP_Lamm_dr) = zero_zero_real ).

% suminf_zero
tff(fact_6559_suminf__zero,axiom,
    suminf_int(aTP_Lamm_lf) = zero_zero_int ).

% suminf_zero
tff(fact_6560_suminf__zero,axiom,
    suminf_complex(aTP_Lamm_lg) = zero_zero_complex ).

% suminf_zero
tff(fact_6561_powr__zero__eq__one,axiom,
    ! [Xb: real] :
      aa_real_real(powr_real(Xb),zero_zero_real) = $ite(Xb = zero_zero_real,zero_zero_real,one_one_real) ).

% powr_zero_eq_one
tff(fact_6562_floor__zero,axiom,
    aa_real_int(archim6058952711729229775r_real,zero_zero_real) = zero_zero_int ).

% floor_zero
tff(fact_6563_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_6564_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_real(gbinomial_real(zero_zero_real),aa_nat_nat(suc,K)) = zero_zero_real ).

% gbinomial_0(2)
tff(fact_6565_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_6566_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_6567_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_6568_gbinomial__0_I1_J,axiom,
    ! [A: int] : aa_nat_int(gbinomial_int(A),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_6569_gbinomial__0_I1_J,axiom,
    ! [A: real] : aa_nat_real(gbinomial_real(A),zero_zero_nat) = one_one_real ).

% gbinomial_0(1)
tff(fact_6570_gbinomial__0_I1_J,axiom,
    ! [A: complex] : aa_nat_complex(gbinomial_complex(A),zero_zero_nat) = one_one_complex ).

% gbinomial_0(1)
tff(fact_6571_zero__le__floor,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% zero_le_floor
tff(fact_6572_floor__less__zero,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% floor_less_zero
tff(fact_6573_numeral__le__floor,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(V)),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(V)),Xb) ) ).

% numeral_le_floor
tff(fact_6574_zero__less__floor,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ).

% zero_less_floor
tff(fact_6575_floor__le__zero,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ).

% floor_le_zero
tff(fact_6576_floor__less__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),numeral_numeral_int(V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),numeral_numeral_real(V)) ) ).

% floor_less_numeral
tff(fact_6577_one__le__floor,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ).

% one_le_floor
tff(fact_6578_floor__less__one,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ).

% floor_less_one
tff(fact_6579_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_ev(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_6580_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_eu(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_6581_numeral__less__floor,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,numeral_numeral_int(V)),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(numeral_numeral_real(V)),one_one_real)),Xb) ) ).

% numeral_less_floor
tff(fact_6582_floor__le__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),numeral_numeral_int(V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(plus_plus_real(numeral_numeral_real(V)),one_one_real)) ) ).

% floor_le_numeral
tff(fact_6583_one__less__floor,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(bit0(one))),Xb) ) ).

% one_less_floor
tff(fact_6584_floor__le__one,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),numeral_numeral_real(bit0(one))) ) ).

% floor_le_one
tff(fact_6585_neg__numeral__le__floor,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(V))),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),Xb) ) ).

% neg_numeral_le_floor
tff(fact_6586_floor__less__neg__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_int_int(uminus_uminus_int,numeral_numeral_int(V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) ) ).

% floor_less_neg_numeral
tff(fact_6587_neg__numeral__less__floor,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(V))),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)),Xb) ) ).

% neg_numeral_less_floor
tff(fact_6588_floor__le__neg__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_int_int(uminus_uminus_int,numeral_numeral_int(V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)) ) ).

% floor_le_neg_numeral
tff(fact_6589_floor__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim6058952711729229775r_real,Y)) ) ).

% floor_mono
tff(fact_6590_of__int__floor__le,axiom,
    ! [Xb: 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,Xb))),Xb) ).

% of_int_floor_le
tff(fact_6591_floor__less__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim6058952711729229775r_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% floor_less_cancel
tff(fact_6592_le__floor__iff,axiom,
    ! [Z: int,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),Xb) ) ).

% le_floor_iff
tff(fact_6593_floor__less__iff,axiom,
    ! [Xb: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,Z)) ) ).

% floor_less_iff
tff(fact_6594_le__floor__add,axiom,
    ! [Xb: 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,Xb)),aa_real_int(archim6058952711729229775r_real,Y))),aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(Xb),Y))) ).

% le_floor_add
tff(fact_6595_floor__add__int,axiom,
    ! [Xb: real,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),Z) = aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(Xb),aa_int_real(ring_1_of_int_real,Z))) ).

% floor_add_int
tff(fact_6596_int__add__floor,axiom,
    ! [Z: int,Xb: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_real_int(archim6058952711729229775r_real,Xb)) = aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Z)),Xb)) ).

% int_add_floor
tff(fact_6597_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_6598_gbinomial__Suc__Suc,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(plus_plus_real(aa_nat_real(gbinomial_real(A),K)),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_6599_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),K) = aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_6600_powr__add,axiom,
    ! [Xb: real,A: real,B: real] : aa_real_real(powr_real(Xb),aa_real_real(plus_plus_real(A),B)) = aa_real_real(times_times_real(aa_real_real(powr_real(Xb),A)),aa_real_real(powr_real(Xb),B)) ).

% powr_add
tff(fact_6601_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_nat2(N),N4)
           => ( aa_nat_int(F,N) = zero_zero_int ) )
       => ( suminf_int(F) = aa_set_nat_int(groups3539618377306564664at_int(F),N4) ) ) ) ).

% suminf_finite
tff(fact_6602_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_nat2(N),N4)
           => ( aa_nat_complex(F,N) = zero_zero_complex ) )
       => ( suminf_complex(F) = groups2073611262835488442omplex(F,N4) ) ) ) ).

% suminf_finite
tff(fact_6603_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_nat2(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_6604_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_nat2(N),N4)
           => ( aa_nat_real(F,N) = zero_zero_real ) )
       => ( suminf_real(F) = aa_set_nat_real(groups6591440286371151544t_real(F),N4) ) ) ) ).

% suminf_finite
tff(fact_6605_one__add__floor,axiom,
    ! [Xb: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int) = aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(Xb),one_one_real)) ).

% one_add_floor
tff(fact_6606_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_6607_gbinomial__addition__formula,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)) = aa_real_real(plus_plus_real(aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),K)) ).

% gbinomial_addition_formula
tff(fact_6608_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(power_power_real(aa_real_real(divide_divide_real(A),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(gbinomial_real(A),K)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_6609_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_6610_gbinomial__mult__1,axiom,
    ! [A: real,K: nat] : aa_real_real(times_times_real(A),aa_nat_real(gbinomial_real(A),K)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(gbinomial_real(A),K))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_6611_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_6612_gbinomial__mult__1_H,axiom,
    ! [A: real,K: nat] : aa_real_real(times_times_real(aa_nat_real(gbinomial_real(A),K)),A) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(gbinomial_real(A),K))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_6613_floor__unique,axiom,
    ! [Z: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Z)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xb) = Z ) ) ) ).

% floor_unique
tff(fact_6614_floor__eq__iff,axiom,
    ! [Xb: real,A: int] :
      ( ( aa_real_int(archim6058952711729229775r_real,Xb) = A )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,A)),one_one_real)) ) ) ).

% floor_eq_iff
tff(fact_6615_floor__split,axiom,
    ! [P: fun_int_o,Ta: real] :
      ( aa_int_o(P,aa_real_int(archim6058952711729229775r_real,Ta))
    <=> ! [I4: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,I4)),Ta)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,I4)),one_one_real)) )
         => aa_int_o(P,I4) ) ) ).

% floor_split
tff(fact_6616_le__mult__floor,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B))),aa_real_int(archim6058952711729229775r_real,aa_real_real(times_times_real(A),B))) ) ) ).

% le_mult_floor
tff(fact_6617_less__floor__iff,axiom,
    ! [Z: int,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Z)),one_one_real)),Xb) ) ).

% less_floor_iff
tff(fact_6618_floor__le__iff,axiom,
    ! [Xb: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Z)),one_one_real)) ) ).

% floor_le_iff
tff(fact_6619_floor__correct,axiom,
    ! [Xb: 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,Xb))),Xb)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int))) ) ).

% floor_correct
tff(fact_6620_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_6621_Suc__times__gbinomial,axiom,
    ! [K: nat,A: real] : aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K))) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_real(gbinomial_real(A),K)) ).

% Suc_times_gbinomial
tff(fact_6622_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_6623_gbinomial__absorption,axiom,
    ! [K: nat,A: real] : aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K))) = aa_real_real(times_times_real(A),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),K)) ).

% gbinomial_absorption
tff(fact_6624_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_6625_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(times_times_real(aa_nat_real(gbinomial_real(A),Mb)),aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),K)) = aa_real_real(times_times_real(aa_nat_real(gbinomial_real(A),K)),aa_nat_real(gbinomial_real(aa_real_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_6626_floor__divide__lower,axiom,
    ! [Q2: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(P2),Q2)))),Q2)),P2) ) ).

% floor_divide_lower
tff(fact_6627_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_6628_gbinomial__rec,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(times_times_real(aa_nat_real(gbinomial_real(A),K)),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_6629_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_6630_gbinomial__factors,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))),aa_nat_real(gbinomial_real(A),K)) ).

% gbinomial_factors
tff(fact_6631_floor__divide__upper,axiom,
    ! [Q2: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,P2),aa_real_real(times_times_real(aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(P2),Q2)))),one_one_real)),Q2)) ) ).

% floor_divide_upper
tff(fact_6632_round__def,axiom,
    ! [Xb: real] : aa_real_int(archim8280529875227126926d_real,Xb) = aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))) ).

% round_def
tff(fact_6633_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_6634_gbinomial__minus,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(uminus_uminus_real,A)),K) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% gbinomial_minus
tff(fact_6635_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_6636_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(gbinomial_real(A),K) = aa_real_real(plus_plus_real(aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_6637_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))
     => ( ! [K3: int,L3: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K3),L3))
           => ( ( ~ ( aa_set_int_o(member_int2(K3),insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & aa_set_int_o(member_int2(L3),insert_int2(zero_zero_int,insert_int2(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(K3),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(L3),numeral_numeral_int(bit0(one)))) )
             => aa_int_o(aa_int_fun_int_o(P,K3),L3) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_6638_gbinomial__sum__up__index,axiom,
    ! [K: nat,Nb: nat] : groups2073611262835488442omplex(aTP_Lamm_mi(K),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_6639_gbinomial__sum__up__index,axiom,
    ! [K: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_mj,K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) = aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_6640_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_6641_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(gbinomial_real(A),K) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_6642_floor__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_6643_floor__log__nat__eq__if,axiom,
    ! [B: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B)
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Nb) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_6644_pi__series,axiom,
    aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(bit0(one)))) = suminf_real(aTP_Lamm_mk) ).

% pi_series
tff(fact_6645_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))
     => ( ! [I2: int,J2: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I2),J2))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2)
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J2) )
             => aa_int_o(aa_int_fun_int_o(P,I2),J2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_6646_gbinomial__partial__row__sum,axiom,
    ! [A: complex,Mb: nat] : groups2073611262835488442omplex(aTP_Lamm_mg(A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),one_one_complex)),numera6690914467698888265omplex(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_6647_gbinomial__partial__row__sum,axiom,
    ! [A: real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_mh(A)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),one_one_real)),numeral_numeral_real(bit0(one)))),aa_nat_real(gbinomial_real(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_6648_arcosh__def,axiom,
    ! [Xb: real] : aa_real_real(arcosh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(Xb),aa_real_real(powr_real(aa_real_real(minus_minus_real(aa_nat_real(power_power_real(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))))) ).

% arcosh_def
tff(fact_6649_round__altdef,axiom,
    ! [Xb: real] :
      aa_real_int(archim8280529875227126926d_real,Xb) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_real_real(archim2898591450579166408c_real,Xb)),aa_real_int(archim7802044766580827645g_real,Xb),aa_real_int(archim6058952711729229775r_real,Xb)) ).

% round_altdef
tff(fact_6650_gbinomial__r__part__sum,axiom,
    ! [Mb: nat] : groups2073611262835488442omplex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_nat_complex(semiri8010041392384452111omplex,Mb))),one_one_complex)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)) ).

% gbinomial_r_part_sum
tff(fact_6651_gbinomial__r__part__sum,axiom,
    ! [Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(gbinomial_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),one_one_real))),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_real(power_power_real(numeral_numeral_real(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)) ).

% gbinomial_r_part_sum
tff(fact_6652_atMost__iff,axiom,
    ! [Ia: $o,K: $o] :
      ( aa_set_o_o(member_o2((Ia)),set_ord_atMost_o((K)))
    <=> aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Ia)),(K)) ) ).

% atMost_iff
tff(fact_6653_atMost__iff,axiom,
    ! [Ia: filter_nat,K: filter_nat] :
      ( member_filter_nat(Ia,set_or9144418160755794905er_nat(K))
    <=> ord_le2510731241096832064er_nat(Ia,K) ) ).

% atMost_iff
tff(fact_6654_atMost__iff,axiom,
    ! [Ia: real,K: real] :
      ( aa_set_real_o(member_real2(Ia),set_ord_atMost_real(K))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ia),K) ) ).

% atMost_iff
tff(fact_6655_atMost__iff,axiom,
    ! [Ia: set_nat,K: set_nat] :
      ( aa_set_set_nat_o(member_set_nat2(Ia),set_or4236626031148496127et_nat(K))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Ia),K) ) ).

% atMost_iff
tff(fact_6656_atMost__iff,axiom,
    ! [Ia: int,K: int] :
      ( aa_set_int_o(member_int2(Ia),set_ord_atMost_int(K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),K) ) ).

% atMost_iff
tff(fact_6657_atMost__iff,axiom,
    ! [Ia: nat,K: nat] :
      ( aa_set_nat_o(member_nat2(Ia),aa_nat_set_nat(set_ord_atMost_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),K) ) ).

% atMost_iff
tff(fact_6658_atMost__subset__iff,axiom,
    ! [Xb: filter_nat,Y: filter_nat] :
      ( ord_le2426478655948331894er_nat(set_or9144418160755794905er_nat(Xb),set_or9144418160755794905er_nat(Y))
    <=> ord_le2510731241096832064er_nat(Xb,Y) ) ).

% atMost_subset_iff
tff(fact_6659_atMost__subset__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_ord_atMost_real(Xb)),set_ord_atMost_real(Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% atMost_subset_iff
tff(fact_6660_atMost__subset__iff,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_or4236626031148496127et_nat(Xb)),set_or4236626031148496127et_nat(Y))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Y) ) ).

% atMost_subset_iff
tff(fact_6661_atMost__subset__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_ord_atMost_int(Xb)),set_ord_atMost_int(Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).

% atMost_subset_iff
tff(fact_6662_atMost__subset__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_atMost_nat,Xb)),aa_nat_set_nat(set_ord_atMost_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).

% atMost_subset_iff
tff(fact_6663_of__real__eq__0__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(real_V1803761363581548252l_real,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_real ) ) ).

% of_real_eq_0_iff
tff(fact_6664_of__real__eq__0__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_complex(real_V4546457046886955230omplex,Xb) = zero_zero_complex )
    <=> ( Xb = zero_zero_real ) ) ).

% of_real_eq_0_iff
tff(fact_6665_of__real__0,axiom,
    aa_real_real(real_V1803761363581548252l_real,zero_zero_real) = zero_zero_real ).

% of_real_0
tff(fact_6666_of__real__0,axiom,
    aa_real_complex(real_V4546457046886955230omplex,zero_zero_real) = zero_zero_complex ).

% of_real_0
tff(fact_6667_of__real__add,axiom,
    ! [Xb: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(plus_plus_real(Xb),Y)) = aa_real_real(plus_plus_real(aa_real_real(real_V1803761363581548252l_real,Xb)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_add
tff(fact_6668_of__real__add,axiom,
    ! [Xb: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(plus_plus_real(Xb),Y)) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,Xb)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_add
tff(fact_6669_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_6670_Icc__subset__Iic__iff,axiom,
    ! [L: filter_nat,H: filter_nat,H2: filter_nat] :
      ( ord_le2426478655948331894er_nat(set_or1955772592623580779er_nat(L,H),set_or9144418160755794905er_nat(H2))
    <=> ( ~ ord_le2510731241096832064er_nat(L,H)
        | ord_le2510731241096832064er_nat(H,H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6671_Icc__subset__Iic__iff,axiom,
    ! [L: set_nat,H: set_nat,H2: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(L,H)),set_or4236626031148496127et_nat(H2))
    <=> ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,L),H)
        | aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6672_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H: nat,H2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L),H)),aa_nat_set_nat(set_ord_atMost_nat,H2))
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),H)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6673_Icc__subset__Iic__iff,axiom,
    ! [L: int,H: int,H2: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_or1266510415728281911st_int(L,H)),set_ord_atMost_int(H2))
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),H)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_6674_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_6675_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_6676_sum_OatMost__Suc,axiom,
    ! [G: fun_na8797796813480665199d_enat,Nb: nat] : groups7108830773950497114d_enat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups7108830773950497114d_enat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_Extended_enat(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_6677_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_6678_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_6679_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_6680_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_complex,Nb: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_6681_prod_OatMost__Suc,axiom,
    ! [G: fun_na8797796813480665199d_enat,Nb: nat] : groups7961826882256487087d_enat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(groups7961826882256487087d_enat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_Extended_enat(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_6682_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_6683_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_6684_atMost__0,axiom,
    aa_nat_set_nat(set_ord_atMost_nat,zero_zero_nat) = insert_nat2(zero_zero_nat,bot_bot_set_nat) ).

% atMost_0
tff(fact_6685_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: real] : bot_bot_set_real != set_ord_atMost_real(H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_6686_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: $o] : bot_bot_set_o != set_ord_atMost_o((H)) ).

% not_empty_eq_Iic_eq_empty
tff(fact_6687_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: int] : bot_bot_set_int != set_ord_atMost_int(H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_6688_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: nat] : bot_bot_set_nat != aa_nat_set_nat(set_ord_atMost_nat,H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_6689_atMost__def,axiom,
    ! [U: filter_nat] : set_or9144418160755794905er_nat(U) = collect_filter_nat(aTP_Lamm_ml(U)) ).

% atMost_def
tff(fact_6690_atMost__def,axiom,
    ! [U: real] : set_ord_atMost_real(U) = collect_real(aa_real_fun_real_o(aTP_Lamm_mm,U)) ).

% atMost_def
tff(fact_6691_atMost__def,axiom,
    ! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_ai,U)) ).

% atMost_def
tff(fact_6692_atMost__def,axiom,
    ! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_mn,U)) ).

% atMost_def
tff(fact_6693_atMost__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ak,U)) ).

% atMost_def
tff(fact_6694_atMost__atLeast0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,Nb) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb) ).

% atMost_atLeast0
tff(fact_6695_lessThan__Suc__atMost,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_ord_atMost_nat,K) ).

% lessThan_Suc_atMost
tff(fact_6696_atMost__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K)) = insert_nat2(aa_nat_nat(suc,K),aa_nat_set_nat(set_ord_atMost_nat,K)) ).

% atMost_Suc
tff(fact_6697_not__Iic__le__Icc,axiom,
    ! [H: int,L2: int,H2: int] : ~ aa_set_int_o(ord_less_eq_set_int(set_ord_atMost_int(H)),set_or1266510415728281911st_int(L2,H2)) ).

% not_Iic_le_Icc
tff(fact_6698_not__Iic__le__Icc,axiom,
    ! [H: real,L2: real,H2: real] : ~ aa_set_real_o(ord_less_eq_set_real(set_ord_atMost_real(H)),set_or1222579329274155063t_real(L2,H2)) ).

% not_Iic_le_Icc
tff(fact_6699_frac__ge__0,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,Xb)) ).

% frac_ge_0
tff(fact_6700_frac__lt__1,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(archim2898591450579166408c_real,Xb)),one_one_real) ).

% frac_lt_1
tff(fact_6701_frac__1__eq,axiom,
    ! [Xb: real] : aa_real_real(archim2898591450579166408c_real,aa_real_real(plus_plus_real(Xb),one_one_real)) = aa_real_real(archim2898591450579166408c_real,Xb) ).

% frac_1_eq
tff(fact_6702_Iic__subset__Iio__iff,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(set_or8332593352340944941d_enat(A)),set_or8419480210114673929d_enat(B))
    <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_6703_Iic__subset__Iio__iff,axiom,
    ! [A: real,B: real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_ord_atMost_real(A)),set_or5984915006950818249n_real(B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_6704_Iic__subset__Iio__iff,axiom,
    ! [A: int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_ord_atMost_int(A)),set_ord_lessThan_int(B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_6705_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_atMost_nat,A)),aa_nat_set_nat(set_ord_lessThan_nat,B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_6706_norm__less__p1,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(aa_real_real(real_V1803761363581548252l_real,aa_real_real(real_V7735802525324610683m_real,Xb))),one_one_real))) ).

% norm_less_p1
tff(fact_6707_norm__less__p1,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,Xb))),one_one_complex))) ).

% norm_less_p1
tff(fact_6708_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_hh(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_6709_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_na8797796813480665199d_enat,Nb: nat] : groups7108830773950497114d_enat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(G,zero_zero_nat)),groups7108830773950497114d_enat(aTP_Lamm_hi(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_6710_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_go(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_6711_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gp(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_6712_sum__telescope,axiom,
    ! [F: fun_nat_int,Ia: nat] : aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_ix(F)),aa_nat_set_nat(set_ord_atMost_nat,Ia)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,aa_nat_nat(suc,Ia))) ).

% sum_telescope
tff(fact_6713_sum__telescope,axiom,
    ! [F: fun_nat_real,Ia: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_iy(F)),aa_nat_set_nat(set_ord_atMost_nat,Ia)) = aa_real_real(minus_minus_real(aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,aa_nat_nat(suc,Ia))) ).

% sum_telescope
tff(fact_6714_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_complex,Nb: nat,D: fun_nat_complex] :
      ( ! [X3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mo(C),X3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mo(D),X3),aa_nat_set_nat(set_ord_atMost_nat,Nb))
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),Nb)
         => ( aa_nat_complex(C,I4) = aa_nat_complex(D,I4) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_6715_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_real,Nb: nat,D: fun_nat_real] :
      ( ! [X3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mp(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mp(D),X3)),aa_nat_set_nat(set_ord_atMost_nat,Nb))
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),Nb)
         => ( aa_nat_real(C,I4) = aa_nat_real(D,I4) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_6716_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_gp(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_6717_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_complex,Nb: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_complex_complex(times_times_complex(aa_nat_complex(G,zero_zero_nat)),groups6464643781859351333omplex(aTP_Lamm_ki(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_6718_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_na8797796813480665199d_enat,Nb: nat] : groups7961826882256487087d_enat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(G,zero_zero_nat)),groups7961826882256487087d_enat(aTP_Lamm_hi(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_6719_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_hh(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_6720_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_go(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_6721_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_mq(A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ms(A),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nested_swap'
tff(fact_6722_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_mt(A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_mv(A),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nested_swap'
tff(fact_6723_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_int,Nb: nat] : aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_mw(A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_my(A),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.nested_swap'
tff(fact_6724_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_mz(A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_na(A),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.nested_swap'
tff(fact_6725_frac__eq,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,Xb) = Xb )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% frac_eq
tff(fact_6726_polyfun__eq__0,axiom,
    ! [C: fun_nat_complex,Nb: nat] :
      ( ! [X3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mo(C),X3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),Nb)
         => ( aa_nat_complex(C,I4) = zero_zero_complex ) ) ) ).

% polyfun_eq_0
tff(fact_6727_polyfun__eq__0,axiom,
    ! [C: fun_nat_real,Nb: nat] :
      ( ! [X3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mp(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),Nb)
         => ( aa_nat_real(C,I4) = zero_zero_real ) ) ) ).

% polyfun_eq_0
tff(fact_6728_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_complex,Nb: nat,K: nat] :
      ( ! [W: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mo(C),W),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( aa_nat_complex(C,K) = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_6729_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_real,Nb: nat,K: nat] :
      ( ! [W: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mp(C),W)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( aa_nat_real(C,K) = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_6730_frac__add,axiom,
    ! [Xb: real,Y: real] :
      aa_real_real(archim2898591450579166408c_real,aa_real_real(plus_plus_real(Xb),Y)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(archim2898591450579166408c_real,Xb)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real),aa_real_real(plus_plus_real(aa_real_real(archim2898591450579166408c_real,Xb)),aa_real_real(archim2898591450579166408c_real,Y)),aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(aa_real_real(archim2898591450579166408c_real,Xb)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real)) ).

% frac_add
tff(fact_6731_sum_OatMost__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_hh(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_6732_sum_OatMost__shift,axiom,
    ! [G: fun_na8797796813480665199d_enat,Nb: nat] : groups7108830773950497114d_enat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(G,zero_zero_nat)),groups7108830773950497114d_enat(aTP_Lamm_hi(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_6733_sum_OatMost__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_go(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_6734_sum_OatMost__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(plus_plus_real(aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gp(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_6735_sum__up__index__split,axiom,
    ! [F: fun_nat_int,Mb: nat,Nb: nat] : aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_6736_sum__up__index__split,axiom,
    ! [F: fun_na8797796813480665199d_enat,Mb: nat,Nb: nat] : groups7108830773950497114d_enat(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups7108830773950497114d_enat(F,aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups7108830773950497114d_enat(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_6737_sum__up__index__split,axiom,
    ! [F: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_6738_sum__up__index__split,axiom,
    ! [F: fun_nat_real,Mb: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_6739_prod_OatMost__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(times_times_real(aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_gp(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_6740_prod_OatMost__shift,axiom,
    ! [G: fun_nat_complex,Nb: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(G,zero_zero_nat)),groups6464643781859351333omplex(aTP_Lamm_ki(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_6741_prod_OatMost__shift,axiom,
    ! [G: fun_na8797796813480665199d_enat,Nb: nat] : groups7961826882256487087d_enat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(G,zero_zero_nat)),groups7961826882256487087d_enat(aTP_Lamm_hi(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_6742_prod_OatMost__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_hh(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_6743_prod_OatMost__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_go(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_6744_atLeast1__atMost__eq__remove0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,Nb),insert_nat2(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_6745_gbinomial__parallel__sum,axiom,
    ! [A: complex,Nb: nat] : groups2073611262835488442omplex(aTP_Lamm_nb(A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,Nb))),one_one_complex)),Nb) ).

% gbinomial_parallel_sum
tff(fact_6746_gbinomial__parallel__sum,axiom,
    ! [A: real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_nc(A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),aa_nat_real(semiri5074537144036343181t_real,Nb))),one_one_real)),Nb) ).

% gbinomial_parallel_sum
tff(fact_6747_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,Nb: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_nd(Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_nf(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.triangle_reindex_eq
tff(fact_6748_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_real,Nb: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_nd(Nb)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_nh(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.triangle_reindex_eq
tff(fact_6749_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_int,Nb: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_nd(Nb)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_nj(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.triangle_reindex_eq
tff(fact_6750_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,Nb: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_nd(Nb)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_nk(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.triangle_reindex_eq
tff(fact_6751_sum__gp__basic,axiom,
    ! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),aa_set_nat_int(groups3539618377306564664at_int(power_power_int(Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(Xb),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_6752_sum__gp__basic,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xb)),groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_6753_sum__gp__basic,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(one_one_real),Xb)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(Xb),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_6754_polyfun__finite__roots,axiom,
    ! [C: fun_nat_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_nl(C),Nb)))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),Nb)
          & ( aa_nat_complex(C,I4) != zero_zero_complex ) ) ) ).

% polyfun_finite_roots
tff(fact_6755_polyfun__finite__roots,axiom,
    ! [C: fun_nat_real,Nb: nat] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_nat_fun_real_o(aTP_Lamm_nm(C),Nb)))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),Nb)
          & ( aa_nat_real(C,I4) != zero_zero_real ) ) ) ).

% polyfun_finite_roots
tff(fact_6756_polyfun__roots__finite,axiom,
    ! [C: fun_nat_complex,K: nat,Nb: nat] :
      ( ( aa_nat_complex(C,K) != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_nl(C),Nb))) ) ) ).

% polyfun_roots_finite
tff(fact_6757_polyfun__roots__finite,axiom,
    ! [C: fun_nat_real,K: nat,Nb: nat] :
      ( ( aa_nat_real(C,K) != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => aa_set_real_o(finite_finite_real,collect_real(aa_nat_fun_real_o(aTP_Lamm_nm(C),Nb))) ) ) ).

% polyfun_roots_finite
tff(fact_6758_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_int,A: int,Nb: nat] :
      ( ( aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_nn(C),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int )
     => ~ ! [B3: fun_nat_int] :
            ~ ! [Z3: int] : aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_nn(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z3),A)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_nn(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_6759_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_complex,A: complex,Nb: nat] :
      ( ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mo(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex )
     => ~ ! [B3: fun_nat_complex] :
            ~ ! [Z3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mo(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z3),A)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mo(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_6760_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_real,A: real,Nb: nat] :
      ( ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mp(C),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real )
     => ~ ! [B3: fun_nat_real] :
            ~ ! [Z3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mp(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(Z3),A)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mp(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_6761_polyfun__linear__factor,axiom,
    ! [C: fun_nat_int,Nb: nat,A: int] :
    ? [B3: fun_nat_int] :
    ! [Z3: int] : aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_nn(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z3),A)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_nn(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_nn(C),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_6762_polyfun__linear__factor,axiom,
    ! [C: fun_nat_complex,Nb: nat,A: complex] :
    ? [B3: fun_nat_complex] :
    ! [Z3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mo(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z3),A)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mo(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mo(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_6763_polyfun__linear__factor,axiom,
    ! [C: fun_nat_real,Nb: nat,A: real] :
    ? [B3: fun_nat_real] :
    ! [Z3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mp(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(Z3),A)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mp(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mp(C),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_6764_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_int(groups3539618377306564664at_int(power_power_int(Xb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),Mb)),aa_set_nat_int(groups3539618377306564664at_int(power_power_int(Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_6765_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_6766_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(Xb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_real_real(times_times_real(aa_nat_real(power_power_real(Xb),Mb)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_6767_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,Nb: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_no(Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_nf(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.triangle_reindex
tff(fact_6768_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_real,Nb: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_no(Nb)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_nh(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.triangle_reindex
tff(fact_6769_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_int,Nb: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_no(Nb)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_nj(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.triangle_reindex
tff(fact_6770_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,Nb: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_no(Nb)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_nk(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.triangle_reindex
tff(fact_6771_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,Nb: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_hq(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_6772_sum_Oin__pairs__0,axiom,
    ! [G: fun_na8797796813480665199d_enat,Nb: nat] : groups7108830773950497114d_enat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))) = groups7108830773950497114d_enat(aTP_Lamm_hr(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_6773_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hs(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_6774_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ht(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_6775_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_int,Nb: nat,B: fun_nat_int,Xb: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I2)
         => ( aa_nat_int(A,I2) = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_int(B,J2) = zero_zero_int ) )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_nn(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_nn(B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_nq(A),B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_6776_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_complex,Nb: nat,B: fun_nat_complex,Xb: complex] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I2)
         => ( aa_nat_complex(A,I2) = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_complex(B,J2) = zero_zero_complex ) )
       => ( aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mo(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mo(B),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_ns(A),B),Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_6777_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_real,Nb: nat,B: fun_nat_real,Xb: real] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I2)
         => ( aa_nat_real(A,I2) = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_real(B,J2) = zero_zero_real ) )
       => ( aa_real_real(times_times_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mp(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mp(B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_nu(A),B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_6778_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_ku(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_6779_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_complex,Nb: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))) = groups6464643781859351333omplex(aTP_Lamm_kv(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_6780_prod_Oin__pairs__0,axiom,
    ! [G: fun_na8797796813480665199d_enat,Nb: nat] : groups7961826882256487087d_enat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))) = groups7961826882256487087d_enat(aTP_Lamm_kw(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_6781_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,Nb: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_kx(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_6782_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ky(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_6783_polyfun__eq__const,axiom,
    ! [C: fun_nat_complex,Nb: nat,K: complex] :
      ( ! [X3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mo(C),X3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = K
    <=> ( ( aa_nat_complex(C,zero_zero_nat) = K )
        & ! [X3: nat] :
            ( aa_set_nat_o(member_nat2(X3),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),Nb))
           => ( aa_nat_complex(C,X3) = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
tff(fact_6784_polyfun__eq__const,axiom,
    ! [C: fun_nat_real,Nb: nat,K: real] :
      ( ! [X3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mp(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = K
    <=> ( ( aa_nat_real(C,zero_zero_nat) = K )
        & ! [X3: nat] :
            ( aa_set_nat_o(member_nat2(X3),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),Nb))
           => ( aa_nat_real(C,X3) = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
tff(fact_6785_polynomial__product__nat,axiom,
    ! [Mb: nat,A: fun_nat_nat,Nb: nat,B: fun_nat_nat,Xb: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I2)
         => ( aa_nat_nat(A,I2) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_nat(B,J2) = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nv(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nv(B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_nx(A),B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_6786_floor__add,axiom,
    ! [Xb: real,Y: real] :
      aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(Xb),Y)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(archim2898591450579166408c_real,Xb)),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,Xb)),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,Xb)),aa_real_int(archim6058952711729229775r_real,Y))),one_one_int)) ).

% floor_add
tff(fact_6787_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_int,H: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ny(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_nz(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_6788_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_oa(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_ob(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_6789_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_na8797796813480665199d_enat,H: fun_na8797796813480665199d_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( groups7108830773950497114d_enat(aa_fun2512298231148344956d_enat(aa_fun1703792696854984886d_enat(aTP_Lamm_oc(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups7108830773950497114d_enat(aa_fun2512298231148344956d_enat(aa_fun1703792696854984886d_enat(aTP_Lamm_od(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_6790_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_oe(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_of(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_6791_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_real,H: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_og(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_oh(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_6792_prod_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_real,H: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_oi(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_oh(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_6793_prod_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_oj(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_ob(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_6794_prod_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_int,H: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ok(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_nz(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_6795_prod_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_ol(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_of(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_6796_gbinomial__partial__sum__poly,axiom,
    ! [Mb: nat,A: complex,Xb: complex,Y: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_om(Mb),A),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_on(Mb),A),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly
tff(fact_6797_gbinomial__partial__sum__poly,axiom,
    ! [Mb: nat,A: real,Xb: real,Y: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_oo(Mb),A),Xb),Y)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_op(Mb),A),Xb),Y)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly
tff(fact_6798_root__polyfun,axiom,
    ! [Nb: nat,Z: complex,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_complex(power_power_complex(Z),Nb) = A )
      <=> ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_oq(Nb),Z),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex ) ) ) ).

% root_polyfun
tff(fact_6799_root__polyfun,axiom,
    ! [Nb: nat,Z: int,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_int(power_power_int(Z),Nb) = A )
      <=> ( aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_or(Nb),Z),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int ) ) ) ).

% root_polyfun
tff(fact_6800_root__polyfun,axiom,
    ! [Nb: nat,Z: real,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_real(power_power_real(Z),Nb) = A )
      <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_os(Nb),Z),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real ) ) ) ).

% root_polyfun
tff(fact_6801_sum__gp0,axiom,
    ! [Xb: complex,Nb: nat] :
      groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = $ite(Xb = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb)))),aa_complex_complex(minus_minus_complex(one_one_complex),Xb))) ).

% sum_gp0
tff(fact_6802_sum__gp0,axiom,
    ! [Xb: real,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(power_power_real(Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = $ite(Xb = one_one_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(Xb),aa_nat_nat(suc,Nb)))),aa_real_real(minus_minus_real(one_one_real),Xb))) ).

% sum_gp0
tff(fact_6803_gbinomial__sum__nat__pow2,axiom,
    ! [Mb: nat] : groups2073611262835488442omplex(aTP_Lamm_ot(Mb),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(bit0(one))),Mb) ).

% gbinomial_sum_nat_pow2
tff(fact_6804_gbinomial__sum__nat__pow2,axiom,
    ! [Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ou,Mb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_real(power_power_real(numeral_numeral_real(bit0(one))),Mb) ).

% gbinomial_sum_nat_pow2
tff(fact_6805_gbinomial__partial__sum__poly__xpos,axiom,
    ! [Mb: nat,A: complex,Xb: complex,Y: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_om(Mb),A),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_ov(Mb),A),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_6806_gbinomial__partial__sum__poly__xpos,axiom,
    ! [Mb: nat,A: real,Xb: real,Y: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_oo(Mb),A),Xb),Y)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_ow(Mb),A),Xb),Y)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_6807_polyfun__diff__alt,axiom,
    ! [Nb: nat,A: fun_nat_int,Xb: int,Y: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_nn(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_nn(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_oy(Nb),A),Xb),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff_alt
tff(fact_6808_polyfun__diff__alt,axiom,
    ! [Nb: nat,A: fun_nat_complex,Xb: complex,Y: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_complex_complex(minus_minus_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mo(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mo(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_pa(Nb),A),Xb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff_alt
tff(fact_6809_polyfun__diff__alt,axiom,
    ! [Nb: nat,A: fun_nat_real,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_real_real(minus_minus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mp(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mp(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(Xb),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_pc(Nb),A),Xb),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff_alt
tff(fact_6810_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: fun_nat_complex,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
     => ? [M7: real] :
        ! [Z3: complex] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,M7),aa_complex_real(real_V1022390504157884413omplex,Z3))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mo(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,Nb)))),aa_real_real(times_times_real(E2),aa_nat_real(power_power_real(aa_complex_real(real_V1022390504157884413omplex,Z3)),aa_nat_nat(suc,Nb)))) ) ) ).

% polyfun_extremal_lemma
tff(fact_6811_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: fun_nat_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
     => ? [M7: real] :
        ! [Z3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,M7),aa_real_real(real_V7735802525324610683m_real,Z3))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mp(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)))),aa_real_real(times_times_real(E2),aa_nat_real(power_power_real(aa_real_real(real_V7735802525324610683m_real,Z3)),aa_nat_nat(suc,Nb)))) ) ) ).

% polyfun_extremal_lemma
tff(fact_6812_polyfun__diff,axiom,
    ! [Nb: nat,A: fun_nat_int,Xb: int,Y: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_nn(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_nn(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_pe(Nb),A),Xb),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff
tff(fact_6813_polyfun__diff,axiom,
    ! [Nb: nat,A: fun_nat_complex,Xb: complex,Y: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_complex_complex(minus_minus_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mo(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mo(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_pg(Nb),A),Xb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff
tff(fact_6814_polyfun__diff,axiom,
    ! [Nb: nat,A: fun_nat_real,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_real_real(minus_minus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mp(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mp(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(Xb),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_pi(Nb),A),Xb),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff
tff(fact_6815_arsinh__def,axiom,
    ! [Xb: real] : aa_real_real(arsinh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(Xb),aa_real_real(powr_real(aa_real_real(plus_plus_real(aa_nat_real(power_power_real(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))))) ).

% arsinh_def
tff(fact_6816_sin__cos__npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))),pi)),numeral_numeral_real(bit0(one)))) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% sin_cos_npi
tff(fact_6817_cos__pi__eq__zero,axiom,
    ! [Mb: nat] : aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb))))),numeral_numeral_real(bit0(one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_6818_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),groups2073611262835488442omplex(aTP_Lamm_pj(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(bit0(one))),Nb) ) ) ).

% choose_even_sum
tff(fact_6819_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_pk,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb) ) ) ).

% choose_even_sum
tff(fact_6820_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_pl,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_real(power_power_real(numeral_numeral_real(bit0(one))),Nb) ) ) ).

% choose_even_sum
tff(fact_6821_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),groups2073611262835488442omplex(aTP_Lamm_pm(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(bit0(one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_6822_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_pn,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_6823_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_po,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_real(power_power_real(numeral_numeral_real(bit0(one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_6824_central__binomial__lower__bound,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_nat_real(power_power_real(numeral_numeral_real(bit0(bit0(one)))),Nb)),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Nb)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)),Nb))) ) ).

% central_binomial_lower_bound
tff(fact_6825_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(set_fo1517530859248394432omplex(aTP_Lamm_pp(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_6826_gbinomial__code,axiom,
    ! [A: real,K: nat] :
      aa_nat_real(gbinomial_real(A),K) = $ite(K = zero_zero_nat,one_one_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_pq(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_6827_sin__zero,axiom,
    aa_real_real(sin_real,zero_zero_real) = zero_zero_real ).

% sin_zero
tff(fact_6828_sin__zero,axiom,
    aa_complex_complex(sin_complex,zero_zero_complex) = zero_zero_complex ).

% sin_zero
tff(fact_6829_binomial__Suc__n,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Nb) = aa_nat_nat(suc,Nb) ).

% binomial_Suc_n
tff(fact_6830_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_6831_cos__zero,axiom,
    aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_6832_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_6833_fact__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).

% fact_0
tff(fact_6834_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_6835_fact__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).

% fact_0
tff(fact_6836_binomial__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,zero_zero_nat)) = Nb ).

% binomial_1
tff(fact_6837_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_6838_binomial__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% binomial_eq_0_iff
tff(fact_6839_binomial__Suc__Suc,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K))) ).

% binomial_Suc_Suc
tff(fact_6840_binomial__n__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_6841_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_6842_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_6843_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_6844_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_6845_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Nb))),aa_nat_complex(semiri5044797733671781792omplex,Nb)) ).

% fact_Suc
tff(fact_6846_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_Extended_enat(semiri4449623510593786356d_enat,aa_nat_nat(suc,Nb)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(suc,Nb))),aa_nat_Extended_enat(semiri4449623510593786356d_enat,Nb)) ).

% fact_Suc
tff(fact_6847_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb))),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_Suc
tff(fact_6848_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb))),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_Suc
tff(fact_6849_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_Suc
tff(fact_6850_zero__less__binomial__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ).

% zero_less_binomial_iff
tff(fact_6851_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_6852_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_6853_sin__cos__squared__add3,axiom,
    ! [Xb: real] : aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Xb))),aa_real_real(times_times_real(aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Xb))) = one_one_real ).

% sin_cos_squared_add3
tff(fact_6854_sin__cos__squared__add3,axiom,
    ! [Xb: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Xb))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,Xb)),aa_complex_complex(sin_complex,Xb))) = one_one_complex ).

% sin_cos_squared_add3
tff(fact_6855_sin__cos__squared__add,axiom,
    ! [Xb: real] : aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one)))) = one_one_real ).

% sin_cos_squared_add
tff(fact_6856_sin__cos__squared__add,axiom,
    ! [Xb: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one)))) = one_one_complex ).

% sin_cos_squared_add
tff(fact_6857_sin__cos__squared__add2,axiom,
    ! [Xb: real] : aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one)))) = one_one_real ).

% sin_cos_squared_add2
tff(fact_6858_sin__cos__squared__add2,axiom,
    ! [Xb: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one)))) = one_one_complex ).

% sin_cos_squared_add2
tff(fact_6859_cos__of__real__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(bit0(one)))) = zero_zero_real ).

% cos_of_real_pi_half
tff(fact_6860_cos__of__real__pi__half,axiom,
    aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(bit0(one)))) = zero_zero_complex ).

% cos_of_real_pi_half
tff(fact_6861_cos__one__sin__zero,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
     => ( aa_real_real(sin_real,Xb) = zero_zero_real ) ) ).

% cos_one_sin_zero
tff(fact_6862_cos__one__sin__zero,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) = one_one_complex )
     => ( aa_complex_complex(sin_complex,Xb) = zero_zero_complex ) ) ).

% cos_one_sin_zero
tff(fact_6863_sin__add,axiom,
    ! [Xb: real,Y: real] : aa_real_real(sin_real,aa_real_real(plus_plus_real(Xb),Y)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(sin_real,Xb)),aa_real_real(cos_real,Y))),aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(sin_real,Y))) ).

% sin_add
tff(fact_6864_sin__add,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,Xb)),aa_complex_complex(cos_complex,Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(sin_complex,Y))) ).

% sin_add
tff(fact_6865_fact__mono__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% fact_mono_nat
tff(fact_6866_fact__ge__self,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_self
tff(fact_6867_fact__nonzero,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) != zero_zero_int ).

% fact_nonzero
tff(fact_6868_fact__nonzero,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,Nb) != zero_zero_complex ).

% fact_nonzero
tff(fact_6869_fact__nonzero,axiom,
    ! [Nb: nat] : aa_nat_Extended_enat(semiri4449623510593786356d_enat,Nb) != zero_z5237406670263579293d_enat ).

% fact_nonzero
tff(fact_6870_fact__nonzero,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) != zero_zero_nat ).

% fact_nonzero
tff(fact_6871_fact__nonzero,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) != zero_zero_real ).

% fact_nonzero
tff(fact_6872_cos__diff,axiom,
    ! [Xb: real,Y: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(Xb),Y)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))),aa_real_real(times_times_real(aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))) ).

% cos_diff
tff(fact_6873_cos__diff,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(Xb),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Y))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,Xb)),aa_complex_complex(sin_complex,Y))) ).

% cos_diff
tff(fact_6874_cos__add,axiom,
    ! [Xb: real,Y: real] : aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),Y)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))),aa_real_real(times_times_real(aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))) ).

% cos_add
tff(fact_6875_cos__add,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Y))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,Xb)),aa_complex_complex(sin_complex,Y))) ).

% cos_add
tff(fact_6876_sin__zero__norm__cos__one,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(cos_real,Xb)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_6877_sin__zero__norm__cos__one,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(sin_complex,Xb) = zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(cos_complex,Xb)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_6878_binomial__fact__lemma,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ) ) ).

% binomial_fact_lemma
tff(fact_6879_binomial__altdef__nat,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_6880_binomial__eq__0,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_6881_Suc__times__binomial__eq,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).

% Suc_times_binomial_eq
tff(fact_6882_Suc__times__binomial,axiom,
    ! [K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ).

% Suc_times_binomial
tff(fact_6883_fact__less__mono__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono_nat
tff(fact_6884_binomial__symmetric,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% binomial_symmetric
tff(fact_6885_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_6886_binomial__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),R2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R2)) ) ).

% binomial_le_pow
tff(fact_6887_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_ge_zero
tff(fact_6888_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_zero
tff(fact_6889_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_ge_zero
tff(fact_6890_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_gt_zero
tff(fact_6891_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_gt_zero
tff(fact_6892_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_gt_zero
tff(fact_6893_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),zero_zero_int) ).

% fact_not_neg
tff(fact_6894_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),zero_zero_nat) ).

% fact_not_neg
tff(fact_6895_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),zero_zero_real) ).

% fact_not_neg
tff(fact_6896_fact__ge__1,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_ge_1
tff(fact_6897_fact__ge__1,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_1
tff(fact_6898_fact__ge__1,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_ge_1
tff(fact_6899_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ).

% fact_mono
tff(fact_6900_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% fact_mono
tff(fact_6901_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ).

% fact_mono
tff(fact_6902_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,Mb)) ) ).

% fact_dvd
tff(fact_6903_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,Mb)) ) ).

% fact_dvd
tff(fact_6904_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,Mb)) ) ).

% fact_dvd
tff(fact_6905_fact__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,Nb)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))) ) ) ).

% fact_binomial
tff(fact_6906_fact__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))) = aa_real_real(divide_divide_real(aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))) ) ) ).

% fact_binomial
tff(fact_6907_binomial__fact,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,Nb)),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,Nb),K)))) ) ) ).

% binomial_fact
tff(fact_6908_binomial__fact,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_real_real(divide_divide_real(aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_real_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,Nb),K)))) ) ) ).

% binomial_fact
tff(fact_6909_zero__less__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ).

% zero_less_binomial
tff(fact_6910_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_6911_fact__ge__Suc__0__nat,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_Suc_0_nat
tff(fact_6912_choose__mult,axiom,
    ! [K: nat,Mb: nat,Nb: 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),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Mb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K))) ) ) ) ).

% choose_mult
tff(fact_6913_binomial__Suc__Suc__eq__times,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))),aa_nat_nat(suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_6914_dvd__fact,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% dvd_fact
tff(fact_6915_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ) ).

% fact_less_mono
tff(fact_6916_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono
tff(fact_6917_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ) ).

% fact_less_mono
tff(fact_6918_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(bit0(one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W2),Z)),numera6690914467698888265omplex(bit0(one)))))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Z),W2)),numera6690914467698888265omplex(bit0(one))))) ).

% cos_diff_cos
tff(fact_6919_cos__diff__cos,axiom,
    ! [W2: real,Z: real] : aa_real_real(minus_minus_real(aa_real_real(cos_real,W2)),aa_real_real(cos_real,Z)) = aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W2),Z)),numeral_numeral_real(bit0(one)))))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Z),W2)),numeral_numeral_real(bit0(one))))) ).

% cos_diff_cos
tff(fact_6920_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(bit0(one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W2),Z)),numera6690914467698888265omplex(bit0(one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W2),Z)),numera6690914467698888265omplex(bit0(one))))) ).

% sin_diff_sin
tff(fact_6921_sin__diff__sin,axiom,
    ! [W2: real,Z: real] : aa_real_real(minus_minus_real(aa_real_real(sin_real,W2)),aa_real_real(sin_real,Z)) = aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W2),Z)),numeral_numeral_real(bit0(one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W2),Z)),numeral_numeral_real(bit0(one))))) ).

% sin_diff_sin
tff(fact_6922_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(bit0(one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W2),Z)),numera6690914467698888265omplex(bit0(one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W2),Z)),numera6690914467698888265omplex(bit0(one))))) ).

% sin_plus_sin
tff(fact_6923_sin__plus__sin,axiom,
    ! [W2: real,Z: real] : aa_real_real(plus_plus_real(aa_real_real(sin_real,W2)),aa_real_real(sin_real,Z)) = aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W2),Z)),numeral_numeral_real(bit0(one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W2),Z)),numeral_numeral_real(bit0(one))))) ).

% sin_plus_sin
tff(fact_6924_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(bit0(one))) ).

% cos_times_sin
tff(fact_6925_cos__times__sin,axiom,
    ! [W2: real,Z: real] : aa_real_real(times_times_real(aa_real_real(cos_real,W2)),aa_real_real(sin_real,Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(W2),Z))),aa_real_real(sin_real,aa_real_real(minus_minus_real(W2),Z)))),numeral_numeral_real(bit0(one))) ).

% cos_times_sin
tff(fact_6926_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(bit0(one))) ).

% sin_times_cos
tff(fact_6927_sin__times__cos,axiom,
    ! [W2: real,Z: real] : aa_real_real(times_times_real(aa_real_real(sin_real,W2)),aa_real_real(cos_real,Z)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(W2),Z))),aa_real_real(sin_real,aa_real_real(minus_minus_real(W2),Z)))),numeral_numeral_real(bit0(one))) ).

% sin_times_cos
tff(fact_6928_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(bit0(one))) ).

% sin_times_sin
tff(fact_6929_sin__times__sin,axiom,
    ! [W2: real,Z: real] : aa_real_real(times_times_real(aa_real_real(sin_real,W2)),aa_real_real(sin_real,Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(cos_real,aa_real_real(minus_minus_real(W2),Z))),aa_real_real(cos_real,aa_real_real(plus_plus_real(W2),Z)))),numeral_numeral_real(bit0(one))) ).

% sin_times_sin
tff(fact_6930_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,Nb))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_6931_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_6932_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_6933_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(modulo_modulo_int(aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,Mb)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_6934_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,Mb)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_6935_fact__le__power,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_6936_fact__le__power,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_6937_fact__le__power,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_6938_sum__choose__upper,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pr,Mb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Mb)) ).

% sum_choose_upper
tff(fact_6939_minus__sin__cos__eq,axiom,
    ! [Xb: real] : aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) = aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(bit0(one))))) ).

% minus_sin_cos_eq
tff(fact_6940_minus__sin__cos__eq,axiom,
    ! [Xb: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(sin_complex,Xb)) = aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(bit0(one))))) ).

% minus_sin_cos_eq
tff(fact_6941_binomial__absorption,axiom,
    ! [K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ).

% binomial_absorption
tff(fact_6942_fact__diff__Suc,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ) ) ).

% fact_diff_Suc
tff(fact_6943_fact__div__fact__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),R2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R2)) ) ).

% fact_div_fact_le_pow
tff(fact_6944_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ).

% choose_dvd
tff(fact_6945_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(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,Nb),K)))),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% choose_dvd
tff(fact_6946_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_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,Nb),K)))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ).

% choose_dvd
tff(fact_6947_fact__eq__fact__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_bc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Nb)),Mb))) ) ) ).

% fact_eq_fact_times
tff(fact_6948_sin__expansion__lemma,axiom,
    ! [Xb: real,Mb: nat] : aa_real_real(sin_real,aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb))),pi)),numeral_numeral_real(bit0(one))))) = aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),pi)),numeral_numeral_real(bit0(one))))) ).

% sin_expansion_lemma
tff(fact_6949_sum__choose__lower,axiom,
    ! [R2: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ps,R2)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R2),Nb))),Nb) ).

% sum_choose_lower
tff(fact_6950_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pt,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_6951_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pt,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),one_one_nat)),Mb) ).

% choose_rising_sum(2)
tff(fact_6952_binomial__code,axiom,
    ! [Nb: nat,K: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K),
        zero_zero_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),one_one_nat)),Nb),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K))) ) ).

% binomial_code
tff(fact_6953_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_6954_binomial__maximum_H,axiom,
    ! [Nb: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)),Nb)) ).

% binomial_maximum'
tff(fact_6955_binomial__mono,axiom,
    ! [K: nat,K6: nat,Nb: 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,bit0(one))),K6)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)) ) ) ).

% binomial_mono
tff(fact_6956_binomial__antimono,axiom,
    ! [K: nat,K6: nat,Nb: 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,Nb),aa_num_nat(numeral_numeral_nat,bit0(one)))),K)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ) ) ).

% binomial_antimono
tff(fact_6957_binomial__maximum,axiom,
    ! [Nb: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% binomial_maximum
tff(fact_6958_binomial__le__pow2,axiom,
    ! [Nb: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)) ).

% binomial_le_pow2
tff(fact_6959_choose__reduce__nat,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_6960_times__binomial__minus1__eq,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_6961_cos__expansion__lemma,axiom,
    ! [Xb: real,Mb: nat] : aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb))),pi)),numeral_numeral_real(bit0(one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),pi)),numeral_numeral_real(bit0(one)))))) ).

% cos_expansion_lemma
tff(fact_6962_fact__div__fact,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_bc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)) ) ) ).

% fact_div_fact
tff(fact_6963_sum__choose__diagonal,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pu(Mb),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Mb) ) ) ).

% sum_choose_diagonal
tff(fact_6964_vandermonde,axiom,
    ! [Mb: nat,Nb: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_pv(Mb),Nb),R2)),aa_nat_set_nat(set_ord_atMost_nat,R2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),R2) ).

% vandermonde
tff(fact_6965_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one))))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_6966_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,Nb: 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,bit0(one))),K6)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)) ) ) ).

% binomial_strict_mono
tff(fact_6967_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,Nb: 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,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_6968_central__binomial__odd,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ) ).

% central_binomial_odd
tff(fact_6969_binomial__addition__formula,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_6970_sin__pi__divide__n__ge__0,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,Nb)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_6971_sin__paired,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aTP_Lamm_pw(Xb)),aa_real_real(sin_real,Xb)) ).

% sin_paired
tff(fact_6972_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_6973_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_Extended_enat(semiri4449623510593786356d_enat,Mb) = $ite(Mb = zero_zero_nat,one_on7984719198319812577d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Mb)),aa_nat_Extended_enat(semiri4449623510593786356d_enat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_6974_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_6975_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_6976_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_real(semiri2265585572941072030t_real,Mb) = $ite(Mb = zero_zero_nat,one_one_real,aa_real_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_6977_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(semiri5044797733671781792omplex,Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_6978_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_Extended_enat(semiri4449623510593786356d_enat,Nb) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb)),aa_nat_Extended_enat(semiri4449623510593786356d_enat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_6979_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_6980_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_6981_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_6982_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(bit0(one))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W2),Z)),numera6690914467698888265omplex(bit0(one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W2),Z)),numera6690914467698888265omplex(bit0(one))))) ).

% cos_plus_cos
tff(fact_6983_cos__plus__cos,axiom,
    ! [W2: real,Z: real] : aa_real_real(plus_plus_real(aa_real_real(cos_real,W2)),aa_real_real(cos_real,Z)) = aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W2),Z)),numeral_numeral_real(bit0(one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W2),Z)),numeral_numeral_real(bit0(one))))) ).

% cos_plus_cos
tff(fact_6984_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(bit0(one))) ).

% cos_times_cos
tff(fact_6985_cos__times__cos,axiom,
    ! [W2: real,Z: real] : aa_real_real(times_times_real(aa_real_real(cos_real,W2)),aa_real_real(cos_real,Z)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(cos_real,aa_real_real(minus_minus_real(W2),Z))),aa_real_real(cos_real,aa_real_real(plus_plus_real(W2),Z)))),numeral_numeral_real(bit0(one))) ).

% cos_times_cos
tff(fact_6986_binomial,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_px(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial
tff(fact_6987_binomial__ring,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(A),B)),Nb) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_py(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_6988_binomial__ring,axiom,
    ! [A: extended_enat,B: extended_enat,Nb: nat] : aa_nat_Extended_enat(power_8040749407984259932d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)),Nb) = groups7108830773950497114d_enat(aa_nat2025370463774033377d_enat(aa_Ext8984936454148280526d_enat(aTP_Lamm_pz(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_6989_binomial__ring,axiom,
    ! [A: int,B: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Nb) = aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_qa(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_6990_binomial__ring,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_px(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_6991_binomial__ring,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(power_power_real(aa_real_real(plus_plus_real(A),B)),Nb) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qb(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_6992_pochhammer__binomial__sum,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),B)),Nb) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_qc(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_6993_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Nb) = aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_qd(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_6994_pochhammer__binomial__sum,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(A),B)),Nb) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qe(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_6995_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_6996_gbinomial__pochhammer_H,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),K) = aa_real_real(divide_divide_real(aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)),aa_nat_real(semiri2265585572941072030t_real,K)) ).

% gbinomial_pochhammer'
tff(fact_6997_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_nat_fun_nat_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun7039663905532832253t_real(aTP_Lamm_qf(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_nat_real(aa_nat_fun_nat_real(Diff,zero_zero_nat),zero_zero_nat) ) ) ) ).

% Maclaurin_zero
tff(fact_6998_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_na6724076748692969250l_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_qg(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_real_real(aa_nat_fun_real_real(Diff,zero_zero_nat),zero_zero_real) ) ) ) ).

% Maclaurin_zero
tff(fact_6999_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_nat_fun_int_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun241848180832411225t_real(aTP_Lamm_qh(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_int_real(aa_nat_fun_int_real(Diff,zero_zero_nat),zero_zero_int) ) ) ) ).

% Maclaurin_zero
tff(fact_7000_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_na5723442673449079972x_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun5746040925223427547t_real(aTP_Lamm_qi(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_complex_real(aa_nat480766339200631653x_real(Diff,zero_zero_nat),zero_zero_complex) ) ) ) ).

% Maclaurin_zero
tff(fact_7001_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_na2978037997607195054t_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun2748096013305118955t_real(aTP_Lamm_qj(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_Ext2541572767299617110t_real(aa_nat6736956760766371991t_real(Diff,zero_zero_nat),zero_z5237406670263579293d_enat) ) ) ) ).

% Maclaurin_zero
tff(fact_7002_Maclaurin__cos__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T5)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),Xb)
            & ( aa_real_real(cos_real,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_qk(Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T5),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(Xb),Nb))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_7003_Maclaurin__minus__cos__expansion,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),T5)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),zero_zero_real)
            & ( aa_real_real(cos_real,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_qk(Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T5),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(Xb),Nb))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_7004_sin__pi__divide__n__gt__0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,Nb)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_7005_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( groups2073611262835488442omplex(aTP_Lamm_ql(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
tff(fact_7006_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_qm,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int ) ) ).

% choose_alternating_linear_sum
tff(fact_7007_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_qn,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real ) ) ).

% choose_alternating_linear_sum
tff(fact_7008_gbinomial__Suc,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)) = aa_real_real(divide_divide_real(aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_qo(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_7009_gbinomial__Suc,axiom,
    ! [A: int,K: nat] : aa_nat_int(gbinomial_int(A),aa_nat_nat(suc,K)) = aa_int_int(divide_divide_int(aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_qp(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_7010_gbinomial__Suc,axiom,
    ! [A: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qq,A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_7011_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,bit0(one))),Mb)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)) ).

% binomial_r_part_sum
tff(fact_7012_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( groups2073611262835488442omplex(aTP_Lamm_qr(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex ) ) ).

% choose_alternating_sum
tff(fact_7013_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_qs,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int ) ) ).

% choose_alternating_sum
tff(fact_7014_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_qt,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real ) ) ).

% choose_alternating_sum
tff(fact_7015_Maclaurin__sin__expansion3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T5)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),Xb)
            & ( aa_real_real(sin_real,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_qu(Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T5),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(Xb),Nb))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_7016_sin__x__sin__y,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_qw(Xb),Y)),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,Xb)),aa_complex_complex(sin_complex,Y))) ).

% sin_x_sin_y
tff(fact_7017_sin__x__sin__y,axiom,
    ! [Xb: real,Y: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_qy(Xb),Y)),aa_real_real(times_times_real(aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))) ).

% sin_x_sin_y
tff(fact_7018_sums__cos__x__plus__y,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_ra(Xb),Y)),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(Xb),Y))) ).

% sums_cos_x_plus_y
tff(fact_7019_sums__cos__x__plus__y,axiom,
    ! [Xb: real,Y: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_rc(Xb),Y)),aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),Y))) ).

% sums_cos_x_plus_y
tff(fact_7020_cos__x__cos__y,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_re(Xb),Y)),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Y))) ).

% cos_x_cos_y
tff(fact_7021_cos__x__cos__y,axiom,
    ! [Xb: real,Y: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_rg(Xb),Y)),aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))) ).

% cos_x_cos_y
tff(fact_7022_sin__coeff__def,axiom,
    ! [X: nat] :
      aa_nat_real(sin_coeff,X) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),X),zero_zero_real,aa_real_real(divide_divide_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_real(semiri2265585572941072030t_real,X))) ).

% sin_coeff_def
tff(fact_7023_tan__double,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),Xb)) != zero_zero_complex )
       => ( aa_complex_complex(tan_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),Xb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_complex_complex(tan_complex,Xb))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(tan_complex,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ) ) ).

% tan_double
tff(fact_7024_tan__double,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),Xb)) != zero_zero_real )
       => ( aa_real_real(tan_real,aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),Xb)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_real_real(tan_real,Xb))),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ) ) ).

% tan_double
tff(fact_7025_of__nat__id,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = Nb ).

% of_nat_id
tff(fact_7026_scaleR__zero__right,axiom,
    ! [A: real] : aa_real_real(real_V1485227260804924795R_real(A),zero_zero_real) = zero_zero_real ).

% scaleR_zero_right
tff(fact_7027_scaleR__zero__right,axiom,
    ! [A: real] : aa_complex_complex(real_V2046097035970521341omplex(A),zero_zero_complex) = zero_zero_complex ).

% scaleR_zero_right
tff(fact_7028_scaleR__cancel__right,axiom,
    ! [A: real,Xb: real,B: real] :
      ( ( aa_real_real(real_V1485227260804924795R_real(A),Xb) = aa_real_real(real_V1485227260804924795R_real(B),Xb) )
    <=> ( ( A = B )
        | ( Xb = zero_zero_real ) ) ) ).

% scaleR_cancel_right
tff(fact_7029_scaleR__cancel__right,axiom,
    ! [A: real,Xb: complex,B: real] :
      ( ( aa_complex_complex(real_V2046097035970521341omplex(A),Xb) = aa_complex_complex(real_V2046097035970521341omplex(B),Xb) )
    <=> ( ( A = B )
        | ( Xb = zero_zero_complex ) ) ) ).

% scaleR_cancel_right
tff(fact_7030_tan__zero,axiom,
    aa_real_real(tan_real,zero_zero_real) = zero_zero_real ).

% tan_zero
tff(fact_7031_tan__zero,axiom,
    aa_complex_complex(tan_complex,zero_zero_complex) = zero_zero_complex ).

% tan_zero
tff(fact_7032_scaleR__zero__left,axiom,
    ! [Xb: real] : aa_real_real(real_V1485227260804924795R_real(zero_zero_real),Xb) = zero_zero_real ).

% scaleR_zero_left
tff(fact_7033_scaleR__zero__left,axiom,
    ! [Xb: complex] : aa_complex_complex(real_V2046097035970521341omplex(zero_zero_real),Xb) = zero_zero_complex ).

% scaleR_zero_left
tff(fact_7034_scaleR__eq__0__iff,axiom,
    ! [A: real,Xb: real] :
      ( ( aa_real_real(real_V1485227260804924795R_real(A),Xb) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( Xb = zero_zero_real ) ) ) ).

% scaleR_eq_0_iff
tff(fact_7035_scaleR__eq__0__iff,axiom,
    ! [A: real,Xb: complex] :
      ( ( aa_complex_complex(real_V2046097035970521341omplex(A),Xb) = zero_zero_complex )
    <=> ( ( A = zero_zero_real )
        | ( Xb = zero_zero_complex ) ) ) ).

% scaleR_eq_0_iff
tff(fact_7036_scaleR__eq__iff,axiom,
    ! [B: real,U: real,A: real] :
      ( ( aa_real_real(plus_plus_real(B),aa_real_real(real_V1485227260804924795R_real(U),A)) = aa_real_real(plus_plus_real(A),aa_real_real(real_V1485227260804924795R_real(U),B)) )
    <=> ( ( A = B )
        | ( U = one_one_real ) ) ) ).

% scaleR_eq_iff
tff(fact_7037_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_7038_scaleR__collapse,axiom,
    ! [U: real,A: real] : aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(minus_minus_real(one_one_real),U)),A)),aa_real_real(real_V1485227260804924795R_real(U),A)) = A ).

% scaleR_collapse
tff(fact_7039_scaleR__half__double,axiom,
    ! [A: real] : aa_real_real(real_V1485227260804924795R_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_real_real(plus_plus_real(A),A)) = A ).

% scaleR_half_double
tff(fact_7040_scaleR__right__imp__eq,axiom,
    ! [Xb: real,A: real,B: real] :
      ( ( Xb != zero_zero_real )
     => ( ( aa_real_real(real_V1485227260804924795R_real(A),Xb) = aa_real_real(real_V1485227260804924795R_real(B),Xb) )
       => ( A = B ) ) ) ).

% scaleR_right_imp_eq
tff(fact_7041_scaleR__right__imp__eq,axiom,
    ! [Xb: complex,A: real,B: real] :
      ( ( Xb != zero_zero_complex )
     => ( ( aa_complex_complex(real_V2046097035970521341omplex(A),Xb) = aa_complex_complex(real_V2046097035970521341omplex(B),Xb) )
       => ( A = B ) ) ) ).

% scaleR_right_imp_eq
tff(fact_7042_scaleR__right__distrib,axiom,
    ! [A: real,Xb: real,Y: real] : aa_real_real(real_V1485227260804924795R_real(A),aa_real_real(plus_plus_real(Xb),Y)) = aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(A),Xb)),aa_real_real(real_V1485227260804924795R_real(A),Y)) ).

% scaleR_right_distrib
tff(fact_7043_scaleR__left__distrib,axiom,
    ! [A: real,B: real,Xb: real] : aa_real_real(real_V1485227260804924795R_real(aa_real_real(plus_plus_real(A),B)),Xb) = aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(A),Xb)),aa_real_real(real_V1485227260804924795R_real(B),Xb)) ).

% scaleR_left_distrib
tff(fact_7044_scaleR__left_Oadd,axiom,
    ! [Xb: real,Y: real,Xaa: real] : aa_real_real(real_V1485227260804924795R_real(aa_real_real(plus_plus_real(Xb),Y)),Xaa) = aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(Xb),Xaa)),aa_real_real(real_V1485227260804924795R_real(Y),Xaa)) ).

% scaleR_left.add
tff(fact_7045_scaleR__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(real_V1485227260804924795R_real(A),C)),aa_real_real(real_V1485227260804924795R_real(B),C)) ) ) ).

% scaleR_right_mono_neg
tff(fact_7046_scaleR__right__mono,axiom,
    ! [A: real,B: real,Xb: 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),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),Xb)),aa_real_real(real_V1485227260804924795R_real(B),Xb)) ) ) ).

% scaleR_right_mono
tff(fact_7047_scaleR__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(real_V1485227260804924795R_real(C),A)),aa_real_real(real_V1485227260804924795R_real(C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_7048_scaleR__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(real_V1485227260804924795R_real(C),A)),aa_real_real(real_V1485227260804924795R_real(C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_7049_scaleR__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(real_V1485227260804924795R_real(C),A)),aa_real_real(real_V1485227260804924795R_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) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_7050_scaleR__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(real_V1485227260804924795R_real(C),A)),aa_real_real(real_V1485227260804924795R_real(C),B)) ) ) ).

% scaleR_left_mono_neg
tff(fact_7051_scaleR__left__mono,axiom,
    ! [Xb: real,Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => ( 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(real_V1485227260804924795R_real(A),Xb)),aa_real_real(real_V1485227260804924795R_real(A),Y)) ) ) ).

% scaleR_left_mono
tff(fact_7052_vector__fraction__eq__iff,axiom,
    ! [U: real,V: real,A: real,Xb: real] :
      ( ( aa_real_real(real_V1485227260804924795R_real(aa_real_real(divide_divide_real(U),V)),A) = Xb )
    <=> $ite(V = zero_zero_real,Xb = zero_zero_real,aa_real_real(real_V1485227260804924795R_real(U),A) = aa_real_real(real_V1485227260804924795R_real(V),Xb)) ) ).

% vector_fraction_eq_iff
tff(fact_7053_vector__fraction__eq__iff,axiom,
    ! [U: real,V: real,A: complex,Xb: complex] :
      ( ( aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(divide_divide_real(U),V)),A) = Xb )
    <=> $ite(V = zero_zero_real,Xb = zero_zero_complex,aa_complex_complex(real_V2046097035970521341omplex(U),A) = aa_complex_complex(real_V2046097035970521341omplex(V),Xb)) ) ).

% vector_fraction_eq_iff
tff(fact_7054_eq__vector__fraction__iff,axiom,
    ! [Xb: real,U: real,V: real,A: real] :
      ( ( Xb = aa_real_real(real_V1485227260804924795R_real(aa_real_real(divide_divide_real(U),V)),A) )
    <=> $ite(V = zero_zero_real,Xb = zero_zero_real,aa_real_real(real_V1485227260804924795R_real(V),Xb) = aa_real_real(real_V1485227260804924795R_real(U),A)) ) ).

% eq_vector_fraction_iff
tff(fact_7055_eq__vector__fraction__iff,axiom,
    ! [Xb: complex,U: real,V: real,A: complex] :
      ( ( Xb = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(divide_divide_real(U),V)),A) )
    <=> $ite(V = zero_zero_real,Xb = zero_zero_complex,aa_complex_complex(real_V2046097035970521341omplex(V),Xb) = aa_complex_complex(real_V2046097035970521341omplex(U),A)) ) ).

% eq_vector_fraction_iff
tff(fact_7056_Real__Vector__Spaces_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(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(A),E2)),C)),aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(minus_minus_real(A),B)),E2)),C)),D) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_7057_Real__Vector__Spaces_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(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(A),E2)),C)),aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(minus_minus_real(B),A)),E2)),D)) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_7058_zero__le__scaleR__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(real_V1485227260804924795R_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_eq_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_eq_real,B),zero_zero_real) )
        | ( A = zero_zero_real ) ) ) ).

% zero_le_scaleR_iff
tff(fact_7059_scaleR__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_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_eq_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_eq_real,zero_zero_real),B) )
        | ( A = zero_zero_real ) ) ) ).

% scaleR_le_0_iff
tff(fact_7060_scaleR__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(real_V1485227260804924795R_real(A),B)) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_7061_scaleR__nonpos__nonneg,axiom,
    ! [A: real,Xb: 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),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),Xb)),zero_zero_real) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_7062_scaleR__nonneg__nonpos,axiom,
    ! [A: real,Xb: 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,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),Xb)),zero_zero_real) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_7063_scaleR__nonneg__nonneg,axiom,
    ! [A: real,Xb: 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),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(real_V1485227260804924795R_real(A),Xb)) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_7064_split__scaleR__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(real_V1485227260804924795R_real(A),B)) ) ).

% split_scaleR_pos_le
tff(fact_7065_split__scaleR__neg__le,axiom,
    ! [A: real,Xb: 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,Xb),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),Xb) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),Xb)),zero_zero_real) ) ).

% split_scaleR_neg_le
tff(fact_7066_scaleR__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(real_V1485227260804924795R_real(A),C)),aa_real_real(real_V1485227260804924795R_real(B),D)) ) ) ) ) ).

% scaleR_mono'
tff(fact_7067_scaleR__mono,axiom,
    ! [A: real,B: real,Xb: real,Y: 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,Xb),Y)
       => ( 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),Xb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),Xb)),aa_real_real(real_V1485227260804924795R_real(B),Y)) ) ) ) ) ).

% scaleR_mono
tff(fact_7068_scaleR__left__le__one__le,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( 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_real_real(real_V1485227260804924795R_real(A),Xb)),Xb) ) ) ).

% scaleR_left_le_one_le
tff(fact_7069_scaleR__2,axiom,
    ! [Xb: real] : aa_real_real(real_V1485227260804924795R_real(numeral_numeral_real(bit0(one))),Xb) = aa_real_real(plus_plus_real(Xb),Xb) ).

% scaleR_2
tff(fact_7070_sin__coeff__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,Nb)) = aa_real_real(divide_divide_real(aa_nat_real(cos_coeff,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))) ).

% sin_coeff_Suc
tff(fact_7071_add__tan__eq,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(Xb),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Y))) ) ) ) ).

% add_tan_eq
tff(fact_7072_add__tan__eq,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)) = aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(Xb),Y))),aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))) ) ) ) ).

% add_tan_eq
tff(fact_7073_lemma__tan__add1,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != 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,Xb)),aa_complex_complex(tan_complex,Y))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(Xb),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_7074_lemma__tan__add1,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))) = aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),Y))),aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_7075_tan__diff,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(Xb),Y)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(minus_minus_complex(Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(tan_complex,Xb)),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,Xb)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_7076_tan__diff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(minus_minus_real(Xb),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(minus_minus_real(Xb),Y)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))),aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_7077_tan__add,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,Xb)),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,Xb)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).

% tan_add
tff(fact_7078_tan__add,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(plus_plus_real(Xb),Y)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))),aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_add
tff(fact_7079_cos__coeff__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,Nb)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,Nb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))) ).

% cos_coeff_Suc
tff(fact_7080_tan__half,axiom,
    ! [Xb: complex] : aa_complex_complex(tan_complex,Xb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),Xb))),aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),Xb))),one_one_complex)) ).

% tan_half
tff(fact_7081_tan__half,axiom,
    ! [Xb: real] : aa_real_real(tan_real,Xb) = aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),Xb))),aa_real_real(plus_plus_real(aa_real_real(cos_real,aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),Xb))),one_one_real)) ).

% tan_half
tff(fact_7082_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(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,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(F)) ) ) ).

% sum_pos_lt_pair
tff(fact_7083_modulo__int__unfold,axiom,
    ! [K: int,Mb: nat,L: int,Nb: 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,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Nb = zero_zero_nat ) ),
        aa_int_int(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),Nb))),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,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))))) ) ).

% modulo_int_unfold
tff(fact_7084_divide__int__unfold,axiom,
    ! [K: int,Mb: nat,L: int,Nb: 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,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb)))))) ) ).

% divide_int_unfold
tff(fact_7085_xor__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))) ).

% xor_Suc_0_eq
tff(fact_7086_Suc__0__xor__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))) ).

% Suc_0_xor_eq
tff(fact_7087_sgn__sgn,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(sgn_sgn_int,A)) = aa_int_int(sgn_sgn_int,A) ).

% sgn_sgn
tff(fact_7088_sgn__sgn,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(sgn_sgn_real,A)) = aa_real_real(sgn_sgn_real,A) ).

% sgn_sgn
tff(fact_7089_sgn__0,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_0
tff(fact_7090_sgn__0,axiom,
    aa_int_int(sgn_sgn_int,zero_zero_int) = zero_zero_int ).

% sgn_0
tff(fact_7091_sgn__0,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_0
tff(fact_7092_sgn__zero,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_zero
tff(fact_7093_sgn__zero,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_zero
tff(fact_7094_sgn__1,axiom,
    aa_complex_complex(sgn_sgn_complex,one_one_complex) = one_one_complex ).

% sgn_1
tff(fact_7095_sgn__1,axiom,
    aa_int_int(sgn_sgn_int,one_one_int) = one_one_int ).

% sgn_1
tff(fact_7096_sgn__1,axiom,
    aa_real_real(sgn_sgn_real,one_one_real) = one_one_real ).

% sgn_1
tff(fact_7097_bit_Oxor__self,axiom,
    ! [Xb: int] : aa_int_int(bit_se6526347334894502574or_int(Xb),Xb) = zero_zero_int ).

% bit.xor_self
tff(fact_7098_xor__self__eq,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(A),A) = zero_zero_int ).

% xor_self_eq
tff(fact_7099_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_7100_xor_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(zero_zero_int),A) = A ).

% xor.left_neutral
tff(fact_7101_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_7102_xor_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(A),zero_zero_int) = A ).

% xor.right_neutral
tff(fact_7103_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_7104_sgn__divide,axiom,
    ! [A: real,B: real] : aa_real_real(sgn_sgn_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,B)) ).

% sgn_divide
tff(fact_7105_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(uminus_uminus_int,aa_int_int(sgn_sgn_int,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_7106_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(sgn_sgn_real,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_7107_summable__single,axiom,
    ! [Ia: nat,F: fun_nat_nat] : summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_lj(Ia),F)) ).

% summable_single
tff(fact_7108_summable__single,axiom,
    ! [Ia: nat,F: fun_nat_real] : summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_lk(Ia),F)) ).

% summable_single
tff(fact_7109_summable__single,axiom,
    ! [Ia: nat,F: fun_nat_int] : summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_ll(Ia),F)) ).

% summable_single
tff(fact_7110_summable__single,axiom,
    ! [Ia: nat,F: fun_nat_complex] : summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lm(Ia),F)) ).

% summable_single
tff(fact_7111_summable__zero,axiom,
    summable_nat(aTP_Lamm_ax) ).

% summable_zero
tff(fact_7112_summable__zero,axiom,
    summable_real(aTP_Lamm_dr) ).

% summable_zero
tff(fact_7113_summable__zero,axiom,
    summable_int(aTP_Lamm_lf) ).

% summable_zero
tff(fact_7114_summable__zero,axiom,
    summable_complex(aTP_Lamm_lg) ).

% summable_zero
tff(fact_7115_summable__iff__shift,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(aa_nat_fun_nat_real(aTP_Lamm_gr(F),K))
    <=> summable_real(F) ) ).

% summable_iff_shift
tff(fact_7116_sgn__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sgn_sgn_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% sgn_less
tff(fact_7117_sgn__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(sgn_sgn_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% sgn_less
tff(fact_7118_sgn__greater,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sgn_sgn_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% sgn_greater
tff(fact_7119_sgn__greater,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(sgn_sgn_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% sgn_greater
tff(fact_7120_divide__sgn,axiom,
    ! [A: real,B: real] : aa_real_real(divide_divide_real(A),aa_real_real(sgn_sgn_real,B)) = aa_real_real(times_times_real(A),aa_real_real(sgn_sgn_real,B)) ).

% divide_sgn
tff(fact_7121_summable__cmult__iff,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_lq(C),F))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_7122_summable__cmult__iff,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lr(C),F))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_cmult_iff
tff(fact_7123_summable__divide__iff,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_rh(F),C))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_divide_iff
tff(fact_7124_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ri(F),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_7125_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_lw(P),F)) ) ).

% summable_If_finite
tff(fact_7126_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_lx(P),F)) ) ).

% summable_If_finite
tff(fact_7127_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_lu(P),F)) ) ).

% summable_If_finite
tff(fact_7128_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_lv(P),F)) ) ).

% summable_If_finite
tff(fact_7129_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_ma(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7130_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_mb(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7131_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_ly(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7132_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_lz(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7133_sgn__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(sgn_sgn_real,A) = one_one_real ) ) ).

% sgn_pos
tff(fact_7134_sgn__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_int(sgn_sgn_int,A) = one_one_int ) ) ).

% sgn_pos
tff(fact_7135_sgn__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,A)) = aa_o_real(zero_n3304061248610475627l_real,A != zero_zero_real) ).

% sgn_mult_self_eq
tff(fact_7136_sgn__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,A)) = aa_o_int(zero_n2684676970156552555ol_int,A != zero_zero_int) ).

% sgn_mult_self_eq
tff(fact_7137_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_7138_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_7139_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_7140_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_7141_sgn__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% sgn_neg
tff(fact_7142_sgn__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% sgn_neg
tff(fact_7143_sgn__of__nat,axiom,
    ! [Nb: nat] : aa_real_real(sgn_sgn_real,aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_o_real(zero_n3304061248610475627l_real,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% sgn_of_nat
tff(fact_7144_sgn__of__nat,axiom,
    ! [Nb: nat] : aa_int_int(sgn_sgn_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% sgn_of_nat
tff(fact_7145_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,bit0(Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_7146_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,bit0(Y)) ).

% xor_nat_numerals(2)
tff(fact_7147_xor__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,bit0(Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% xor_nat_numerals(3)
tff(fact_7148_xor__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,bit0(Xb)) ).

% xor_nat_numerals(4)
tff(fact_7149_xor__numerals_I6_J,axiom,
    ! [Xb: num,Y: num] : aa_int_int(bit_se6526347334894502574or_int(numeral_numeral_int(aa_num_num(bit1,Xb))),numeral_numeral_int(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,numeral_numeral_int(bit0(one))),aa_int_int(bit_se6526347334894502574or_int(numeral_numeral_int(Xb)),numeral_numeral_int(Y)))) ).

% xor_numerals(6)
tff(fact_7150_xor__numerals_I6_J,axiom,
    ! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_num_nat(numeral_numeral_nat,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,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% xor_numerals(6)
tff(fact_7151_xor__numerals_I4_J,axiom,
    ! [Xb: num,Y: num] : aa_int_int(bit_se6526347334894502574or_int(numeral_numeral_int(bit0(Xb))),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,numeral_numeral_int(bit0(one))),aa_int_int(bit_se6526347334894502574or_int(numeral_numeral_int(Xb)),numeral_numeral_int(Y)))) ).

% xor_numerals(4)
tff(fact_7152_xor__numerals_I4_J,axiom,
    ! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,bit0(Xb))),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,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% xor_numerals(4)
tff(fact_7153_sgn__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sgn_sgn_real,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_real ) ) ).

% sgn_zero_iff
tff(fact_7154_sgn__zero__iff,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(sgn_sgn_complex,Xb) = zero_zero_complex )
    <=> ( Xb = zero_zero_complex ) ) ).

% sgn_zero_iff
tff(fact_7155_sgn__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% sgn_eq_0_iff
tff(fact_7156_sgn__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% sgn_eq_0_iff
tff(fact_7157_sgn__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(sgn_sgn_complex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% sgn_eq_0_iff
tff(fact_7158_sgn__0__0,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% sgn_0_0
tff(fact_7159_sgn__0__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% sgn_0_0
tff(fact_7160_sgn__mult,axiom,
    ! [A: int,B: int] : aa_int_int(sgn_sgn_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(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,B)) ).

% sgn_mult
tff(fact_7161_sgn__mult,axiom,
    ! [A: real,B: real] : aa_real_real(sgn_sgn_real,aa_real_real(times_times_real(A),B)) = aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,B)) ).

% sgn_mult
tff(fact_7162_sgn__mult,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(sgn_sgn_complex,B)) ).

% sgn_mult
tff(fact_7163_same__sgn__sgn__add,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) = aa_int_int(sgn_sgn_int,A) )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(sgn_sgn_int,A) ) ) ).

% same_sgn_sgn_add
tff(fact_7164_same__sgn__sgn__add,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B) = aa_real_real(sgn_sgn_real,A) )
     => ( aa_real_real(sgn_sgn_real,aa_real_real(plus_plus_real(A),B)) = aa_real_real(sgn_sgn_real,A) ) ) ).

% same_sgn_sgn_add
tff(fact_7165_summable__const__iff,axiom,
    ! [C: real] :
      ( summable_real(aTP_Lamm_rj(C))
    <=> ( C = zero_zero_real ) ) ).

% summable_const_iff
tff(fact_7166_summable__const__iff,axiom,
    ! [C: complex] :
      ( summable_complex(aTP_Lamm_rk(C))
    <=> ( C = zero_zero_complex ) ) ).

% summable_const_iff
tff(fact_7167_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_7168_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_7169_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_7170_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_7171_summable__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ey(F),G)) ) ) ).

% summable_add
tff(fact_7172_summable__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_ln(F),G)) ) ) ).

% summable_add
tff(fact_7173_summable__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_fa(F),G)) ) ) ).

% summable_add
tff(fact_7174_summable__Suc__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_gp(F))
    <=> summable_real(F) ) ).

% summable_Suc_iff
tff(fact_7175_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => summable_real(aa_nat_fun_nat_real(aTP_Lamm_gr(F),K)) ) ).

% summable_ignore_initial_segment
tff(fact_7176_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_7177_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_7178_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_7179_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_nat2(N),N4)
           => ( aa_nat_nat(F,N) = zero_zero_nat ) )
       => summable_nat(F) ) ) ).

% summable_finite
tff(fact_7180_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_nat2(N),N4)
           => ( aa_nat_real(F,N) = zero_zero_real ) )
       => summable_real(F) ) ) ).

% summable_finite
tff(fact_7181_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_nat2(N),N4)
           => ( aa_nat_int(F,N) = zero_zero_int ) )
       => summable_int(F) ) ) ).

% summable_finite
tff(fact_7182_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_nat2(N),N4)
           => ( aa_nat_complex(F,N) = zero_zero_complex ) )
       => summable_complex(F) ) ) ).

% summable_finite
tff(fact_7183_sgn__not__eq__imp,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) != aa_int_int(sgn_sgn_int,A) )
     => ( ( aa_int_int(sgn_sgn_int,A) != zero_zero_int )
       => ( ( aa_int_int(sgn_sgn_int,B) != zero_zero_int )
         => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,aa_int_int(sgn_sgn_int,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_7184_sgn__not__eq__imp,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B) != aa_real_real(sgn_sgn_real,A) )
     => ( ( aa_real_real(sgn_sgn_real,A) != zero_zero_real )
       => ( ( aa_real_real(sgn_sgn_real,B) != zero_zero_real )
         => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,aa_real_real(sgn_sgn_real,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_7185_sgn__minus__1,axiom,
    aa_complex_complex(sgn_sgn_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% sgn_minus_1
tff(fact_7186_sgn__minus__1,axiom,
    aa_int_int(sgn_sgn_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% sgn_minus_1
tff(fact_7187_sgn__minus__1,axiom,
    aa_real_real(sgn_sgn_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sgn_minus_1
tff(fact_7188_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_7189_summable__mult__D,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_lq(C),F))
     => ( ( C != zero_zero_real )
       => summable_real(F) ) ) ).

% summable_mult_D
tff(fact_7190_summable__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lr(C),F))
     => ( ( C != zero_zero_complex )
       => summable_complex(F) ) ) ).

% summable_mult_D
tff(fact_7191_summable__zero__power,axiom,
    summable_real(power_power_real(zero_zero_real)) ).

% summable_zero_power
tff(fact_7192_summable__zero__power,axiom,
    summable_int(power_power_int(zero_zero_int)) ).

% summable_zero_power
tff(fact_7193_summable__zero__power,axiom,
    summable_complex(power_power_complex(zero_zero_complex)) ).

% summable_zero_power
tff(fact_7194_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_ey(F),G)) ) ) ) ).

% suminf_add
tff(fact_7195_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_ln(F),G)) ) ) ) ).

% suminf_add
tff(fact_7196_suminf__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => ( aa_real_real(plus_plus_real(suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_fa(F),G)) ) ) ) ).

% suminf_add
tff(fact_7197_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_7198_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_7199_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_7200_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_7201_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_7202_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_7203_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_7204_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_7205_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_7206_sgn__1__pos,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = one_one_real )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% sgn_1_pos
tff(fact_7207_sgn__1__pos,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = one_one_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% sgn_1_pos
tff(fact_7208_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_rl(F)) ).

% summable_zero_power'
tff(fact_7209_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_ev(F)) ).

% summable_zero_power'
tff(fact_7210_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_eu(F)) ).

% summable_zero_power'
tff(fact_7211_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_ev(F)) ).

% summable_0_powser
tff(fact_7212_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_eu(F)) ).

% summable_0_powser
tff(fact_7213_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_mp(F),Z))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_rm(F),Z)) ) ).

% powser_split_head(3)
tff(fact_7214_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_mo(F),Z))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_rn(F),Z)) ) ).

% powser_split_head(3)
tff(fact_7215_summable__powser__split__head,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_rm(F),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_mp(F),Z)) ) ).

% summable_powser_split_head
tff(fact_7216_summable__powser__split__head,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_rn(F),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_mo(F),Z)) ) ).

% summable_powser_split_head
tff(fact_7217_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_ro(F),Mb),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_mp(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7218_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,Mb: nat,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_rp(F),Mb),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_mo(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7219_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_rq(F)) ) ) ).

% summable_norm_comparison_test
tff(fact_7220_suminf__pos2,axiom,
    ! [F: fun_nat_real,Ia: 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,Ia))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F)) ) ) ) ).

% suminf_pos2
tff(fact_7221_suminf__pos2,axiom,
    ! [F: fun_nat_nat,Ia: 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,Ia))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F)) ) ) ) ).

% suminf_pos2
tff(fact_7222_suminf__pos2,axiom,
    ! [F: fun_nat_int,Ia: 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,Ia))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F)) ) ) ) ).

% suminf_pos2
tff(fact_7223_suminf__pos__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F))
        <=> ? [I4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I4)) ) ) ) ).

% suminf_pos_iff
tff(fact_7224_suminf__pos__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F))
        <=> ? [I4: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I4)) ) ) ) ).

% suminf_pos_iff
tff(fact_7225_suminf__pos__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F))
        <=> ? [I4: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I4)) ) ) ) ).

% suminf_pos_iff
tff(fact_7226_suminf__le__const,axiom,
    ! [F: fun_nat_int,Xb: int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,suminf_int(F)),Xb) ) ) ).

% suminf_le_const
tff(fact_7227_suminf__le__const,axiom,
    ! [F: fun_nat_nat,Xb: nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,suminf_nat(F)),Xb) ) ) ).

% suminf_le_const
tff(fact_7228_suminf__le__const,axiom,
    ! [F: fun_nat_real,Xb: real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(F)),Xb) ) ) ).

% suminf_le_const
tff(fact_7229_sgn__1__neg,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% sgn_1_neg
tff(fact_7230_sgn__1__neg,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% sgn_1_neg
tff(fact_7231_sgn__if,axiom,
    ! [Xb: int] :
      aa_int_int(sgn_sgn_int,Xb) = $ite(
        Xb = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% sgn_if
tff(fact_7232_sgn__if,axiom,
    ! [Xb: real] :
      aa_real_real(sgn_sgn_real,Xb) = $ite(
        Xb = zero_zero_real,
        zero_zero_real,
        $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% sgn_if
tff(fact_7233_zsgn__def,axiom,
    ! [Ia: int] :
      aa_int_int(sgn_sgn_int,Ia) = $ite(
        Ia = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ia),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% zsgn_def
tff(fact_7234_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_int,Xb: 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,aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
       => summable_int(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_7235_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_nat,Xb: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
       => summable_nat(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_7236_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_real,Xb: 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,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
       => summable_real(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_7237_norm__sgn,axiom,
    ! [Xb: real] :
      aa_real_real(real_V7735802525324610683m_real,aa_real_real(sgn_sgn_real,Xb)) = $ite(Xb = zero_zero_real,zero_zero_real,one_one_real) ).

% norm_sgn
tff(fact_7238_norm__sgn,axiom,
    ! [Xb: complex] :
      aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(sgn_sgn_complex,Xb)) = $ite(Xb = zero_zero_complex,zero_zero_real,one_one_real) ).

% norm_sgn
tff(fact_7239_bounded__imp__summable,axiom,
    ! [A: fun_nat_int,B2: int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(A,N))
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_nat_int(groups3539618377306564664at_int(A),aa_nat_set_nat(set_ord_atMost_nat,N))),B2)
       => summable_int(A) ) ) ).

% bounded_imp_summable
tff(fact_7240_bounded__imp__summable,axiom,
    ! [A: fun_nat_nat,B2: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(A,N))
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(set_ord_atMost_nat,N))),B2)
       => summable_nat(A) ) ) ).

% bounded_imp_summable
tff(fact_7241_bounded__imp__summable,axiom,
    ! [A: fun_nat_real,B2: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(A),aa_nat_set_nat(set_ord_atMost_nat,N))),B2)
       => summable_real(A) ) ) ).

% bounded_imp_summable
tff(fact_7242_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_gp(F)) = aa_real_real(minus_minus_real(suminf_real(F)),aa_nat_real(F,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_7243_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_nat2(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,aa_set_nat_int(groups3539618377306564664at_int(F),I5)),suminf_int(F)) ) ) ) ).

% sum_le_suminf
tff(fact_7244_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_nat2(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_7245_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_nat2(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,aa_set_nat_real(groups6591440286371151544t_real(F),I5)),suminf_real(F)) ) ) ) ).

% sum_le_suminf
tff(fact_7246_suminf__split__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(F) = aa_real_real(plus_plus_real(suminf_real(aa_nat_fun_nat_real(aTP_Lamm_gr(F),K))),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_split_initial_segment
tff(fact_7247_suminf__minus__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(aa_nat_fun_nat_real(aTP_Lamm_gr(F),K)) = aa_real_real(minus_minus_real(suminf_real(F)),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_minus_initial_segment
tff(fact_7248_sum__less__suminf,axiom,
    ! [F: fun_nat_int,Nb: nat] :
      ( summable_int(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,M)) )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_int(F)) ) ) ).

% sum_less_suminf
tff(fact_7249_sum__less__suminf,axiom,
    ! [F: fun_nat_nat,Nb: nat] :
      ( summable_nat(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,M)) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_nat(F)) ) ) ).

% sum_less_suminf
tff(fact_7250_sum__less__suminf,axiom,
    ! [F: fun_nat_real,Nb: nat] :
      ( summable_real(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,M)) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_real(F)) ) ) ).

% sum_less_suminf
tff(fact_7251_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_mp(F),Z))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_mp(F),Z)) = aa_real_real(plus_plus_real(aa_nat_real(F,zero_zero_nat)),aa_real_real(times_times_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_rm(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_7252_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_mo(F),Z))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_mo(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_rn(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_7253_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_mp(F),Z))
     => ( aa_real_real(times_times_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_rm(F),Z))),Z) = aa_real_real(minus_minus_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_mp(F),Z))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_7254_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_mo(F),Z))
     => ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_rn(F),Z))),Z) = aa_complex_complex(minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_mo(F),Z))),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_7255_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] :
              ~ ! [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),M2)
                 => ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N5)))),E2) ) ) ) ).

% summable_partial_sum_bound
tff(fact_7256_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] :
              ~ ! [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),M2)
                 => ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N5)))),E2) ) ) ) ).

% summable_partial_sum_bound
tff(fact_7257_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_gr(F),N5)))),R2) ) ) ) ).

% suminf_exist_split
tff(fact_7258_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_rr(F),N5)))),R2) ) ) ) ).

% suminf_exist_split
tff(fact_7259_summable__ratio__test,axiom,
    ! [C: real,N4: nat,F: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),one_one_real)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N4),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,aa_nat_nat(suc,N)))),aa_real_real(times_times_real(C),aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N)))) )
       => summable_real(F) ) ) ).

% summable_ratio_test
tff(fact_7260_summable__ratio__test,axiom,
    ! [C: real,N4: nat,F: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),one_one_real)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N4),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,aa_nat_nat(suc,N)))),aa_real_real(times_times_real(C),aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N)))) )
       => summable_complex(F) ) ) ).

% summable_ratio_test
tff(fact_7261_sum__less__suminf2,axiom,
    ! [F: fun_nat_int,Nb: nat,Ia: nat] :
      ( summable_int(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,M)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ia)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,Ia))
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_int(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_7262_sum__less__suminf2,axiom,
    ! [F: fun_nat_nat,Nb: nat,Ia: nat] :
      ( summable_nat(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,M)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ia)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,Ia))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_nat(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_7263_sum__less__suminf2,axiom,
    ! [F: fun_nat_real,Nb: nat,Ia: nat] :
      ( summable_real(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,M)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ia)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,Ia))
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_real(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_7264_xor__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = $ite(
        Mb = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one))))))) ) ).

% xor_nat_unfold
tff(fact_7265_xor__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb) != ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% xor_nat_rec
tff(fact_7266_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,numeral_numeral_int(bit0(one))),A)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))) ).

% one_xor_eq
tff(fact_7267_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,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,bit0(one))),A))) ).

% one_xor_eq
tff(fact_7268_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,numeral_numeral_int(bit0(one))),A)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))) ).

% xor_one_eq
tff(fact_7269_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,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,bit0(one))),A))) ).

% xor_one_eq
tff(fact_7270_summable__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => summable_real(aTP_Lamm_rs(Xb)) ) ).

% summable_arctan_series
tff(fact_7271_diffs__equiv,axiom,
    ! [C: fun_nat_complex,Xb: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_rt(C),Xb))
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_ru(C),Xb)),suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_rt(C),Xb))) ) ).

% diffs_equiv
tff(fact_7272_diffs__equiv,axiom,
    ! [C: fun_nat_real,Xb: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_rv(C),Xb))
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_rw(C),Xb)),suminf_real(aa_real_fun_nat_real(aTP_Lamm_rv(C),Xb))) ) ).

% diffs_equiv
tff(fact_7273_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_7274_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_7275_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_7276_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(product_Pair_nat_nat),P2) = P2 ).

% case_prod_Pair_iden
tff(fact_7277_case__prod__Pair__iden,axiom,
    ! [P2: produc859450856879609959at_nat] : produc6744312701629110395at_nat(produc6161850002892822231at_nat,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_7278_case__prod__Pair__iden,axiom,
    ! [P2: produc9072475918466114483BT_nat] : produc2645369811736392845BT_nat(produc738532404422230701BT_nat,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_7279_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_int_int] : produc4245557441103728435nt_int(product_Pair_int_int,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_7280_case__prod__Pair__iden,axiom,
    ! [P2: produc7272778201969148633d_enat] : produc4174022389229927035d_enat(produc581526299967858633d_enat,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_7281_the__elem__eq,axiom,
    ! [Xb: real] : the_elem_real(insert_real2(Xb,bot_bot_set_real)) = Xb ).

% the_elem_eq
tff(fact_7282_the__elem__eq,axiom,
    ! [Xb: $o] :
      ( aa_set_o_o(the_elem_o,insert_o2((Xb),bot_bot_set_o))
    <=> (Xb) ) ).

% the_elem_eq
tff(fact_7283_the__elem__eq,axiom,
    ! [Xb: nat] : aa_set_nat_nat(the_elem_nat,insert_nat2(Xb,bot_bot_set_nat)) = Xb ).

% the_elem_eq
tff(fact_7284_the__elem__eq,axiom,
    ! [Xb: int] : the_elem_int(insert_int2(Xb,bot_bot_set_int)) = Xb ).

% the_elem_eq
tff(fact_7285_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_7286_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_7287_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_7288_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_7289_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_7290_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_7291_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_7292_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_7293_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_7294_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_7295_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_7296_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_7297_abs__0,axiom,
    aa_complex_complex(abs_abs_complex,zero_zero_complex) = zero_zero_complex ).

% abs_0
tff(fact_7298_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_7299_abs__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,A)) = aa_real_real(times_times_real(A),A) ).

% abs_mult_self_eq
tff(fact_7300_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_7301_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_7302_abs__1,axiom,
    aa_complex_complex(abs_abs_complex,one_one_complex) = one_one_complex ).

% abs_1
tff(fact_7303_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_7304_abs__add__abs,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) = aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_add_abs
tff(fact_7305_abs__divide,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_divide
tff(fact_7306_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_7307_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_7308_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_7309_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_7310_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_7311_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_7312_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_7313_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_7314_abs__of__nat,axiom,
    ! [Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_nat_real(semiri5074537144036343181t_real,Nb) ).

% abs_of_nat
tff(fact_7315_abs__of__nat,axiom,
    ! [Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_int(semiri1314217659103216013at_int,Nb) ).

% abs_of_nat
tff(fact_7316_of__int__abs,axiom,
    ! [Xb: int] : aa_int_int(ring_1_of_int_int,aa_int_int(abs_abs_int,Xb)) = aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Xb)) ).

% of_int_abs
tff(fact_7317_of__int__abs,axiom,
    ! [Xb: int] : aa_int_real(ring_1_of_int_real,aa_int_int(abs_abs_int,Xb)) = aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Xb)) ).

% of_int_abs
tff(fact_7318_abs__bool__eq,axiom,
    ! [P: $o] : aa_real_real(abs_abs_real,aa_o_real(zero_n3304061248610475627l_real,(P))) = aa_o_real(zero_n3304061248610475627l_real,(P)) ).

% abs_bool_eq
tff(fact_7319_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_7320_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_7321_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_7322_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_7323_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_7324_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_7325_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_7326_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_7327_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_7328_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,aa_set_nat_real(groups6591440286371151544t_real(F),A2))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_rx(F)),A2)) ).

% sum_abs
tff(fact_7329_zero__le__divide__abs__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(A),aa_real_real(abs_abs_real,B)))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_7330_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(A),aa_real_real(abs_abs_real,B))),zero_zero_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_7331_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_7332_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_7333_abs__sgn__eq__1,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = one_one_real ) ) ).

% abs_sgn_eq_1
tff(fact_7334_abs__sgn__eq__1,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = one_one_int ) ) ).

% abs_sgn_eq_1
tff(fact_7335_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(abs_abs_real,A)) = aa_o_real(zero_n3304061248610475627l_real,A != zero_zero_real) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_7336_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(abs_abs_complex,A)) = aa_o_complex(zero_n1201886186963655149omplex,A != zero_zero_complex) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_7337_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(abs_abs_int,A)) = aa_o_int(zero_n2684676970156552555ol_int,A != zero_zero_int) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_7338_sgn__abs,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = aa_o_real(zero_n3304061248610475627l_real,A != zero_zero_real) ).

% sgn_abs
tff(fact_7339_sgn__abs,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(sgn_sgn_complex,A)) = aa_o_complex(zero_n1201886186963655149omplex,A != zero_zero_complex) ).

% sgn_abs
tff(fact_7340_sgn__abs,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = aa_o_int(zero_n2684676970156552555ol_int,A != zero_zero_int) ).

% sgn_abs
tff(fact_7341_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),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_rx(F)),A2)) ).

% sum_abs_ge_zero
tff(fact_7342_zero__less__power__abs__iff,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),Nb))
    <=> ( ( A != zero_zero_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_7343_zero__less__power__abs__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb))
    <=> ( ( A != zero_zero_int )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_7344_norm__of__real__add1,axiom,
    ! [Xb: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(aa_real_real(real_V1803761363581548252l_real,Xb)),one_one_real)) = aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(Xb),one_one_real)) ).

% norm_of_real_add1
tff(fact_7345_norm__of__real__add1,axiom,
    ! [Xb: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,Xb)),one_one_complex)) = aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(Xb),one_one_real)) ).

% norm_of_real_add1
tff(fact_7346_norm__of__real__addn,axiom,
    ! [Xb: real,B: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(aa_real_real(real_V1803761363581548252l_real,Xb)),numeral_numeral_real(B))) = aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(Xb),numeral_numeral_real(B))) ).

% norm_of_real_addn
tff(fact_7347_norm__of__real__addn,axiom,
    ! [Xb: real,B: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,Xb)),numera6690914467698888265omplex(B))) = aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(Xb),numeral_numeral_real(B))) ).

% norm_of_real_addn
tff(fact_7348_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_7349_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_7350_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_7351_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_7352_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_7353_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_7354_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_7355_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_7356_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_7357_abs__eq__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_int_int(abs_abs_int,Xb) = aa_int_int(abs_abs_int,Y) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% abs_eq_iff
tff(fact_7358_abs__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(abs_abs_real,Xb) = aa_real_real(abs_abs_real,Y) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% abs_eq_iff
tff(fact_7359_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_7360_abs__minus__commute,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B)) = aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),A)) ).

% abs_minus_commute
tff(fact_7361_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_7362_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_7363_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_7364_abs__mult,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(times_times_real(A),B)) = aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_mult
tff(fact_7365_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_7366_sgn__power__injE,axiom,
    ! [A: real,Nb: nat,Xb: real,B: real] :
      ( ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,A)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),Nb)) = Xb )
     => ( ( Xb = aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,B)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,B)),Nb)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_7367_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_7368_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_7369_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_7370_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_7371_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_7372_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_7373_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(A),B))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq
tff(fact_7374_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_7375_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(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(times_times_real(C),D)) ) ) ).

% abs_mult_less
tff(fact_7376_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_7377_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B))) ).

% abs_triangle_ineq2
tff(fact_7378_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_7379_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B))) ).

% abs_triangle_ineq3
tff(fact_7380_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_7381_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_7382_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_7383_nonzero__abs__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% nonzero_abs_divide
tff(fact_7384_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_7385_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_7386_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_7387_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_7388_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_7389_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_7390_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_7391_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_7392_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_7393_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_7394_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: int] : aa_int_int(abs_abs_int,K) = aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_7395_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: real] : aa_real_real(abs_abs_real,K) = aa_real_real(times_times_real(K),aa_real_real(sgn_sgn_real,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_7396_abs__mult__sgn,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(sgn_sgn_int,A)) = A ).

% abs_mult_sgn
tff(fact_7397_abs__mult__sgn,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(sgn_sgn_real,A)) = A ).

% abs_mult_sgn
tff(fact_7398_abs__mult__sgn,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(sgn_sgn_complex,A)) = A ).

% abs_mult_sgn
tff(fact_7399_sgn__mult__abs,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(abs_abs_int,A)) = A ).

% sgn_mult_abs
tff(fact_7400_sgn__mult__abs,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,A)),aa_real_real(abs_abs_real,A)) = A ).

% sgn_mult_abs
tff(fact_7401_sgn__mult__abs,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(abs_abs_complex,A)) = A ).

% sgn_mult_abs
tff(fact_7402_mult__sgn__abs,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,Xb)),aa_int_int(abs_abs_int,Xb)) = Xb ).

% mult_sgn_abs
tff(fact_7403_mult__sgn__abs,axiom,
    ! [Xb: real] : aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Xb)),aa_real_real(abs_abs_real,Xb)) = Xb ).

% mult_sgn_abs
tff(fact_7404_same__sgn__abs__add,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) = aa_int_int(sgn_sgn_int,A) )
     => ( 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)) ) ) ).

% same_sgn_abs_add
tff(fact_7405_same__sgn__abs__add,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B) = aa_real_real(sgn_sgn_real,A) )
     => ( aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% same_sgn_abs_add
tff(fact_7406_dense__eq0__I,axiom,
    ! [Xb: 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,Xb)),E) )
     => ( Xb = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_7407_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(times_times_real(A),B)) = aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% abs_eq_mult
tff(fact_7408_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_7409_abs__mult__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(times_times_real(aa_real_real(abs_abs_real,Y)),Xb) = aa_real_real(abs_abs_real,aa_real_real(times_times_real(Y),Xb)) ) ) ).

% abs_mult_pos
tff(fact_7410_abs__mult__pos,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),Xb) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xb)) ) ) ).

% abs_mult_pos
tff(fact_7411_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_7412_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_7413_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_7414_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_7415_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_7416_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_7417_abs__div__pos,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,Xb)),Y) = aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(Xb),Y)) ) ) ).

% abs_div_pos
tff(fact_7418_zero__le__power__abs,axiom,
    ! [A: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),Nb)) ).

% zero_le_power_abs
tff(fact_7419_zero__le__power__abs,axiom,
    ! [A: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb)) ).

% zero_le_power_abs
tff(fact_7420_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_7421_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_7422_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_7423_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_7424_abs__if__raw,axiom,
    ! [X: int] :
      aa_int_int(abs_abs_int,X) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X),zero_zero_int),aa_int_int(uminus_uminus_int,X),X) ).

% abs_if_raw
tff(fact_7425_abs__if__raw,axiom,
    ! [X: real] :
      aa_real_real(abs_abs_real,X) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real),aa_real_real(uminus_uminus_real,X),X) ).

% abs_if_raw
tff(fact_7426_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq4
tff(fact_7427_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_7428_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),aa_real_real(plus_plus_real(C),D)))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),C))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_7429_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_7430_abs__diff__le__iff,axiom,
    ! [Xb: real,A: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xb),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),R2)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(plus_plus_real(A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_7431_abs__diff__le__iff,axiom,
    ! [Xb: 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,Xb),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)),Xb)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_7432_abs__diff__less__iff,axiom,
    ! [Xb: real,A: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xb),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),R2)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(plus_plus_real(A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_7433_abs__diff__less__iff,axiom,
    ! [Xb: 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,Xb),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)),Xb)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_7434_abs__sgn__eq,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = $ite(A = zero_zero_real,zero_zero_real,one_one_real) ).

% abs_sgn_eq
tff(fact_7435_abs__sgn__eq,axiom,
    ! [A: int] :
      aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = $ite(A = zero_zero_int,zero_zero_int,one_one_int) ).

% abs_sgn_eq
tff(fact_7436_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_rx(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_7437_abs__add__one__gt__zero,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(one_one_real),aa_real_real(abs_abs_real,Xb))) ).

% abs_add_one_gt_zero
tff(fact_7438_abs__add__one__gt__zero,axiom,
    ! [Xb: 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,Xb))) ).

% abs_add_one_gt_zero
tff(fact_7439_of__int__leD,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% of_int_leD
tff(fact_7440_of__int__leD,axiom,
    ! [Nb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Xb) ) ) ).

% of_int_leD
tff(fact_7441_of__int__lessD,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% of_int_lessD
tff(fact_7442_of__int__lessD,axiom,
    ! [Nb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Xb) ) ) ).

% of_int_lessD
tff(fact_7443_round__diff__minimal,axiom,
    ! [Z: real,Mb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Z))))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z),aa_int_real(ring_1_of_int_real,Mb)))) ).

% round_diff_minimal
tff(fact_7444_diffs__def,axiom,
    ! [C: fun_nat_complex,X: nat] : aa_nat_complex(diffs_complex(C),X) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,X))),aa_nat_complex(C,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_7445_diffs__def,axiom,
    ! [C: fun_nat_real,X: nat] : aa_nat_real(diffs_real(C),X) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,X))),aa_nat_real(C,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_7446_diffs__def,axiom,
    ! [C: fun_nat_int,X: nat] : aa_nat_int(diffs_int(C),X) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,X))),aa_nat_int(C,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_7447_abs__le__square__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ).

% abs_le_square_iff
tff(fact_7448_abs__le__square__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),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(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ).

% abs_le_square_iff
tff(fact_7449_power2__le__iff__abs__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7450_power2__le__iff__abs__le,axiom,
    ! [Y: int,Xb: 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(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7451_abs__sqrt__wlog,axiom,
    ! [P: fun_real_fun_real_o,Xb: real] :
      ( ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X4)
         => aa_real_o(aa_real_fun_real_o(P,X4),aa_nat_real(power_power_real(X4),aa_num_nat(numeral_numeral_nat,bit0(one)))) )
     => aa_real_o(aa_real_fun_real_o(P,aa_real_real(abs_abs_real,Xb)),aa_nat_real(power_power_real(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ).

% abs_sqrt_wlog
tff(fact_7452_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_o,Xb: int] :
      ( ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
         => aa_int_o(aa_int_fun_int_o(P,X4),aa_nat_int(power_power_int(X4),aa_num_nat(numeral_numeral_nat,bit0(one)))) )
     => aa_int_o(aa_int_fun_int_o(P,aa_int_int(abs_abs_int,Xb)),aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ).

% abs_sqrt_wlog
tff(fact_7453_abs__square__le__1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real) ) ).

% abs_square_le_1
tff(fact_7454_abs__square__le__1,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),one_one_int) ) ).

% abs_square_le_1
tff(fact_7455_abs__square__less__1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real) ) ).

% abs_square_less_1
tff(fact_7456_abs__square__less__1,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Xb)),one_one_int) ) ).

% abs_square_less_1
tff(fact_7457_power__mono__even,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),Nb)),aa_nat_real(power_power_real(B),Nb)) ) ) ).

% power_mono_even
tff(fact_7458_power__mono__even,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono_even
tff(fact_7459_convex__sum__bound__le,axiom,
    ! [I5: set_real,Xb: fun_real_real,A: fun_real_real,B: real,Delta: real] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real2(I2),I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(Xb,I2)) )
     => ( ( groups8097168146408367636l_real(Xb,I5) = one_one_real )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real2(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(A,I2)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ry(Xb),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7460_convex__sum__bound__le,axiom,
    ! [I5: set_o,Xb: fun_o_real,A: fun_o_real,B: real,Delta: real] :
      ( ! [I2: $o] :
          ( aa_set_o_o(member_o2((I2)),I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(Xb,(I2))) )
     => ( ( groups8691415230153176458o_real(Xb,I5) = one_one_real )
       => ( ! [I2: $o] :
              ( aa_set_o_o(member_o2((I2)),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_o_real(A,(I2))),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups8691415230153176458o_real(aa_fun688608109223240316o_real(aTP_Lamm_rz(Xb),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7461_convex__sum__bound__le,axiom,
    ! [I5: set_set_nat,Xb: fun_set_nat_real,A: fun_set_nat_real,B: real,Delta: real] :
      ( ! [I2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat2(I2),I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_set_nat_real(Xb,I2)) )
     => ( ( groups5107569545109728110t_real(Xb,I5) = one_one_real )
       => ( ! [I2: set_nat] :
              ( aa_set_set_nat_o(member_set_nat2(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_set_nat_real(A,I2)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups5107569545109728110t_real(aa_fun3569254276315353500t_real(aTP_Lamm_sa(Xb),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7462_convex__sum__bound__le,axiom,
    ! [I5: set_int,Xb: fun_int_real,A: fun_int_real,B: real,Delta: real] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int2(I2),I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(Xb,I2)) )
     => ( ( groups8778361861064173332t_real(Xb,I5) = one_one_real )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int2(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_int_real(A,I2)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_sb(Xb),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7463_convex__sum__bound__le,axiom,
    ! [I5: set_real,Xb: fun_real_int,A: fun_real_int,B: int,Delta: int] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real2(I2),I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(Xb,I2)) )
     => ( ( groups1932886352136224148al_int(Xb,I5) = one_one_int )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real2(I2),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,I2)),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_sc(Xb),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7464_convex__sum__bound__le,axiom,
    ! [I5: set_o,Xb: fun_o_int,A: fun_o_int,B: int,Delta: int] :
      ( ! [I2: $o] :
          ( aa_set_o_o(member_o2((I2)),I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(Xb,(I2))) )
     => ( ( groups8505340233167759370_o_int(Xb,I5) = one_one_int )
       => ( ! [I2: $o] :
              ( aa_set_o_o(member_o2((I2)),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_o_int(A,(I2))),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,groups8505340233167759370_o_int(aa_fun4727766026823762812_o_int(aTP_Lamm_sd(Xb),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7465_convex__sum__bound__le,axiom,
    ! [I5: set_set_nat,Xb: fun_set_nat_int,A: fun_set_nat_int,B: int,Delta: int] :
      ( ! [I2: set_nat] :
          ( aa_set_set_nat_o(member_set_nat2(I2),I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_set_nat_int(Xb,I2)) )
     => ( ( groups8292507037921071086at_int(Xb,I5) = one_one_int )
       => ( ! [I2: set_nat] :
              ( aa_set_set_nat_o(member_set_nat2(I2),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_set_nat_int(A,I2)),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,groups8292507037921071086at_int(aa_fun3600621996379952540at_int(aTP_Lamm_se(Xb),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7466_convex__sum__bound__le,axiom,
    ! [I5: set_nat,Xb: fun_nat_int,A: fun_nat_int,B: int,Delta: int] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat2(I2),I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(Xb,I2)) )
     => ( ( aa_set_nat_int(groups3539618377306564664at_int(Xb),I5) = one_one_int )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat2(I2),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,I2)),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,aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_sf(Xb),A)),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7467_convex__sum__bound__le,axiom,
    ! [I5: set_int,Xb: fun_int_int,A: fun_int_int,B: int,Delta: int] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int2(I2),I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(Xb,I2)) )
     => ( ( groups4538972089207619220nt_int(Xb,I5) = one_one_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int2(I2),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_int_int(A,I2)),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,groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_sg(Xb),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7468_convex__sum__bound__le,axiom,
    ! [I5: set_nat,Xb: fun_nat_real,A: fun_nat_real,B: real,Delta: real] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat2(I2),I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Xb,I2)) )
     => ( ( aa_set_nat_real(groups6591440286371151544t_real(Xb),I5) = one_one_real )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat2(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(A,I2)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_sh(Xb),A)),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7469_dbl__dec__def,axiom,
    ! [Xb: complex] : aa_complex_complex(neg_nu6511756317524482435omplex,Xb) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(Xb),Xb)),one_one_complex) ).

% dbl_dec_def
tff(fact_7470_dbl__dec__def,axiom,
    ! [Xb: int] : aa_int_int(neg_nu3811975205180677377ec_int,Xb) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xb)),one_one_int) ).

% dbl_dec_def
tff(fact_7471_dbl__dec__def,axiom,
    ! [Xb: real] : aa_real_real(neg_nu6075765906172075777c_real,Xb) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(Xb),Xb)),one_one_real) ).

% dbl_dec_def
tff(fact_7472_of__int__round__abs__le,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))),Xb))),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))) ).

% of_int_round_abs_le
tff(fact_7473_round__unique_H,axiom,
    ! [Xb: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xb),aa_int_real(ring_1_of_int_real,Nb)))),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))
     => ( aa_real_int(archim8280529875227126926d_real,Xb) = Nb ) ) ).

% round_unique'
tff(fact_7474_monoseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => topolo6980174941875973593q_real(aTP_Lamm_si(Xb)) ) ).

% monoseq_arctan_series
tff(fact_7475_arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_real(arctan,Xb) = suminf_real(aTP_Lamm_rs(Xb)) ) ) ).

% arctan_series
tff(fact_7476_Maclaurin__exp__lt,axiom,
    ! [Xb: real,Nb: nat] :
      ( ( Xb != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T5))
            & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(exp_real,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_sj(Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(exp_real,T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(Xb),Nb))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_7477_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(minus_minus_real(one_one_real),aa_real_real(divide_divide_real(Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_7478_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(plus_plus_real(one_one_real),aa_real_real(divide_divide_real(Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,Xb)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_7479_exp__zero,axiom,
    aa_real_real(exp_real,zero_zero_real) = one_one_real ).

% exp_zero
tff(fact_7480_exp__zero,axiom,
    aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).

% exp_zero
tff(fact_7481_zdvd1__eq,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),one_one_int)
    <=> ( aa_int_int(abs_abs_int,Xb) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_7482_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_7483_exp__not__eq__zero,axiom,
    ! [Xb: real] : aa_real_real(exp_real,Xb) != zero_zero_real ).

% exp_not_eq_zero
tff(fact_7484_exp__not__eq__zero,axiom,
    ! [Xb: complex] : aa_complex_complex(exp_complex,Xb) != zero_zero_complex ).

% exp_not_eq_zero
tff(fact_7485_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_7486_exp__add__commuting,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(times_times_real(Xb),Y) = aa_real_real(times_times_real(Y),Xb) )
     => ( aa_real_real(exp_real,aa_real_real(plus_plus_real(Xb),Y)) = aa_real_real(times_times_real(aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y)) ) ) ).

% exp_add_commuting
tff(fact_7487_exp__add__commuting,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(times_times_complex(Xb),Y) = aa_complex_complex(times_times_complex(Y),Xb) )
     => ( aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,Y)) ) ) ).

% exp_add_commuting
tff(fact_7488_mult__exp__exp,axiom,
    ! [Xb: real,Y: real] : aa_real_real(times_times_real(aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y)) = aa_real_real(exp_real,aa_real_real(plus_plus_real(Xb),Y)) ).

% mult_exp_exp
tff(fact_7489_mult__exp__exp,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,Y)) = aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) ).

% mult_exp_exp
tff(fact_7490_abs__zmult__eq__1,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb)) = one_one_int )
     => ( aa_int_int(abs_abs_int,Mb) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_7491_zabs__def,axiom,
    ! [Ia: int] :
      aa_int_int(abs_abs_int,Ia) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,Ia),zero_zero_int),aa_int_int(uminus_uminus_int,Ia),Ia) ).

% zabs_def
tff(fact_7492_dvd__imp__le__int,axiom,
    ! [Ia: int,D: int] :
      ( ( Ia != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),Ia)
       => 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,Ia)) ) ) ).

% dvd_imp_le_int
tff(fact_7493_zdvd__mult__cancel1,axiom,
    ! [Mb: int,Nb: 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),Nb)),Mb)
      <=> ( aa_int_int(abs_abs_int,Nb) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_7494_powr__def,axiom,
    ! [Xb: real,A: real] :
      aa_real_real(powr_real(Xb),A) = $ite(Xb = zero_zero_real,zero_zero_real,aa_real_real(exp_real,aa_real_real(times_times_real(A),aa_real_real(ln_ln_real,Xb)))) ).

% powr_def
tff(fact_7495_exp__divide__power__eq,axiom,
    ! [Nb: nat,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,aa_complex_complex(divide1717551699836669952omplex(Xb),aa_nat_complex(semiri8010041392384452111omplex,Nb)))),Nb) = aa_complex_complex(exp_complex,Xb) ) ) ).

% exp_divide_power_eq
tff(fact_7496_exp__divide__power__eq,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(power_power_real(aa_real_real(exp_real,aa_real_real(divide_divide_real(Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb) = aa_real_real(exp_real,Xb) ) ) ).

% exp_divide_power_eq
tff(fact_7497_nat__intermed__int__val,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),I2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Mb)),K)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Nb))
           => ? [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),I2)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Nb)
                & ( aa_nat_int(F,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_7498_nat__ivt__aux,axiom,
    ! [Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Nb))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Nb)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_7499_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Nb))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Nb)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_7500_monoI1,axiom,
    ! [X5: fun_nat_filter_nat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => ord_le2510731241096832064er_nat(aa_nat_filter_nat(X5,M),aa_nat_filter_nat(X5,N)) )
     => topolo6427056007704750605er_nat(X5) ) ).

% monoI1
tff(fact_7501_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_7502_monoI1,axiom,
    ! [X5: fun_nat_set_nat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(X5,M)),aa_nat_set_nat(X5,N)) )
     => topolo7278393974255667507et_nat(X5) ) ).

% monoI1
tff(fact_7503_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_7504_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_7505_monoI2,axiom,
    ! [X5: fun_nat_filter_nat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => ord_le2510731241096832064er_nat(aa_nat_filter_nat(X5,N),aa_nat_filter_nat(X5,M)) )
     => topolo6427056007704750605er_nat(X5) ) ).

% monoI2
tff(fact_7506_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_7507_monoI2,axiom,
    ! [X5: fun_nat_set_nat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(X5,N)),aa_nat_set_nat(X5,M)) )
     => topolo7278393974255667507et_nat(X5) ) ).

% monoI2
tff(fact_7508_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_7509_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_7510_monoseq__def,axiom,
    ! [X5: fun_nat_filter_nat] :
      ( topolo6427056007704750605er_nat(X5)
    <=> ( ! [M5: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N3)
           => ord_le2510731241096832064er_nat(aa_nat_filter_nat(X5,M5),aa_nat_filter_nat(X5,N3)) )
        | ! [M5: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N3)
           => ord_le2510731241096832064er_nat(aa_nat_filter_nat(X5,N3),aa_nat_filter_nat(X5,M5)) ) ) ) ).

% monoseq_def
tff(fact_7511_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_7512_monoseq__def,axiom,
    ! [X5: fun_nat_set_nat] :
      ( topolo7278393974255667507et_nat(X5)
    <=> ( ! [M5: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N3)
           => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(X5,M5)),aa_nat_set_nat(X5,N3)) )
        | ! [M5: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N3)
           => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(X5,N3)),aa_nat_set_nat(X5,M5)) ) ) ) ).

% monoseq_def
tff(fact_7513_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_7514_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_7515_monoseq__Suc,axiom,
    ! [X5: fun_nat_filter_nat] :
      ( topolo6427056007704750605er_nat(X5)
    <=> ( ! [N3: nat] : ord_le2510731241096832064er_nat(aa_nat_filter_nat(X5,N3),aa_nat_filter_nat(X5,aa_nat_nat(suc,N3)))
        | ! [N3: nat] : ord_le2510731241096832064er_nat(aa_nat_filter_nat(X5,aa_nat_nat(suc,N3)),aa_nat_filter_nat(X5,N3)) ) ) ).

% monoseq_Suc
tff(fact_7516_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_7517_monoseq__Suc,axiom,
    ! [X5: fun_nat_set_nat] :
      ( topolo7278393974255667507et_nat(X5)
    <=> ( ! [N3: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(X5,N3)),aa_nat_set_nat(X5,aa_nat_nat(suc,N3)))
        | ! [N3: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(X5,aa_nat_nat(suc,N3))),aa_nat_set_nat(X5,N3)) ) ) ).

% monoseq_Suc
tff(fact_7518_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_7519_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_7520_mono__SucI2,axiom,
    ! [X5: fun_nat_filter_nat] :
      ( ! [N: nat] : ord_le2510731241096832064er_nat(aa_nat_filter_nat(X5,aa_nat_nat(suc,N)),aa_nat_filter_nat(X5,N))
     => topolo6427056007704750605er_nat(X5) ) ).

% mono_SucI2
tff(fact_7521_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_7522_mono__SucI2,axiom,
    ! [X5: fun_nat_set_nat] :
      ( ! [N: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(X5,aa_nat_nat(suc,N))),aa_nat_set_nat(X5,N))
     => topolo7278393974255667507et_nat(X5) ) ).

% mono_SucI2
tff(fact_7523_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_7524_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_7525_mono__SucI1,axiom,
    ! [X5: fun_nat_filter_nat] :
      ( ! [N: nat] : ord_le2510731241096832064er_nat(aa_nat_filter_nat(X5,N),aa_nat_filter_nat(X5,aa_nat_nat(suc,N)))
     => topolo6427056007704750605er_nat(X5) ) ).

% mono_SucI1
tff(fact_7526_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_7527_mono__SucI1,axiom,
    ! [X5: fun_nat_set_nat] :
      ( ! [N: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(X5,N)),aa_nat_set_nat(X5,aa_nat_nat(suc,N)))
     => topolo7278393974255667507et_nat(X5) ) ).

% mono_SucI1
tff(fact_7528_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_7529_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_7530_exp__first__two__terms,axiom,
    ! [Xb: real] : aa_real_real(exp_real,Xb) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(one_one_real),Xb)),suminf_real(aTP_Lamm_sk(Xb))) ).

% exp_first_two_terms
tff(fact_7531_exp__first__two__terms,axiom,
    ! [Xb: complex] : aa_complex_complex(exp_complex,Xb) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(one_one_complex),Xb)),suminf_complex(aTP_Lamm_sl(Xb))) ).

% exp_first_two_terms
tff(fact_7532_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_7533_exp__first__terms,axiom,
    ! [Xb: complex,K: nat] : aa_complex_complex(exp_complex,Xb) = aa_complex_complex(plus_plus_complex(groups2073611262835488442omplex(aTP_Lamm_sm(Xb),aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_sn(Xb),K))) ).

% exp_first_terms
tff(fact_7534_exp__first__terms,axiom,
    ! [Xb: real,K: nat] : aa_real_real(exp_real,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_so(Xb)),aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(aa_nat_fun_nat_real(aTP_Lamm_sp(Xb),K))) ).

% exp_first_terms
tff(fact_7535_cosh__zero__iff,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(cosh_complex,Xb) = zero_zero_complex )
    <=> ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% cosh_zero_iff
tff(fact_7536_cosh__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cosh_real,Xb) = zero_zero_real )
    <=> ( aa_nat_real(power_power_real(aa_real_real(exp_real,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% cosh_zero_iff
tff(fact_7537_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,numeral_numeral_int(bit0(one)),Bs)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Bs))
        & aa_nat_o(nth_o(Bs),Nb) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_7538_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,groups9119017779487936845_o_nat(zero_n2687167440665602831ol_nat,aa_num_nat(numeral_numeral_nat,bit0(one)),Bs)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Bs))
        & aa_nat_o(nth_o(Bs),Nb) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_7539_inverse__inverse__eq,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,aa_real_real(inverse_inverse_real,A)) = A ).

% inverse_inverse_eq
tff(fact_7540_inverse__eq__iff__eq,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B) )
    <=> ( A = B ) ) ).

% inverse_eq_iff_eq
tff(fact_7541_inverse__zero,axiom,
    aa_complex_complex(invers8013647133539491842omplex,zero_zero_complex) = zero_zero_complex ).

% inverse_zero
tff(fact_7542_inverse__zero,axiom,
    aa_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

% inverse_zero
tff(fact_7543_inverse__nonzero__iff__nonzero,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_7544_inverse__nonzero__iff__nonzero,axiom,
    ! [A: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_7545_inverse__mult__distrib,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) ).

% inverse_mult_distrib
tff(fact_7546_inverse__mult__distrib,axiom,
    ! [A: real,B: real] : aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(A),B)) = aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) ).

% inverse_mult_distrib
tff(fact_7547_inverse__1,axiom,
    aa_complex_complex(invers8013647133539491842omplex,one_one_complex) = one_one_complex ).

% inverse_1
tff(fact_7548_inverse__1,axiom,
    aa_real_real(inverse_inverse_real,one_one_real) = one_one_real ).

% inverse_1
tff(fact_7549_inverse__eq__1__iff,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,Xb) = one_one_complex )
    <=> ( Xb = one_one_complex ) ) ).

% inverse_eq_1_iff
tff(fact_7550_inverse__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(inverse_inverse_real,Xb) = one_one_real )
    <=> ( Xb = one_one_real ) ) ).

% inverse_eq_1_iff
tff(fact_7551_inverse__divide,axiom,
    ! [A: real,B: real] : aa_real_real(inverse_inverse_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(B),A) ).

% inverse_divide
tff(fact_7552_inverse__minus__eq,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,A)) ).

% inverse_minus_eq
tff(fact_7553_abs__inverse,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(abs_abs_real,A)) ).

% abs_inverse
tff(fact_7554_inverse__sgn,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,aa_real_real(sgn_sgn_real,A)) = aa_real_real(sgn_sgn_real,A) ).

% inverse_sgn
tff(fact_7555_sgn__inverse,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(sgn_sgn_real,A)) ).

% sgn_inverse
tff(fact_7556_nat__int,axiom,
    ! [Nb: nat] : aa_int_nat(nat2,aa_nat_int(semiri1314217659103216013at_int,Nb)) = Nb ).

% nat_int
tff(fact_7557_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_7558_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_7559_inverse__less__iff__less,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,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% inverse_less_iff_less
tff(fact_7560_inverse__less__iff__less__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(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_7561_inverse__negative__iff__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% inverse_negative_iff_negative
tff(fact_7562_inverse__positive__iff__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% inverse_positive_iff_positive
tff(fact_7563_cosh__0,axiom,
    aa_real_real(cosh_real,zero_zero_real) = one_one_real ).

% cosh_0
tff(fact_7564_cosh__0,axiom,
    aa_complex_complex(cosh_complex,zero_zero_complex) = one_one_complex ).

% cosh_0
tff(fact_7565_nat__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,numeral_numeral_int(K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_numeral
tff(fact_7566_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_7567_inverse__le__iff__le,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_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% inverse_le_iff_le
tff(fact_7568_inverse__le__iff__le__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_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_7569_right__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,A)) = one_one_complex ) ) ).

% right_inverse
tff(fact_7570_right__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(times_times_real(A),aa_real_real(inverse_inverse_real,A)) = one_one_real ) ) ).

% right_inverse
tff(fact_7571_left__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),A) = one_one_complex ) ) ).

% left_inverse
tff(fact_7572_left__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).

% left_inverse
tff(fact_7573_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_7574_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_7575_nat__0__iff,axiom,
    ! [Ia: int] :
      ( ( aa_int_nat(nat2,Ia) = zero_zero_nat )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),zero_zero_int) ) ).

% nat_0_iff
tff(fact_7576_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_7577_nat__neg__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,numeral_numeral_int(K))) = zero_zero_nat ).

% nat_neg_numeral
tff(fact_7578_nat__zminus__int,axiom,
    ! [Nb: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_7579_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_7580_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_7581_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,Z)) = aa_int_real(ring_1_of_int_real,Z) ) ) ).

% of_nat_nat
tff(fact_7582_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = aa_int_int(ring_1_of_int_int,Z) ) ) ).

% of_nat_nat
tff(fact_7583_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,numeral_numeral_int(V)),numeral_numeral_int(V3))) ).

% diff_nat_numeral
tff(fact_7584_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = aa_int_nat(nat2,Y) )
    <=> ( aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_7585_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_7586_dvd__nat__abs__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),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,Nb)),K) ) ).

% dvd_nat_abs_iff
tff(fact_7587_nat__abs__dvd__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),Nb)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% nat_abs_dvd_iff
tff(fact_7588_nat__ceiling__le__eq,axiom,
    ! [Xb: real,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xb))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,A)) ) ).

% nat_ceiling_le_eq
tff(fact_7589_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_7590_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,numeral_numeral_int(V)),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_7591_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_7592_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb)),A) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_7593_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_7594_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(numeral_numeral_int(Xb)),Nb)),A) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_7595_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: complex,Xb: complex] :
      ( ( aa_complex_complex(times_times_complex(Y),Xb) = aa_complex_complex(times_times_complex(Xb),Y) )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,Y)),Xb) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(invers8013647133539491842omplex,Y)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_7596_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: real,Xb: real] :
      ( ( aa_real_real(times_times_real(Y),Xb) = aa_real_real(times_times_real(Xb),Y) )
     => ( aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,Y)),Xb) = aa_real_real(times_times_real(Xb),aa_real_real(inverse_inverse_real,Y)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_7597_nonzero__norm__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(real_V7735802525324610683m_real,A)) ) ) ).

% nonzero_norm_inverse
tff(fact_7598_nonzero__norm__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = aa_real_real(inverse_inverse_real,aa_complex_real(real_V1022390504157884413omplex,A)) ) ) ).

% nonzero_norm_inverse
tff(fact_7599_inverse__eq__imp__eq,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B) )
     => ( A = B ) ) ).

% inverse_eq_imp_eq
tff(fact_7600_nonzero__imp__inverse__nonzero,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) != zero_zero_complex ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_7601_nonzero__imp__inverse__nonzero,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,A) != zero_zero_real ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_7602_nonzero__inverse__inverse__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_7603_nonzero__inverse__inverse__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,aa_real_real(inverse_inverse_real,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_7604_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(invers8013647133539491842omplex,B) )
     => ( ( A != zero_zero_complex )
       => ( ( B != zero_zero_complex )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_7605_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B) )
     => ( ( A != zero_zero_real )
       => ( ( B != zero_zero_real )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_7606_inverse__zero__imp__zero,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = zero_zero_complex )
     => ( A = zero_zero_complex ) ) ).

% inverse_zero_imp_zero
tff(fact_7607_inverse__zero__imp__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = zero_zero_real )
     => ( A = zero_zero_real ) ) ).

% inverse_zero_imp_zero
tff(fact_7608_field__class_Ofield__inverse__zero,axiom,
    aa_complex_complex(invers8013647133539491842omplex,zero_zero_complex) = zero_zero_complex ).

% field_class.field_inverse_zero
tff(fact_7609_field__class_Ofield__inverse__zero,axiom,
    aa_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

% field_class.field_inverse_zero
tff(fact_7610_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: real,Xb: complex] :
      ( ( A != zero_zero_real )
     => ( ( Xb != zero_zero_complex )
       => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(real_V2046097035970521341omplex(A),Xb)) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,A)),aa_complex_complex(invers8013647133539491842omplex,Xb)) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_7611_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: real,Xb: real] :
      ( ( A != zero_zero_real )
     => ( ( Xb != zero_zero_real )
       => ( aa_real_real(inverse_inverse_real,aa_real_real(real_V1485227260804924795R_real(A),Xb)) = aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,Xb)) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_7612_inverse__less__imp__less,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,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,B),A) ) ) ).

% inverse_less_imp_less
tff(fact_7613_less__imp__inverse__less,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,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% less_imp_inverse_less
tff(fact_7614_inverse__less__imp__less__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
     => ( 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,B),A) ) ) ).

% inverse_less_imp_less_neg
tff(fact_7615_less__imp__inverse__less__neg,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),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% less_imp_inverse_less_neg
tff(fact_7616_inverse__negative__imp__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
     => ( ( A != zero_zero_real )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% inverse_negative_imp_negative
tff(fact_7617_inverse__positive__imp__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
     => ( ( A != zero_zero_real )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ).

% inverse_positive_imp_positive
tff(fact_7618_negative__imp__inverse__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real) ) ).

% negative_imp_inverse_negative
tff(fact_7619_positive__imp__inverse__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A)) ) ).

% positive_imp_inverse_positive
tff(fact_7620_nonzero__inverse__mult__distrib,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,B)),aa_complex_complex(invers8013647133539491842omplex,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_7621_nonzero__inverse__mult__distrib,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(A),B)) = aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_7622_nonzero__inverse__minus__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(invers8013647133539491842omplex,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_7623_nonzero__inverse__minus__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_7624_inverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = one_one_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) = B ) ) ).

% inverse_unique
tff(fact_7625_inverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(times_times_real(A),B) = one_one_real )
     => ( aa_real_real(inverse_inverse_real,A) = B ) ) ).

% inverse_unique
tff(fact_7626_divide__inverse__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,B)),A) ).

% divide_inverse_commute
tff(fact_7627_divide__inverse__commute,axiom,
    ! [A: real,B: real] : aa_real_real(divide_divide_real(A),B) = aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,B)),A) ).

% divide_inverse_commute
tff(fact_7628_divide__inverse,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,B)) ).

% divide_inverse
tff(fact_7629_divide__inverse,axiom,
    ! [A: real,B: real] : aa_real_real(divide_divide_real(A),B) = aa_real_real(times_times_real(A),aa_real_real(inverse_inverse_real,B)) ).

% divide_inverse
tff(fact_7630_field__class_Ofield__divide__inverse,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,B)) ).

% field_class.field_divide_inverse
tff(fact_7631_field__class_Ofield__divide__inverse,axiom,
    ! [A: real,B: real] : aa_real_real(divide_divide_real(A),B) = aa_real_real(times_times_real(A),aa_real_real(inverse_inverse_real,B)) ).

% field_class.field_divide_inverse
tff(fact_7632_inverse__eq__divide,axiom,
    ! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ).

% inverse_eq_divide
tff(fact_7633_inverse__eq__divide,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,A) = aa_real_real(divide_divide_real(one_one_real),A) ).

% inverse_eq_divide
tff(fact_7634_power__mult__power__inverse__commute,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),Nb)),aa_nat_complex(power_power_complex(Xb),Mb)) ).

% power_mult_power_inverse_commute
tff(fact_7635_power__mult__power__inverse__commute,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] : aa_real_real(times_times_real(aa_nat_real(power_power_real(Xb),Mb)),aa_nat_real(power_power_real(aa_real_real(inverse_inverse_real,Xb)),Nb)) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(inverse_inverse_real,Xb)),Nb)),aa_nat_real(power_power_real(Xb),Mb)) ).

% power_mult_power_inverse_commute
tff(fact_7636_power__mult__inverse__distrib,axiom,
    ! [Xb: complex,Mb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_complex_complex(invers8013647133539491842omplex,Xb)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),aa_nat_complex(power_power_complex(Xb),Mb)) ).

% power_mult_inverse_distrib
tff(fact_7637_power__mult__inverse__distrib,axiom,
    ! [Xb: real,Mb: nat] : aa_real_real(times_times_real(aa_nat_real(power_power_real(Xb),Mb)),aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,Xb)),aa_nat_real(power_power_real(Xb),Mb)) ).

% power_mult_inverse_distrib
tff(fact_7638_mult__inverse__of__nat__commute,axiom,
    ! [Xaa: nat,Xb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri8010041392384452111omplex,Xaa))),Xb) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri8010041392384452111omplex,Xaa))) ).

% mult_inverse_of_nat_commute
tff(fact_7639_mult__inverse__of__nat__commute,axiom,
    ! [Xaa: nat,Xb: real] : aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,Xaa))),Xb) = aa_real_real(times_times_real(Xb),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,Xaa))) ).

% mult_inverse_of_nat_commute
tff(fact_7640_nonzero__abs__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(abs_abs_real,A)) ) ) ).

% nonzero_abs_inverse
tff(fact_7641_mult__inverse__of__int__commute,axiom,
    ! [Xaa: int,Xb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(ring_17405671764205052669omplex,Xaa))),Xb) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(ring_17405671764205052669omplex,Xaa))) ).

% mult_inverse_of_int_commute
tff(fact_7642_mult__inverse__of__int__commute,axiom,
    ! [Xaa: int,Xb: real] : aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,aa_int_real(ring_1_of_int_real,Xaa))),Xb) = aa_real_real(times_times_real(Xb),aa_real_real(inverse_inverse_real,aa_int_real(ring_1_of_int_real,Xaa))) ).

% mult_inverse_of_int_commute
tff(fact_7643_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_7644_nat__mono,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ).

% nat_mono
tff(fact_7645_ex__nat,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_13: nat] : aa_nat_o(P,X_13)
    <=> ? [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
          & aa_nat_o(P,aa_int_nat(nat2,X3)) ) ) ).

% ex_nat
tff(fact_7646_all__nat,axiom,
    ! [P: fun_nat_o] :
      ( ! [X_13: nat] : aa_nat_o(P,X_13)
    <=> ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
         => aa_nat_o(P,aa_int_nat(nat2,X3)) ) ) ).

% all_nat
tff(fact_7647_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_7648_inverse__le__imp__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,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_eq_real,B),A) ) ) ).

% inverse_le_imp_le
tff(fact_7649_le__imp__inverse__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_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% le_imp_inverse_le
tff(fact_7650_inverse__le__imp__le__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
     => ( 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_eq_real,B),A) ) ) ).

% inverse_le_imp_le_neg
tff(fact_7651_le__imp__inverse__le__neg,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,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% le_imp_inverse_le_neg
tff(fact_7652_inverse__le__1__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,Xb)),one_one_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% inverse_le_1_iff
tff(fact_7653_one__less__inverse,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(inverse_inverse_real,A)) ) ) ).

% one_less_inverse
tff(fact_7654_one__less__inverse__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(inverse_inverse_real,Xb))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% one_less_inverse_iff
tff(fact_7655_field__class_Ofield__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),A) = one_one_complex ) ) ).

% field_class.field_inverse
tff(fact_7656_field__class_Ofield__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).

% field_class.field_inverse
tff(fact_7657_division__ring__inverse__add,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(plus_plus_complex(A),B))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_7658_division__ring__inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(plus_plus_real(A),B))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_7659_inverse__add,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),aa_complex_complex(invers8013647133539491842omplex,A))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% inverse_add
tff(fact_7660_inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),aa_real_real(inverse_inverse_real,A))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% inverse_add
tff(fact_7661_division__ring__inverse__diff,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(minus_minus_complex(B),A))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_7662_division__ring__inverse__diff,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(minus_minus_real(B),A))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_7663_nonzero__inverse__eq__divide,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_7664_nonzero__inverse__eq__divide,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,A) = aa_real_real(divide_divide_real(one_one_real),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_7665_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_7666_of__nat__ceiling,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,R2)))) ).

% of_nat_ceiling
tff(fact_7667_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_7668_nat__le__iff,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xb)),Nb)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% nat_le_iff
tff(fact_7669_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_7670_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_7671_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_7672_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_7673_nat__plus__as__int,axiom,
    ! [X: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_plus_as_int
tff(fact_7674_cosh__def,axiom,
    ! [Xb: real] : aa_real_real(cosh_real,Xb) = aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,numeral_numeral_real(bit0(one)))),aa_real_real(plus_plus_real(aa_real_real(exp_real,Xb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb)))) ).

% cosh_def
tff(fact_7675_inverse__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ) ).

% inverse_le_iff
tff(fact_7676_inverse__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ) ).

% inverse_less_iff
tff(fact_7677_one__le__inverse,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(inverse_inverse_real,A)) ) ) ).

% one_le_inverse
tff(fact_7678_inverse__less__1__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,Xb)),one_one_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% inverse_less_1_iff
tff(fact_7679_one__le__inverse__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(inverse_inverse_real,Xb))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).

% one_le_inverse_iff
tff(fact_7680_inverse__diff__inverse,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(minus_minus_complex(A),B))),aa_complex_complex(invers8013647133539491842omplex,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_7681_inverse__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(minus_minus_real(A),B))),aa_real_real(inverse_inverse_real,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_7682_reals__Archimedean,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)))),Xb) ) ).

% reals_Archimedean
tff(fact_7683_of__nat__floor,axiom,
    ! [R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,R2)))),R2) ) ).

% of_nat_floor
tff(fact_7684_real__vector__eq__affinity,axiom,
    ! [Mb: real,Y: real,Xb: real,C: real] :
      ( ( Mb != zero_zero_real )
     => ( ( Y = aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(Mb),Xb)),C) )
      <=> ( aa_real_real(minus_minus_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,Mb)),Y)),aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,Mb)),C)) = Xb ) ) ) ).

% real_vector_eq_affinity
tff(fact_7685_real__vector__affinity__eq,axiom,
    ! [Mb: real,Xb: real,C: real,Y: real] :
      ( ( Mb != zero_zero_real )
     => ( ( aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(Mb),Xb)),C) = Y )
      <=> ( Xb = aa_real_real(minus_minus_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,Mb)),Y)),aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,Mb)),C)) ) ) ) ).

% real_vector_affinity_eq
tff(fact_7686_pos__divideR__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(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(real_V1485227260804924795R_real(C),A)) ) ) ).

% pos_divideR_le_eq
tff(fact_7687_pos__le__divideR__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(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(C),A)),B) ) ) ).

% pos_le_divideR_eq
tff(fact_7688_neg__divideR__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(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(C),A)),B) ) ) ).

% neg_divideR_le_eq
tff(fact_7689_neg__le__divideR__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(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(real_V1485227260804924795R_real(C),A)) ) ) ).

% neg_le_divideR_eq
tff(fact_7690_pos__divideR__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(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(real_V1485227260804924795R_real(C),A)) ) ) ).

% pos_divideR_less_eq
tff(fact_7691_pos__less__divideR__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(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V1485227260804924795R_real(C),A)),B) ) ) ).

% pos_less_divideR_eq
tff(fact_7692_neg__divideR__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(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V1485227260804924795R_real(C),A)),B) ) ) ).

% neg_divideR_less_eq
tff(fact_7693_neg__less__divideR__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(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(real_V1485227260804924795R_real(C),A)) ) ) ).

% neg_less_divideR_eq
tff(fact_7694_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_7695_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_7696_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_7697_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_7698_split__nat,axiom,
    ! [P: fun_nat_o,Ia: int] :
      ( aa_nat_o(P,aa_int_nat(nat2,Ia))
    <=> ( ! [N3: nat] :
            ( ( Ia = aa_nat_int(semiri1314217659103216013at_int,N3) )
           => aa_nat_o(P,N3) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ia),zero_zero_int)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_nat
tff(fact_7699_le__mult__nat__floor,axiom,
    ! [A: real,B: real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,B)))),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,aa_real_real(times_times_real(A),B)))) ).

% le_mult_nat_floor
tff(fact_7700_le__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K) ) ) ).

% le_nat_iff
tff(fact_7701_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_7702_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_7703_Suc__as__int,axiom,
    ! [X: nat] : aa_nat_nat(suc,X) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X)),one_one_int)) ).

% Suc_as_int
tff(fact_7704_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_7705_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_7706_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_7707_nat__diff__distrib_H,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( 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,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_7708_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_7709_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_7710_nat__floor__neg,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_7711_nat__power__eq,axiom,
    ! [Z: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),Nb) ) ) ).

% nat_power_eq
tff(fact_7712_floor__eq3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = Nb ) ) ) ).

% floor_eq3
tff(fact_7713_le__nat__floor,axiom,
    ! [Xb: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))) ) ).

% le_nat_floor
tff(fact_7714_ex__inverse__of__nat__less,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N))),Xb) ) ) ).

% ex_inverse_of_nat_less
tff(fact_7715_power__diff__conv__inverse,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_complex(power_power_complex(Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),Mb)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_7716_power__diff__conv__inverse,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_real(power_power_real(Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) = aa_real_real(times_times_real(aa_nat_real(power_power_real(Xb),Nb)),aa_nat_real(power_power_real(aa_real_real(inverse_inverse_real,Xb)),Mb)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_7717_nat__2,axiom,
    aa_int_nat(nat2,numeral_numeral_int(bit0(one))) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% nat_2
tff(fact_7718_cosh__converges,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aTP_Lamm_sq(Xb)),aa_real_real(cosh_real,Xb)) ).

% cosh_converges
tff(fact_7719_cosh__converges,axiom,
    ! [Xb: complex] : aa_complex_o(sums_complex(aTP_Lamm_sr(Xb)),aa_complex_complex(cosh_complex,Xb)) ).

% cosh_converges
tff(fact_7720_pos__le__minus__divideR__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(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(C),A)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_7721_pos__minus__divideR__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(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(real_V1485227260804924795R_real(C),A)) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_7722_neg__le__minus__divideR__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(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(real_V1485227260804924795R_real(C),A)) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_7723_neg__minus__divideR__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(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(C),A)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_7724_pos__less__minus__divideR__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(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V1485227260804924795R_real(C),A)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_7725_pos__minus__divideR__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(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(real_V1485227260804924795R_real(C),A)) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_7726_neg__less__minus__divideR__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(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(real_V1485227260804924795R_real(C),A)) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_7727_neg__minus__divideR__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(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V1485227260804924795R_real(C),A)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_7728_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_7729_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_7730_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_7731_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_7732_floor__eq4,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = Nb ) ) ) ).

% floor_eq4
tff(fact_7733_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_7734_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_real(ring_1_of_int_real,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_7735_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_int(ring_1_of_int_int,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_7736_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_7737_cosh__field__def,axiom,
    ! [Z: real] : aa_real_real(cosh_real,Z) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(exp_real,Z)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Z)))),numeral_numeral_real(bit0(one))) ).

% cosh_field_def
tff(fact_7738_exp__series__add__commuting,axiom,
    ! [Xb: complex,Y: complex,Nb: nat] :
      ( ( aa_complex_complex(times_times_complex(Xb),Y) = aa_complex_complex(times_times_complex(Y),Xb) )
     => ( aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Xb),Y)),Nb)) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ss(Xb),Y),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ) ) ).

% exp_series_add_commuting
tff(fact_7739_exp__series__add__commuting,axiom,
    ! [Xb: real,Y: real,Nb: nat] :
      ( ( aa_real_real(times_times_real(Xb),Y) = aa_real_real(times_times_real(Y),Xb) )
     => ( aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Xb),Y)),Nb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_st(Xb),Y),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ) ) ).

% exp_series_add_commuting
tff(fact_7740_exp__first__term,axiom,
    ! [Xb: real] : aa_real_real(exp_real,Xb) = aa_real_real(plus_plus_real(one_one_real),suminf_real(aTP_Lamm_su(Xb))) ).

% exp_first_term
tff(fact_7741_exp__first__term,axiom,
    ! [Xb: complex] : aa_complex_complex(exp_complex,Xb) = aa_complex_complex(plus_plus_complex(one_one_complex),suminf_complex(aTP_Lamm_sv(Xb))) ).

% exp_first_term
tff(fact_7742_tan__sec,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(tan_complex,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one)))) = aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(cos_complex,Xb))),aa_num_nat(numeral_numeral_nat,bit0(one))) ) ) ).

% tan_sec
tff(fact_7743_tan__sec,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( aa_real_real(plus_plus_real(one_one_real),aa_nat_real(power_power_real(aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one)))) = aa_nat_real(power_power_real(aa_real_real(inverse_inverse_real,aa_real_real(cos_real,Xb))),aa_num_nat(numeral_numeral_nat,bit0(one))) ) ) ).

% tan_sec
tff(fact_7744_sinh__converges,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aTP_Lamm_sw(Xb)),aa_real_real(sinh_real,Xb)) ).

% sinh_converges
tff(fact_7745_sinh__converges,axiom,
    ! [Xb: complex] : aa_complex_o(sums_complex(aTP_Lamm_sx(Xb)),aa_complex_complex(sinh_complex,Xb)) ).

% sinh_converges
tff(fact_7746_cosh__double,axiom,
    ! [Xb: complex] : aa_complex_complex(cosh_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),Xb)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cosh_complex,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sinh_complex,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% cosh_double
tff(fact_7747_cosh__double,axiom,
    ! [Xb: real] : aa_real_real(cosh_real,aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),Xb)) = aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_real_real(cosh_real,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(power_power_real(aa_real_real(sinh_real,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% cosh_double
tff(fact_7748_tanh__add,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(cosh_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cosh_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(tanh_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tanh_complex,Xb)),aa_complex_complex(tanh_complex,Y))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tanh_complex,Xb)),aa_complex_complex(tanh_complex,Y)))) ) ) ) ).

% tanh_add
tff(fact_7749_tanh__add,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cosh_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cosh_real,Y) != zero_zero_real )
       => ( aa_real_real(tanh_real,aa_real_real(plus_plus_real(Xb),Y)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(tanh_real,Xb)),aa_real_real(tanh_real,Y))),aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tanh_real,Xb)),aa_real_real(tanh_real,Y)))) ) ) ) ).

% tanh_add
tff(fact_7750_sum__diff1_H__aux,axiom,
    ! [F3: set_complex,I5: set_complex,F: fun_complex_complex,Ia: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_sy(I5),F))),F3)
       => ( groups808145749697022017omplex(F,minus_811609699411566653omplex(I5,insert_complex(Ia,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(Ia),I5),aa_complex_complex(minus_minus_complex(groups808145749697022017omplex(F,I5)),aa_complex_complex(F,Ia)),groups808145749697022017omplex(F,I5)) ) ) ) ).

% sum_diff1'_aux
tff(fact_7751_sum__diff1_H__aux,axiom,
    ! [F3: set_Extended_enat,I5: set_Extended_enat,F: fun_Ex3741802288643809319omplex,Ia: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F3)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(collec4429806609662206161d_enat(aa_fun807264594178108256enat_o(aTP_Lamm_sz(I5),F))),F3)
       => ( groups4395127934049735793omplex(F,minus_925952699566721837d_enat(I5,insert_Extended_enat(Ia,bot_bo7653980558646680370d_enat))) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(Ia),I5),aa_complex_complex(minus_minus_complex(groups4395127934049735793omplex(F,I5)),aa_Ext8748583412541189208omplex(F,Ia)),groups4395127934049735793omplex(F,I5)) ) ) ) ).

% sum_diff1'_aux
tff(fact_7752_sum__diff1_H__aux,axiom,
    ! [F3: set_real,I5: set_real,F: fun_real_complex,Ia: real] :
      ( aa_set_real_o(finite_finite_real,F3)
     => ( aa_set_real_o(ord_less_eq_set_real(collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_ta(I5),F))),F3)
       => ( groups5683813829254066239omplex(F,minus_minus_set_real(I5,insert_real2(Ia,bot_bot_set_real))) = $ite(aa_set_real_o(member_real2(Ia),I5),aa_complex_complex(minus_minus_complex(groups5683813829254066239omplex(F,I5)),aa_real_complex(F,Ia)),groups5683813829254066239omplex(F,I5)) ) ) ) ).

% sum_diff1'_aux
tff(fact_7753_sum__diff1_H__aux,axiom,
    ! [F3: set_o,I5: set_o,F: fun_o_complex,Ia: $o] :
      ( aa_set_o_o(finite_finite_o,F3)
     => ( aa_set_o_o(ord_less_eq_set_o(collect_o(aa_fun8807775440140275936un_o_o(aTP_Lamm_tb(I5),F))),F3)
       => ( groups3443914341975893411omplex(F,minus_minus_set_o(I5,insert_o2((Ia),bot_bot_set_o))) = $ite(aa_set_o_o(member_o2((Ia)),I5),aa_complex_complex(minus_minus_complex(groups3443914341975893411omplex(F,I5)),aa_o_complex(F,(Ia))),groups3443914341975893411omplex(F,I5)) ) ) ) ).

% sum_diff1'_aux
tff(fact_7754_sum__diff1_H__aux,axiom,
    ! [F3: set_int,I5: set_int,F: fun_int_complex,Ia: int] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( aa_set_int_o(ord_less_eq_set_int(collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_tc(I5),F))),F3)
       => ( groups267424677133301183omplex(F,minus_minus_set_int(I5,insert_int2(Ia,bot_bot_set_int))) = $ite(aa_set_int_o(member_int2(Ia),I5),aa_complex_complex(minus_minus_complex(groups267424677133301183omplex(F,I5)),aa_int_complex(F,Ia)),groups267424677133301183omplex(F,I5)) ) ) ) ).

% sum_diff1'_aux
tff(fact_7755_sum__diff1_H__aux,axiom,
    ! [F3: set_complex,I5: set_complex,F: fun_complex_int,Ia: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aa_fun8831066929681621998plex_o(aTP_Lamm_td(I5),F))),F3)
       => ( groups2909182065852811199ex_int(F,minus_811609699411566653omplex(I5,insert_complex(Ia,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(Ia),I5),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups2909182065852811199ex_int(F,I5)),aa_complex_int(F,Ia)),groups2909182065852811199ex_int(F,I5)) ) ) ) ).

% sum_diff1'_aux
tff(fact_7756_sum__diff1_H__aux,axiom,
    ! [F3: set_Extended_enat,I5: set_Extended_enat,F: fun_Ex3482447378695272101at_int,Ia: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F3)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(collec4429806609662206161d_enat(aa_fun3959557771777301982enat_o(aTP_Lamm_te(I5),F))),F3)
       => ( groups4847789403509879791at_int(F,minus_925952699566721837d_enat(I5,insert_Extended_enat(Ia,bot_bo7653980558646680370d_enat))) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(Ia),I5),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4847789403509879791at_int(F,I5)),aa_Extended_enat_int(F,Ia)),groups4847789403509879791at_int(F,I5)) ) ) ) ).

% sum_diff1'_aux
tff(fact_7757_sum__diff1_H__aux,axiom,
    ! [F3: set_real,I5: set_real,F: fun_real_int,Ia: real] :
      ( aa_set_real_o(finite_finite_real,F3)
     => ( aa_set_real_o(ord_less_eq_set_real(collect_real(aa_fun429334026828149234real_o(aTP_Lamm_tf(I5),F))),F3)
       => ( groups3901808747961969597al_int(F,minus_minus_set_real(I5,insert_real2(Ia,bot_bot_set_real))) = $ite(aa_set_real_o(member_real2(Ia),I5),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3901808747961969597al_int(F,I5)),aa_real_int(F,Ia)),groups3901808747961969597al_int(F,I5)) ) ) ) ).

% sum_diff1'_aux
tff(fact_7758_sum__diff1_H__aux,axiom,
    ! [F3: set_o,I5: set_o,F: fun_o_int,Ia: $o] :
      ( aa_set_o_o(finite_finite_o,F3)
     => ( aa_set_o_o(ord_less_eq_set_o(collect_o(aa_fun_o_int_fun_o_o(aTP_Lamm_tg(I5),F))),F3)
       => ( groups4553916814277028129_o_int(F,minus_minus_set_o(I5,insert_o2((Ia),bot_bot_set_o))) = $ite(aa_set_o_o(member_o2((Ia)),I5),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4553916814277028129_o_int(F,I5)),aa_o_int(F,(Ia))),groups4553916814277028129_o_int(F,I5)) ) ) ) ).

% sum_diff1'_aux
tff(fact_7759_sum__diff1_H__aux,axiom,
    ! [F3: set_int,I5: set_int,F: fun_int_int,Ia: int] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( aa_set_int_o(ord_less_eq_set_int(collect_int(aa_fun8155382727227203058_int_o(aTP_Lamm_th(I5),F))),F3)
       => ( groups2983280209131991357nt_int(F,minus_minus_set_int(I5,insert_int2(Ia,bot_bot_set_int))) = $ite(aa_set_int_o(member_int2(Ia),I5),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups2983280209131991357nt_int(F,I5)),aa_int_int(F,Ia)),groups2983280209131991357nt_int(F,I5)) ) ) ) ).

% sum_diff1'_aux
tff(fact_7760_or__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = $ite(
        Mb = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one))))))) ) ).

% or_nat_unfold
tff(fact_7761_or_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se1409905431419307370or_int(A),zero_zero_int) = A ).

% or.right_neutral
tff(fact_7762_or_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),zero_zero_nat) = A ).

% or.right_neutral
tff(fact_7763_or_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se1409905431419307370or_int(zero_zero_int),A) = A ).

% or.left_neutral
tff(fact_7764_or_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,zero_zero_nat),A) = A ).

% or.left_neutral
tff(fact_7765_sinh__0,axiom,
    aa_real_real(sinh_real,zero_zero_real) = zero_zero_real ).

% sinh_0
tff(fact_7766_sinh__0,axiom,
    aa_complex_complex(sinh_complex,zero_zero_complex) = zero_zero_complex ).

% sinh_0
tff(fact_7767_tanh__0,axiom,
    aa_real_real(tanh_real,zero_zero_real) = zero_zero_real ).

% tanh_0
tff(fact_7768_tanh__0,axiom,
    aa_complex_complex(tanh_complex,zero_zero_complex) = zero_zero_complex ).

% tanh_0
tff(fact_7769_sum_Oempty_H,axiom,
    ! [P2: fun_real_nat] : groups3904299218471019873al_nat(P2,bot_bot_set_real) = zero_zero_nat ).

% sum.empty'
tff(fact_7770_sum_Oempty_H,axiom,
    ! [P2: fun_real_real] : groups97945582718554045l_real(P2,bot_bot_set_real) = zero_zero_real ).

% sum.empty'
tff(fact_7771_sum_Oempty_H,axiom,
    ! [P2: fun_real_int] : groups3901808747961969597al_int(P2,bot_bot_set_real) = zero_zero_int ).

% sum.empty'
tff(fact_7772_sum_Oempty_H,axiom,
    ! [P2: fun_real_complex] : groups5683813829254066239omplex(P2,bot_bot_set_real) = zero_zero_complex ).

% sum.empty'
tff(fact_7773_sum_Oempty_H,axiom,
    ! [P2: fun_re6065469302365115155d_enat] : groups8946708780592123413d_enat(P2,bot_bot_set_real) = zero_z5237406670263579293d_enat ).

% sum.empty'
tff(fact_7774_sum_Oempty_H,axiom,
    ! [P2: fun_o_nat] : groups4556407284786078405_o_nat(P2,bot_bot_set_o) = zero_zero_nat ).

% sum.empty'
tff(fact_7775_sum_Oempty_H,axiom,
    ! [P2: fun_o_real] : groups627172608727702305o_real(P2,bot_bot_set_o) = zero_zero_real ).

% sum.empty'
tff(fact_7776_sum_Oempty_H,axiom,
    ! [P2: fun_o_int] : groups4553916814277028129_o_int(P2,bot_bot_set_o) = zero_zero_int ).

% sum.empty'
tff(fact_7777_sum_Oempty_H,axiom,
    ! [P2: fun_o_complex] : groups3443914341975893411omplex(P2,bot_bot_set_o) = zero_zero_complex ).

% sum.empty'
tff(fact_7778_sum_Oempty_H,axiom,
    ! [P2: fun_o_Extended_enat] : groups8422889724626310577d_enat(P2,bot_bot_set_o) = zero_z5237406670263579293d_enat ).

% sum.empty'
tff(fact_7779_or__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% or_nat_numerals(4)
tff(fact_7780_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_7781_sum_Oinsert_H,axiom,
    ! [I5: set_o,P2: fun_o_nat,Ia: $o] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_ce(I5),P2)))
     => ( groups4556407284786078405_o_nat(P2,insert_o2((Ia),I5)) = $ite(aa_set_o_o(member_o2((Ia)),I5),groups4556407284786078405_o_nat(P2,I5),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(P2,(Ia))),groups4556407284786078405_o_nat(P2,I5))) ) ) ).

% sum.insert'
tff(fact_7782_sum_Oinsert_H,axiom,
    ! [I5: set_real,P2: fun_real_nat,Ia: real] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun7775777430665565006real_o(aTP_Lamm_cg(I5),P2)))
     => ( groups3904299218471019873al_nat(P2,insert_real2(Ia,I5)) = $ite(aa_set_real_o(member_real2(Ia),I5),groups3904299218471019873al_nat(P2,I5),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(P2,Ia)),groups3904299218471019873al_nat(P2,I5))) ) ) ).

% sum.insert'
tff(fact_7783_sum_Oinsert_H,axiom,
    ! [I5: set_nat,P2: fun_nat_nat,Ia: nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_ci(I5),P2)))
     => ( aa_set_nat_nat(groups1986416967739987077at_nat(P2),insert_nat2(Ia,I5)) = $ite(aa_set_nat_o(member_nat2(Ia),I5),aa_set_nat_nat(groups1986416967739987077at_nat(P2),I5),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(P2,Ia)),aa_set_nat_nat(groups1986416967739987077at_nat(P2),I5))) ) ) ).

% sum.insert'
tff(fact_7784_sum_Oinsert_H,axiom,
    ! [I5: set_complex,P2: fun_complex_nat,Ia: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun8723867720292580170plex_o(aTP_Lamm_ck(I5),P2)))
     => ( groups2911672536361861475ex_nat(P2,insert_complex(Ia,I5)) = $ite(aa_set_complex_o(member_complex(Ia),I5),groups2911672536361861475ex_nat(P2,I5),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(P2,Ia)),groups2911672536361861475ex_nat(P2,I5))) ) ) ).

% sum.insert'
tff(fact_7785_sum_Oinsert_H,axiom,
    ! [I5: set_int,P2: fun_int_nat,Ia: int] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_cm(I5),P2)))
     => ( groups2985770679641041633nt_nat(P2,insert_int2(Ia,I5)) = $ite(aa_set_int_o(member_int2(Ia),I5),groups2985770679641041633nt_nat(P2,I5),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(P2,Ia)),groups2985770679641041633nt_nat(P2,I5))) ) ) ).

% sum.insert'
tff(fact_7786_sum_Oinsert_H,axiom,
    ! [I5: set_Extended_enat,P2: fun_Ex7660298398204468809at_nat,Ia: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun3064223910257113474enat_o(aTP_Lamm_co(I5),P2)))
     => ( groups4850279874018930067at_nat(P2,insert_Extended_enat(Ia,I5)) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(Ia),I5),groups4850279874018930067at_nat(P2,I5),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Extended_enat_nat(P2,Ia)),groups4850279874018930067at_nat(P2,I5))) ) ) ).

% sum.insert'
tff(fact_7787_sum_Oinsert_H,axiom,
    ! [I5: set_o,P2: fun_o_real,Ia: $o] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun1386971751305049950un_o_o(aTP_Lamm_cq(I5),P2)))
     => ( groups627172608727702305o_real(P2,insert_o2((Ia),I5)) = $ite(aa_set_o_o(member_o2((Ia)),I5),groups627172608727702305o_real(P2,I5),aa_real_real(plus_plus_real(aa_o_real(P2,(Ia))),groups627172608727702305o_real(P2,I5))) ) ) ).

% sum.insert'
tff(fact_7788_sum_Oinsert_H,axiom,
    ! [I5: set_real,P2: fun_real_real,Ia: real] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_cs(I5),P2)))
     => ( groups97945582718554045l_real(P2,insert_real2(Ia,I5)) = $ite(aa_set_real_o(member_real2(Ia),I5),groups97945582718554045l_real(P2,I5),aa_real_real(plus_plus_real(aa_real_real(P2,Ia)),groups97945582718554045l_real(P2,I5))) ) ) ).

% sum.insert'
tff(fact_7789_sum_Oinsert_H,axiom,
    ! [I5: set_nat,P2: fun_nat_real,Ia: nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_cu(I5),P2)))
     => ( aa_set_nat_real(groups8560362682196896993t_real(P2),insert_nat2(Ia,I5)) = $ite(aa_set_nat_o(member_nat2(Ia),I5),aa_set_nat_real(groups8560362682196896993t_real(P2),I5),aa_real_real(plus_plus_real(aa_nat_real(P2,Ia)),aa_set_nat_real(groups8560362682196896993t_real(P2),I5))) ) ) ).

% sum.insert'
tff(fact_7790_sum_Oinsert_H,axiom,
    ! [I5: set_complex,P2: fun_complex_real,Ia: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_cw(I5),P2)))
     => ( groups5737402329758386879x_real(P2,insert_complex(Ia,I5)) = $ite(aa_set_complex_o(member_complex(Ia),I5),groups5737402329758386879x_real(P2,I5),aa_real_real(plus_plus_real(aa_complex_real(P2,Ia)),groups5737402329758386879x_real(P2,I5))) ) ) ).

% sum.insert'
tff(fact_7791_or__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,bit0(Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% or_nat_numerals(3)
tff(fact_7792_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,bit0(Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_7793_or__numerals_I4_J,axiom,
    ! [Xb: num,Y: num] : aa_int_int(bit_se1409905431419307370or_int(numeral_numeral_int(bit0(Xb))),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,numeral_numeral_int(bit0(one))),aa_int_int(bit_se1409905431419307370or_int(numeral_numeral_int(Xb)),numeral_numeral_int(Y)))) ).

% or_numerals(4)
tff(fact_7794_or__numerals_I4_J,axiom,
    ! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,bit0(Xb))),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,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% or_numerals(4)
tff(fact_7795_or__numerals_I6_J,axiom,
    ! [Xb: num,Y: num] : aa_int_int(bit_se1409905431419307370or_int(numeral_numeral_int(aa_num_num(bit1,Xb))),numeral_numeral_int(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,numeral_numeral_int(bit0(one))),aa_int_int(bit_se1409905431419307370or_int(numeral_numeral_int(Xb)),numeral_numeral_int(Y)))) ).

% or_numerals(6)
tff(fact_7796_or__numerals_I6_J,axiom,
    ! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_num_nat(numeral_numeral_nat,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,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% or_numerals(6)
tff(fact_7797_or__numerals_I7_J,axiom,
    ! [Xb: num,Y: num] : aa_int_int(bit_se1409905431419307370or_int(numeral_numeral_int(aa_num_num(bit1,Xb))),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,numeral_numeral_int(bit0(one))),aa_int_int(bit_se1409905431419307370or_int(numeral_numeral_int(Xb)),numeral_numeral_int(Y)))) ).

% or_numerals(7)
tff(fact_7798_or__numerals_I7_J,axiom,
    ! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),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,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% or_numerals(7)
tff(fact_7799_bit_Odisj__zero__right,axiom,
    ! [Xb: int] : aa_int_int(bit_se1409905431419307370or_int(Xb),zero_zero_int) = Xb ).

% bit.disj_zero_right
tff(fact_7800_or__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(bit_se1409905431419307370or_int(A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & ( B = zero_zero_int ) ) ) ).

% or_eq_0_iff
tff(fact_7801_or__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% or_eq_0_iff
tff(fact_7802_sum_Onon__neutral_H,axiom,
    ! [G: fun_o_nat,I5: set_o] : groups4556407284786078405_o_nat(G,collect_o(aa_set_o_fun_o_o(aTP_Lamm_ti(G),I5))) = groups4556407284786078405_o_nat(G,I5) ).

% sum.non_neutral'
tff(fact_7803_sum_Onon__neutral_H,axiom,
    ! [G: fun_real_nat,I5: set_real] : groups3904299218471019873al_nat(G,collect_real(aa_set2323979645938365349real_o(aTP_Lamm_tj(G),I5))) = groups3904299218471019873al_nat(G,I5) ).

% sum.non_neutral'
tff(fact_7804_sum_Onon__neutral_H,axiom,
    ! [G: fun_nat_nat,I5: set_nat] : aa_set_nat_nat(groups1986416967739987077at_nat(G),collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_tk(G),I5))) = aa_set_nat_nat(groups1986416967739987077at_nat(G),I5) ).

% sum.non_neutral'
tff(fact_7805_sum_Onon__neutral_H,axiom,
    ! [G: fun_int_nat,I5: set_int] : groups2985770679641041633nt_nat(G,collect_int(aa_set_int_fun_int_o(aTP_Lamm_tl(G),I5))) = groups2985770679641041633nt_nat(G,I5) ).

% sum.non_neutral'
tff(fact_7806_sum_Onon__neutral_H,axiom,
    ! [G: fun_o_real,I5: set_o] : groups627172608727702305o_real(G,collect_o(aa_set_o_fun_o_o(aTP_Lamm_tm(G),I5))) = groups627172608727702305o_real(G,I5) ).

% sum.non_neutral'
tff(fact_7807_sum_Onon__neutral_H,axiom,
    ! [G: fun_real_real,I5: set_real] : groups97945582718554045l_real(G,collect_real(aa_set2323979645938365349real_o(aTP_Lamm_tn(G),I5))) = groups97945582718554045l_real(G,I5) ).

% sum.non_neutral'
tff(fact_7808_sum_Onon__neutral_H,axiom,
    ! [G: fun_nat_real,I5: set_nat] : aa_set_nat_real(groups8560362682196896993t_real(G),collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_to(G),I5))) = aa_set_nat_real(groups8560362682196896993t_real(G),I5) ).

% sum.non_neutral'
tff(fact_7809_sum_Onon__neutral_H,axiom,
    ! [G: fun_int_real,I5: set_int] : groups1523912220035142973t_real(G,collect_int(aa_set_int_fun_int_o(aTP_Lamm_tp(G),I5))) = groups1523912220035142973t_real(G,I5) ).

% sum.non_neutral'
tff(fact_7810_sum_Onon__neutral_H,axiom,
    ! [G: fun_o_int,I5: set_o] : groups4553916814277028129_o_int(G,collect_o(aa_set_o_fun_o_o(aTP_Lamm_tq(G),I5))) = groups4553916814277028129_o_int(G,I5) ).

% sum.non_neutral'
tff(fact_7811_sum_Onon__neutral_H,axiom,
    ! [G: fun_real_int,I5: set_real] : groups3901808747961969597al_int(G,collect_real(aa_set2323979645938365349real_o(aTP_Lamm_tr(G),I5))) = groups3901808747961969597al_int(G,I5) ).

% sum.non_neutral'
tff(fact_7812_disjunctive__add,axiom,
    ! [A: int,B: int] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(bit_se1146084159140164899it_int(A),N)
          | ~ aa_nat_o(bit_se1146084159140164899it_int(B),N) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(bit_se1409905431419307370or_int(A),B) ) ) ).

% disjunctive_add
tff(fact_7813_disjunctive__add,axiom,
    ! [A: nat,B: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N)
          | ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),N) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B) ) ) ).

% disjunctive_add
tff(fact_7814_sum_Odistrib__triv_H,axiom,
    ! [I5: set_nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_nat(groups1986416967739987077at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ey(G),H)),I5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups1986416967739987077at_nat(G),I5)),aa_set_nat_nat(groups1986416967739987077at_nat(H),I5)) ) ) ).

% sum.distrib_triv'
tff(fact_7815_sum_Odistrib__triv_H,axiom,
    ! [I5: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( groups2911672536361861475ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_ts(G),H),I5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups2911672536361861475ex_nat(G,I5)),groups2911672536361861475ex_nat(H,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_7816_sum_Odistrib__triv_H,axiom,
    ! [I5: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( groups2985770679641041633nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_tt(G),H),I5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups2985770679641041633nt_nat(G,I5)),groups2985770679641041633nt_nat(H,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_7817_sum_Odistrib__triv_H,axiom,
    ! [I5: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,H: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,I5)
     => ( groups4850279874018930067at_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_tu(G),H),I5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4850279874018930067at_nat(G,I5)),groups4850279874018930067at_nat(H,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_7818_sum_Odistrib__triv_H,axiom,
    ! [I5: set_nat,G: fun_nat_int,H: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_int(groups1983926497230936801at_int(aa_fun5565043122120104496at_int(aTP_Lamm_ln(G),H)),I5) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups1983926497230936801at_int(G),I5)),aa_set_nat_int(groups1983926497230936801at_int(H),I5)) ) ) ).

% sum.distrib_triv'
tff(fact_7819_sum_Odistrib__triv_H,axiom,
    ! [I5: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( groups2909182065852811199ex_int(aa_fun4286351638413746156ex_int(aTP_Lamm_tv(G),H),I5) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups2909182065852811199ex_int(G,I5)),groups2909182065852811199ex_int(H,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_7820_sum_Odistrib__triv_H,axiom,
    ! [I5: set_int,G: fun_int_int,H: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( groups2983280209131991357nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_tw(G),H),I5) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups2983280209131991357nt_int(G,I5)),groups2983280209131991357nt_int(H,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_7821_sum_Odistrib__triv_H,axiom,
    ! [I5: set_Extended_enat,G: fun_Ex3482447378695272101at_int,H: fun_Ex3482447378695272101at_int] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,I5)
     => ( groups4847789403509879791at_int(aa_fun1695976788104708604at_int(aTP_Lamm_tx(G),H),I5) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4847789403509879791at_int(G,I5)),groups4847789403509879791at_int(H,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_7822_sum_Odistrib__triv_H,axiom,
    ! [I5: set_nat,G: fun_nat_real,H: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_real(groups8560362682196896993t_real(aa_fun1839206346889086768t_real(aTP_Lamm_fa(G),H)),I5) = aa_real_real(plus_plus_real(aa_set_nat_real(groups8560362682196896993t_real(G),I5)),aa_set_nat_real(groups8560362682196896993t_real(H),I5)) ) ) ).

% sum.distrib_triv'
tff(fact_7823_sum_Odistrib__triv_H,axiom,
    ! [I5: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( groups5737402329758386879x_real(aa_fun8933909628229591020x_real(aTP_Lamm_ty(G),H),I5) = aa_real_real(plus_plus_real(groups5737402329758386879x_real(G,I5)),groups5737402329758386879x_real(H,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_7824_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
     => ( ! [X4: real] :
            ( aa_set_real_o(member_real2(X4),minus_minus_set_real(T2,S3))
           => ( aa_real_nat(G,X4) = zero_zero_nat ) )
       => ( ! [X4: real] :
              ( aa_set_real_o(member_real2(X4),S3)
             => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
         => ( groups3904299218471019873al_nat(G,T2) = groups3904299218471019873al_nat(H,S3) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_7825_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_o,T2: set_o,G: fun_o_nat,H: fun_o_nat] :
      ( aa_set_o_o(ord_less_eq_set_o(S3),T2)
     => ( ! [X4: $o] :
            ( aa_set_o_o(member_o2((X4)),minus_minus_set_o(T2,S3))
           => ( aa_o_nat(G,(X4)) = zero_zero_nat ) )
       => ( ! [X4: $o] :
              ( aa_set_o_o(member_o2((X4)),S3)
             => ( aa_o_nat(G,(X4)) = aa_o_nat(H,(X4)) ) )
         => ( groups4556407284786078405_o_nat(G,T2) = groups4556407284786078405_o_nat(H,S3) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_7826_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_int,T2: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
     => ( ! [X4: int] :
            ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
           => ( aa_int_nat(G,X4) = zero_zero_nat ) )
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),S3)
             => ( aa_int_nat(G,X4) = aa_int_nat(H,X4) ) )
         => ( groups2985770679641041633nt_nat(G,T2) = groups2985770679641041633nt_nat(H,S3) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_7827_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T2: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
     => ( ! [X4: real] :
            ( aa_set_real_o(member_real2(X4),minus_minus_set_real(T2,S3))
           => ( aa_real_real(G,X4) = zero_zero_real ) )
       => ( ! [X4: real] :
              ( aa_set_real_o(member_real2(X4),S3)
             => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
         => ( groups97945582718554045l_real(G,T2) = groups97945582718554045l_real(H,S3) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_7828_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_o,T2: set_o,G: fun_o_real,H: fun_o_real] :
      ( aa_set_o_o(ord_less_eq_set_o(S3),T2)
     => ( ! [X4: $o] :
            ( aa_set_o_o(member_o2((X4)),minus_minus_set_o(T2,S3))
           => ( aa_o_real(G,(X4)) = zero_zero_real ) )
       => ( ! [X4: $o] :
              ( aa_set_o_o(member_o2((X4)),S3)
             => ( aa_o_real(G,(X4)) = aa_o_real(H,(X4)) ) )
         => ( groups627172608727702305o_real(G,T2) = groups627172608727702305o_real(H,S3) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_7829_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_int,T2: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
     => ( ! [X4: int] :
            ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
           => ( aa_int_real(G,X4) = zero_zero_real ) )
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),S3)
             => ( aa_int_real(G,X4) = aa_int_real(H,X4) ) )
         => ( groups1523912220035142973t_real(G,T2) = groups1523912220035142973t_real(H,S3) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_7830_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T2: set_real,G: fun_real_int,H: fun_real_int] :
      ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
     => ( ! [X4: real] :
            ( aa_set_real_o(member_real2(X4),minus_minus_set_real(T2,S3))
           => ( aa_real_int(G,X4) = zero_zero_int ) )
       => ( ! [X4: real] :
              ( aa_set_real_o(member_real2(X4),S3)
             => ( aa_real_int(G,X4) = aa_real_int(H,X4) ) )
         => ( groups3901808747961969597al_int(G,T2) = groups3901808747961969597al_int(H,S3) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_7831_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_o,T2: set_o,G: fun_o_int,H: fun_o_int] :
      ( aa_set_o_o(ord_less_eq_set_o(S3),T2)
     => ( ! [X4: $o] :
            ( aa_set_o_o(member_o2((X4)),minus_minus_set_o(T2,S3))
           => ( aa_o_int(G,(X4)) = zero_zero_int ) )
       => ( ! [X4: $o] :
              ( aa_set_o_o(member_o2((X4)),S3)
             => ( aa_o_int(G,(X4)) = aa_o_int(H,(X4)) ) )
         => ( groups4553916814277028129_o_int(G,T2) = groups4553916814277028129_o_int(H,S3) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_7832_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_int,T2: set_int,G: fun_int_int,H: fun_int_int] :
      ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
     => ( ! [X4: int] :
            ( aa_set_int_o(member_int2(X4),minus_minus_set_int(T2,S3))
           => ( aa_int_int(G,X4) = zero_zero_int ) )
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),S3)
             => ( aa_int_int(G,X4) = aa_int_int(H,X4) ) )
         => ( groups2983280209131991357nt_int(G,T2) = groups2983280209131991357nt_int(H,S3) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_7833_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
     => ( ! [X4: real] :
            ( aa_set_real_o(member_real2(X4),minus_minus_set_real(T2,S3))
           => ( aa_real_complex(G,X4) = zero_zero_complex ) )
       => ( ! [X4: real] :
              ( aa_set_real_o(member_real2(X4),S3)
             => ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
         => ( groups5683813829254066239omplex(G,T2) = groups5683813829254066239omplex(H,S3) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_7834_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T2: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real2(I2),minus_minus_set_real(T2,S3))
           => ( aa_real_nat(H,I2) = zero_zero_nat ) )
       => ( ! [X4: real] :
              ( aa_set_real_o(member_real2(X4),S3)
             => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
         => ( groups3904299218471019873al_nat(G,S3) = groups3904299218471019873al_nat(H,T2) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_7835_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_o,T2: set_o,H: fun_o_nat,G: fun_o_nat] :
      ( aa_set_o_o(ord_less_eq_set_o(S3),T2)
     => ( ! [I2: $o] :
            ( aa_set_o_o(member_o2((I2)),minus_minus_set_o(T2,S3))
           => ( aa_o_nat(H,(I2)) = zero_zero_nat ) )
       => ( ! [X4: $o] :
              ( aa_set_o_o(member_o2((X4)),S3)
             => ( aa_o_nat(G,(X4)) = aa_o_nat(H,(X4)) ) )
         => ( groups4556407284786078405_o_nat(G,S3) = groups4556407284786078405_o_nat(H,T2) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_7836_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_int,T2: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int2(I2),minus_minus_set_int(T2,S3))
           => ( aa_int_nat(H,I2) = zero_zero_nat ) )
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),S3)
             => ( aa_int_nat(G,X4) = aa_int_nat(H,X4) ) )
         => ( groups2985770679641041633nt_nat(G,S3) = groups2985770679641041633nt_nat(H,T2) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_7837_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T2: set_real,H: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real2(I2),minus_minus_set_real(T2,S3))
           => ( aa_real_real(H,I2) = zero_zero_real ) )
       => ( ! [X4: real] :
              ( aa_set_real_o(member_real2(X4),S3)
             => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
         => ( groups97945582718554045l_real(G,S3) = groups97945582718554045l_real(H,T2) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_7838_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_o,T2: set_o,H: fun_o_real,G: fun_o_real] :
      ( aa_set_o_o(ord_less_eq_set_o(S3),T2)
     => ( ! [I2: $o] :
            ( aa_set_o_o(member_o2((I2)),minus_minus_set_o(T2,S3))
           => ( aa_o_real(H,(I2)) = zero_zero_real ) )
       => ( ! [X4: $o] :
              ( aa_set_o_o(member_o2((X4)),S3)
             => ( aa_o_real(G,(X4)) = aa_o_real(H,(X4)) ) )
         => ( groups627172608727702305o_real(G,S3) = groups627172608727702305o_real(H,T2) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_7839_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_int,T2: set_int,H: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int2(I2),minus_minus_set_int(T2,S3))
           => ( aa_int_real(H,I2) = zero_zero_real ) )
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),S3)
             => ( aa_int_real(G,X4) = aa_int_real(H,X4) ) )
         => ( groups1523912220035142973t_real(G,S3) = groups1523912220035142973t_real(H,T2) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_7840_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T2: set_real,H: fun_real_int,G: fun_real_int] :
      ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real2(I2),minus_minus_set_real(T2,S3))
           => ( aa_real_int(H,I2) = zero_zero_int ) )
       => ( ! [X4: real] :
              ( aa_set_real_o(member_real2(X4),S3)
             => ( aa_real_int(G,X4) = aa_real_int(H,X4) ) )
         => ( groups3901808747961969597al_int(G,S3) = groups3901808747961969597al_int(H,T2) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_7841_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_o,T2: set_o,H: fun_o_int,G: fun_o_int] :
      ( aa_set_o_o(ord_less_eq_set_o(S3),T2)
     => ( ! [I2: $o] :
            ( aa_set_o_o(member_o2((I2)),minus_minus_set_o(T2,S3))
           => ( aa_o_int(H,(I2)) = zero_zero_int ) )
       => ( ! [X4: $o] :
              ( aa_set_o_o(member_o2((X4)),S3)
             => ( aa_o_int(G,(X4)) = aa_o_int(H,(X4)) ) )
         => ( groups4553916814277028129_o_int(G,S3) = groups4553916814277028129_o_int(H,T2) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_7842_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_int,T2: set_int,H: fun_int_int,G: fun_int_int] :
      ( aa_set_int_o(ord_less_eq_set_int(S3),T2)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int2(I2),minus_minus_set_int(T2,S3))
           => ( aa_int_int(H,I2) = zero_zero_int ) )
       => ( ! [X4: int] :
              ( aa_set_int_o(member_int2(X4),S3)
             => ( aa_int_int(G,X4) = aa_int_int(H,X4) ) )
         => ( groups2983280209131991357nt_int(G,S3) = groups2983280209131991357nt_int(H,T2) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_7843_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T2: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( aa_set_real_o(ord_less_eq_set_real(S3),T2)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real2(I2),minus_minus_set_real(T2,S3))
           => ( aa_real_complex(H,I2) = zero_zero_complex ) )
       => ( ! [X4: real] :
              ( aa_set_real_o(member_real2(X4),S3)
             => ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
         => ( groups5683813829254066239omplex(G,S3) = groups5683813829254066239omplex(H,T2) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_7844_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),T2)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),minus_minus_set_nat(T2,S3))
           => ( aa_nat_nat(G,X4) = zero_zero_nat ) )
       => ( aa_set_nat_nat(groups1986416967739987077at_nat(G),T2) = aa_set_nat_nat(groups1986416967739987077at_nat(G),S3) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_7845_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T2: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),T2)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),minus_minus_set_nat(T2,S3))
           => ( aa_nat_real(G,X4) = zero_zero_real ) )
       => ( aa_set_nat_real(groups8560362682196896993t_real(G),T2) = aa_set_nat_real(groups8560362682196896993t_real(G),S3) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_7846_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),T2)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),minus_minus_set_nat(T2,S3))
           => ( aa_nat_int(G,X4) = zero_zero_int ) )
       => ( aa_set_nat_int(groups1983926497230936801at_int(G),T2) = aa_set_nat_int(groups1983926497230936801at_int(G),S3) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_7847_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T2: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),T2)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),minus_minus_set_nat(T2,S3))
           => ( aa_nat_complex(G,X4) = zero_zero_complex ) )
       => ( groups8515261248781899619omplex(G,T2) = groups8515261248781899619omplex(G,S3) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_7848_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T2: set_nat,G: fun_na8797796813480665199d_enat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),T2)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),minus_minus_set_nat(T2,S3))
           => ( aa_nat_Extended_enat(G,X4) = zero_z5237406670263579293d_enat ) )
       => ( groups707763781290628081d_enat(G,T2) = groups707763781290628081d_enat(G,S3) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_7849_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T2: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),T2)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),minus_minus_set_nat(T2,S3))
           => ( aa_nat_nat(G,X4) = zero_zero_nat ) )
       => ( aa_set_nat_nat(groups1986416967739987077at_nat(G),S3) = aa_set_nat_nat(groups1986416967739987077at_nat(G),T2) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_7850_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T2: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),T2)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),minus_minus_set_nat(T2,S3))
           => ( aa_nat_real(G,X4) = zero_zero_real ) )
       => ( aa_set_nat_real(groups8560362682196896993t_real(G),S3) = aa_set_nat_real(groups8560362682196896993t_real(G),T2) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_7851_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),T2)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),minus_minus_set_nat(T2,S3))
           => ( aa_nat_int(G,X4) = zero_zero_int ) )
       => ( aa_set_nat_int(groups1983926497230936801at_int(G),S3) = aa_set_nat_int(groups1983926497230936801at_int(G),T2) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_7852_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T2: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),T2)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),minus_minus_set_nat(T2,S3))
           => ( aa_nat_complex(G,X4) = zero_zero_complex ) )
       => ( groups8515261248781899619omplex(G,S3) = groups8515261248781899619omplex(G,T2) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_7853_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T2: set_nat,G: fun_na8797796813480665199d_enat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S3),T2)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),minus_minus_set_nat(T2,S3))
           => ( aa_nat_Extended_enat(G,X4) = zero_z5237406670263579293d_enat ) )
       => ( groups707763781290628081d_enat(G,S3) = groups707763781290628081d_enat(G,T2) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_7854_sinh__add,axiom,
    ! [Xb: real,Y: real] : aa_real_real(sinh_real,aa_real_real(plus_plus_real(Xb),Y)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Y))),aa_real_real(times_times_real(aa_real_real(cosh_real,Xb)),aa_real_real(sinh_real,Y))) ).

% sinh_add
tff(fact_7855_sinh__add,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(sinh_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(sinh_complex,Xb)),aa_complex_complex(cosh_complex,Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cosh_complex,Xb)),aa_complex_complex(sinh_complex,Y))) ).

% sinh_add
tff(fact_7856_cosh__add,axiom,
    ! [Xb: real,Y: real] : aa_real_real(cosh_real,aa_real_real(plus_plus_real(Xb),Y)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))),aa_real_real(times_times_real(aa_real_real(sinh_real,Xb)),aa_real_real(sinh_real,Y))) ).

% cosh_add
tff(fact_7857_cosh__add,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(cosh_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cosh_complex,Xb)),aa_complex_complex(cosh_complex,Y))),aa_complex_complex(times_times_complex(aa_complex_complex(sinh_complex,Xb)),aa_complex_complex(sinh_complex,Y))) ).

% cosh_add
tff(fact_7858_bit_Ocomplement__unique,axiom,
    ! [A: int,Xb: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(A),Xb) = zero_zero_int )
     => ( ( aa_int_int(bit_se1409905431419307370or_int(A),Xb) = aa_int_int(uminus_uminus_int,one_one_int) )
       => ( ( aa_int_int(bit_se725231765392027082nd_int(A),Y) = zero_zero_int )
         => ( ( aa_int_int(bit_se1409905431419307370or_int(A),Y) = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( Xb = Y ) ) ) ) ) ).

% bit.complement_unique
tff(fact_7859_sinh__plus__cosh,axiom,
    ! [Xb: real] : aa_real_real(plus_plus_real(aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Xb)) = aa_real_real(exp_real,Xb) ).

% sinh_plus_cosh
tff(fact_7860_cosh__plus__sinh,axiom,
    ! [Xb: real] : aa_real_real(plus_plus_real(aa_real_real(cosh_real,Xb)),aa_real_real(sinh_real,Xb)) = aa_real_real(exp_real,Xb) ).

% cosh_plus_sinh
tff(fact_7861_sum_Odistrib_H,axiom,
    ! [I5: set_o,G: fun_o_nat,H: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_ce(I5),G)))
     => ( aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_ce(I5),H)))
       => ( groups4556407284786078405_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_tz(G),H),I5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4556407284786078405_o_nat(G,I5)),groups4556407284786078405_o_nat(H,I5)) ) ) ) ).

% sum.distrib'
tff(fact_7862_sum_Odistrib_H,axiom,
    ! [I5: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun7775777430665565006real_o(aTP_Lamm_cg(I5),G)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun7775777430665565006real_o(aTP_Lamm_cg(I5),H)))
       => ( groups3904299218471019873al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ua(G),H),I5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups3904299218471019873al_nat(G,I5)),groups3904299218471019873al_nat(H,I5)) ) ) ) ).

% sum.distrib'
tff(fact_7863_sum_Odistrib_H,axiom,
    ! [I5: set_nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_ci(I5),G)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_ci(I5),H)))
       => ( aa_set_nat_nat(groups1986416967739987077at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ey(G),H)),I5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups1986416967739987077at_nat(G),I5)),aa_set_nat_nat(groups1986416967739987077at_nat(H),I5)) ) ) ) ).

% sum.distrib'
tff(fact_7864_sum_Odistrib_H,axiom,
    ! [I5: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun8723867720292580170plex_o(aTP_Lamm_ck(I5),G)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun8723867720292580170plex_o(aTP_Lamm_ck(I5),H)))
       => ( groups2911672536361861475ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_ts(G),H),I5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups2911672536361861475ex_nat(G,I5)),groups2911672536361861475ex_nat(H,I5)) ) ) ) ).

% sum.distrib'
tff(fact_7865_sum_Odistrib_H,axiom,
    ! [I5: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_cm(I5),G)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_cm(I5),H)))
       => ( groups2985770679641041633nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_tt(G),H),I5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups2985770679641041633nt_nat(G,I5)),groups2985770679641041633nt_nat(H,I5)) ) ) ) ).

% sum.distrib'
tff(fact_7866_sum_Odistrib_H,axiom,
    ! [I5: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,H: fun_Ex7660298398204468809at_nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun3064223910257113474enat_o(aTP_Lamm_co(I5),G)))
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun3064223910257113474enat_o(aTP_Lamm_co(I5),H)))
       => ( groups4850279874018930067at_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_tu(G),H),I5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4850279874018930067at_nat(G,I5)),groups4850279874018930067at_nat(H,I5)) ) ) ) ).

% sum.distrib'
tff(fact_7867_sum_Odistrib_H,axiom,
    ! [I5: set_o,G: fun_o_real,H: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun1386971751305049950un_o_o(aTP_Lamm_cq(I5),G)))
     => ( aa_set_o_o(finite_finite_o,collect_o(aa_fun1386971751305049950un_o_o(aTP_Lamm_cq(I5),H)))
       => ( groups627172608727702305o_real(aa_fun688608109223240316o_real(aTP_Lamm_ub(G),H),I5) = aa_real_real(plus_plus_real(groups627172608727702305o_real(G,I5)),groups627172608727702305o_real(H,I5)) ) ) ) ).

% sum.distrib'
tff(fact_7868_sum_Odistrib_H,axiom,
    ! [I5: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_cs(I5),G)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_cs(I5),H)))
       => ( groups97945582718554045l_real(aa_fun6946558556880649704l_real(aTP_Lamm_uc(G),H),I5) = aa_real_real(plus_plus_real(groups97945582718554045l_real(G,I5)),groups97945582718554045l_real(H,I5)) ) ) ) ).

% sum.distrib'
tff(fact_7869_sum_Odistrib_H,axiom,
    ! [I5: set_nat,G: fun_nat_real,H: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_cu(I5),G)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_cu(I5),H)))
       => ( aa_set_nat_real(groups8560362682196896993t_real(aa_fun1839206346889086768t_real(aTP_Lamm_fa(G),H)),I5) = aa_real_real(plus_plus_real(aa_set_nat_real(groups8560362682196896993t_real(G),I5)),aa_set_nat_real(groups8560362682196896993t_real(H),I5)) ) ) ) ).

% sum.distrib'
tff(fact_7870_sum_Odistrib_H,axiom,
    ! [I5: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_cw(I5),G)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_cw(I5),H)))
       => ( groups5737402329758386879x_real(aa_fun8933909628229591020x_real(aTP_Lamm_ty(G),H),I5) = aa_real_real(plus_plus_real(groups5737402329758386879x_real(G,I5)),groups5737402329758386879x_real(H,I5)) ) ) ) ).

% sum.distrib'
tff(fact_7871_sum_OG__def,axiom,
    ! [P2: fun_o_nat,I5: set_o] :
      groups4556407284786078405_o_nat(P2,I5) = $ite(aa_set_o_o(finite_finite_o,collect_o(aa_set_o_fun_o_o(aTP_Lamm_ti(P2),I5))),groups8507830703676809646_o_nat(P2,collect_o(aa_set_o_fun_o_o(aTP_Lamm_ti(P2),I5))),zero_zero_nat) ).

% sum.G_def
tff(fact_7872_sum_OG__def,axiom,
    ! [P2: fun_real_nat,I5: set_real] :
      groups3904299218471019873al_nat(P2,I5) = $ite(aa_set_real_o(finite_finite_real,collect_real(aa_set2323979645938365349real_o(aTP_Lamm_tj(P2),I5))),groups1935376822645274424al_nat(P2,collect_real(aa_set2323979645938365349real_o(aTP_Lamm_tj(P2),I5))),zero_zero_nat) ).

% sum.G_def
tff(fact_7873_sum_OG__def,axiom,
    ! [P2: fun_complex_nat,I5: set_complex] :
      groups2911672536361861475ex_nat(P2,I5) = $ite(aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_ud(P2),I5))),groups5693394587270226106ex_nat(P2,collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_ud(P2),I5))),zero_zero_nat) ).

% sum.G_def
tff(fact_7874_sum_OG__def,axiom,
    ! [P2: fun_int_nat,I5: set_int] :
      groups2985770679641041633nt_nat(P2,I5) = $ite(aa_set_int_o(finite_finite_int,collect_int(aa_set_int_fun_int_o(aTP_Lamm_tl(P2),I5))),groups4541462559716669496nt_nat(P2,collect_int(aa_set_int_fun_int_o(aTP_Lamm_tl(P2),I5))),zero_zero_nat) ).

% sum.G_def
tff(fact_7875_sum_OG__def,axiom,
    ! [P2: fun_Ex7660298398204468809at_nat,I5: set_Extended_enat] :
      groups4850279874018930067at_nat(P2,I5) = $ite(aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_set3572156908926473983enat_o(aTP_Lamm_ue(P2),I5))),groups2027974829824023292at_nat(P2,collec4429806609662206161d_enat(aa_set3572156908926473983enat_o(aTP_Lamm_ue(P2),I5))),zero_zero_nat) ).

% sum.G_def
tff(fact_7876_sum_OG__def,axiom,
    ! [P2: fun_o_real,I5: set_o] :
      groups627172608727702305o_real(P2,I5) = $ite(aa_set_o_o(finite_finite_o,collect_o(aa_set_o_fun_o_o(aTP_Lamm_tm(P2),I5))),groups8691415230153176458o_real(P2,collect_o(aa_set_o_fun_o_o(aTP_Lamm_tm(P2),I5))),zero_zero_real) ).

% sum.G_def
tff(fact_7877_sum_OG__def,axiom,
    ! [P2: fun_real_real,I5: set_real] :
      groups97945582718554045l_real(P2,I5) = $ite(aa_set_real_o(finite_finite_real,collect_real(aa_set2323979645938365349real_o(aTP_Lamm_tn(P2),I5))),groups8097168146408367636l_real(P2,collect_real(aa_set2323979645938365349real_o(aTP_Lamm_tn(P2),I5))),zero_zero_real) ).

% sum.G_def
tff(fact_7878_sum_OG__def,axiom,
    ! [P2: fun_complex_real,I5: set_complex] :
      groups5737402329758386879x_real(P2,I5) = $ite(aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_uf(P2),I5))),groups5808333547571424918x_real(P2,collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_uf(P2),I5))),zero_zero_real) ).

% sum.G_def
tff(fact_7879_sum_OG__def,axiom,
    ! [P2: fun_int_real,I5: set_int] :
      groups1523912220035142973t_real(P2,I5) = $ite(aa_set_int_o(finite_finite_int,collect_int(aa_set_int_fun_int_o(aTP_Lamm_tp(P2),I5))),groups8778361861064173332t_real(P2,collect_int(aa_set_int_fun_int_o(aTP_Lamm_tp(P2),I5))),zero_zero_real) ).

% sum.G_def
tff(fact_7880_sum_OG__def,axiom,
    ! [P2: fun_Ex1451620480498224165t_real,I5: set_Extended_enat] :
      groups1070518202123951855t_real(P2,I5) = $ite(aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_set3572156908926473983enat_o(aTP_Lamm_ug(P2),I5))),groups4148127829035722712t_real(P2,collec4429806609662206161d_enat(aa_set3572156908926473983enat_o(aTP_Lamm_ug(P2),I5))),zero_zero_real) ).

% sum.G_def
tff(fact_7881_mask__Suc__exp,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,Nb)) = aa_int_int(bit_se1409905431419307370or_int(aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),Nb)),aa_nat_int(bit_se2000444600071755411sk_int,Nb)) ).

% mask_Suc_exp
tff(fact_7882_mask__Suc__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ).

% mask_Suc_exp
tff(fact_7883_or__one__eq,axiom,
    ! [A: int] : aa_int_int(bit_se1409905431419307370or_int(A),one_one_int) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))) ).

% or_one_eq
tff(fact_7884_or__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))) ).

% or_one_eq
tff(fact_7885_one__or__eq,axiom,
    ! [A: int] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))) ).

% one_or_eq
tff(fact_7886_one__or__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))) ).

% one_or_eq
tff(fact_7887_mask__Suc__double,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,Nb)) = aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_nat_int(bit_se2000444600071755411sk_int,Nb))) ).

% mask_Suc_double
tff(fact_7888_mask__Suc__double,axiom,
    ! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb))) ).

% mask_Suc_double
tff(fact_7889_sinh__zero__iff,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(sinh_complex,Xb) = zero_zero_complex )
    <=> aa_set_complex_o(member_complex(aa_complex_complex(exp_complex,Xb)),insert_complex(one_one_complex,insert_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex),bot_bot_set_complex))) ) ).

% sinh_zero_iff
tff(fact_7890_sinh__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sinh_real,Xb) = zero_zero_real )
    <=> aa_set_real_o(member_real2(aa_real_real(exp_real,Xb)),insert_real2(one_one_real,insert_real2(aa_real_real(uminus_uminus_real,one_one_real),bot_bot_set_real))) ) ).

% sinh_zero_iff
tff(fact_7891_tanh__altdef,axiom,
    ! [Xb: real] : aa_real_real(tanh_real,Xb) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,Xb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb)))),aa_real_real(plus_plus_real(aa_real_real(exp_real,Xb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb)))) ).

% tanh_altdef
tff(fact_7892_sum__diff1_H,axiom,
    ! [I5: set_complex,F: fun_complex_complex,Ia: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_sy(I5),F)))
     => ( groups808145749697022017omplex(F,minus_811609699411566653omplex(I5,insert_complex(Ia,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(Ia),I5),aa_complex_complex(minus_minus_complex(groups808145749697022017omplex(F,I5)),aa_complex_complex(F,Ia)),groups808145749697022017omplex(F,I5)) ) ) ).

% sum_diff1'
tff(fact_7893_sum__diff1_H,axiom,
    ! [I5: set_Extended_enat,F: fun_Ex3741802288643809319omplex,Ia: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun807264594178108256enat_o(aTP_Lamm_sz(I5),F)))
     => ( groups4395127934049735793omplex(F,minus_925952699566721837d_enat(I5,insert_Extended_enat(Ia,bot_bo7653980558646680370d_enat))) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(Ia),I5),aa_complex_complex(minus_minus_complex(groups4395127934049735793omplex(F,I5)),aa_Ext8748583412541189208omplex(F,Ia)),groups4395127934049735793omplex(F,I5)) ) ) ).

% sum_diff1'
tff(fact_7894_sum__diff1_H,axiom,
    ! [I5: set_real,F: fun_real_complex,Ia: real] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_ta(I5),F)))
     => ( groups5683813829254066239omplex(F,minus_minus_set_real(I5,insert_real2(Ia,bot_bot_set_real))) = $ite(aa_set_real_o(member_real2(Ia),I5),aa_complex_complex(minus_minus_complex(groups5683813829254066239omplex(F,I5)),aa_real_complex(F,Ia)),groups5683813829254066239omplex(F,I5)) ) ) ).

% sum_diff1'
tff(fact_7895_sum__diff1_H,axiom,
    ! [I5: set_o,F: fun_o_complex,Ia: $o] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun8807775440140275936un_o_o(aTP_Lamm_tb(I5),F)))
     => ( groups3443914341975893411omplex(F,minus_minus_set_o(I5,insert_o2((Ia),bot_bot_set_o))) = $ite(aa_set_o_o(member_o2((Ia)),I5),aa_complex_complex(minus_minus_complex(groups3443914341975893411omplex(F,I5)),aa_o_complex(F,(Ia))),groups3443914341975893411omplex(F,I5)) ) ) ).

% sum_diff1'
tff(fact_7896_sum__diff1_H,axiom,
    ! [I5: set_int,F: fun_int_complex,Ia: int] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_tc(I5),F)))
     => ( groups267424677133301183omplex(F,minus_minus_set_int(I5,insert_int2(Ia,bot_bot_set_int))) = $ite(aa_set_int_o(member_int2(Ia),I5),aa_complex_complex(minus_minus_complex(groups267424677133301183omplex(F,I5)),aa_int_complex(F,Ia)),groups267424677133301183omplex(F,I5)) ) ) ).

% sum_diff1'
tff(fact_7897_sum__diff1_H,axiom,
    ! [I5: set_complex,F: fun_complex_int,Ia: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun8831066929681621998plex_o(aTP_Lamm_td(I5),F)))
     => ( groups2909182065852811199ex_int(F,minus_811609699411566653omplex(I5,insert_complex(Ia,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(Ia),I5),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups2909182065852811199ex_int(F,I5)),aa_complex_int(F,Ia)),groups2909182065852811199ex_int(F,I5)) ) ) ).

% sum_diff1'
tff(fact_7898_sum__diff1_H,axiom,
    ! [I5: set_Extended_enat,F: fun_Ex3482447378695272101at_int,Ia: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun3959557771777301982enat_o(aTP_Lamm_te(I5),F)))
     => ( groups4847789403509879791at_int(F,minus_925952699566721837d_enat(I5,insert_Extended_enat(Ia,bot_bo7653980558646680370d_enat))) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(Ia),I5),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4847789403509879791at_int(F,I5)),aa_Extended_enat_int(F,Ia)),groups4847789403509879791at_int(F,I5)) ) ) ).

% sum_diff1'
tff(fact_7899_sum__diff1_H,axiom,
    ! [I5: set_real,F: fun_real_int,Ia: real] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun429334026828149234real_o(aTP_Lamm_tf(I5),F)))
     => ( groups3901808747961969597al_int(F,minus_minus_set_real(I5,insert_real2(Ia,bot_bot_set_real))) = $ite(aa_set_real_o(member_real2(Ia),I5),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3901808747961969597al_int(F,I5)),aa_real_int(F,Ia)),groups3901808747961969597al_int(F,I5)) ) ) ).

% sum_diff1'
tff(fact_7900_sum__diff1_H,axiom,
    ! [I5: set_o,F: fun_o_int,Ia: $o] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_int_fun_o_o(aTP_Lamm_tg(I5),F)))
     => ( groups4553916814277028129_o_int(F,minus_minus_set_o(I5,insert_o2((Ia),bot_bot_set_o))) = $ite(aa_set_o_o(member_o2((Ia)),I5),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4553916814277028129_o_int(F,I5)),aa_o_int(F,(Ia))),groups4553916814277028129_o_int(F,I5)) ) ) ).

% sum_diff1'
tff(fact_7901_sum__diff1_H,axiom,
    ! [I5: set_int,F: fun_int_int,Ia: int] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun8155382727227203058_int_o(aTP_Lamm_th(I5),F)))
     => ( groups2983280209131991357nt_int(F,minus_minus_set_int(I5,insert_int2(Ia,bot_bot_set_int))) = $ite(aa_set_int_o(member_int2(Ia),I5),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups2983280209131991357nt_int(F,I5)),aa_int_int(F,Ia)),groups2983280209131991357nt_int(F,I5)) ) ) ).

% sum_diff1'
tff(fact_7902_or__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))) ).

% or_Suc_0_eq
tff(fact_7903_Suc__0__or__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))) ).

% Suc_0_or_eq
tff(fact_7904_or__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Mb)
            | ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% or_nat_rec
tff(fact_7905_cosh__square__eq,axiom,
    ! [Xb: real] : aa_nat_real(power_power_real(aa_real_real(cosh_real,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_real_real(plus_plus_real(aa_nat_real(power_power_real(aa_real_real(sinh_real,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real) ).

% cosh_square_eq
tff(fact_7906_cosh__square__eq,axiom,
    ! [Xb: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(cosh_complex,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(sinh_complex,Xb)),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_complex) ).

% cosh_square_eq
tff(fact_7907_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_7908_Cauchy__iff2,axiom,
    ! [X5: fun_nat_real] :
      ( topolo4055970368930404560y_real(X5)
    <=> ! [J3: nat] :
        ? [M8: nat] :
        ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),M5)
         => ! [N3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),N3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(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_7909_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,bit0(one)))),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K)),one_one_nat) ).

% sum_power2
tff(fact_7910_is__singleton__the__elem,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(is_singleton_real,A2)
    <=> ( A2 = insert_real2(the_elem_real(A2),bot_bot_set_real) ) ) ).

% is_singleton_the_elem
tff(fact_7911_is__singleton__the__elem,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(is_singleton_o,A2)
    <=> ( A2 = insert_o2(aa_set_o_o(the_elem_o,A2),bot_bot_set_o) ) ) ).

% is_singleton_the_elem
tff(fact_7912_is__singleton__the__elem,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A2)
    <=> ( A2 = insert_nat2(aa_set_nat_nat(the_elem_nat,A2),bot_bot_set_nat) ) ) ).

% is_singleton_the_elem
tff(fact_7913_is__singleton__the__elem,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(is_singleton_int,A2)
    <=> ( A2 = insert_int2(the_elem_int(A2),bot_bot_set_int) ) ) ).

% is_singleton_the_elem
tff(fact_7914_length__subseqs,axiom,
    ! [Xs: list_VEBT_VEBT] : size_s8217280938318005548T_VEBT(subseqs_VEBT_VEBT(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),size_s6755466524823107622T_VEBT(Xs)) ).

% length_subseqs
tff(fact_7915_length__subseqs,axiom,
    ! [Xs: list_int] : size_s533118279054570080st_int(subseqs_int(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),size_size_list_int(Xs)) ).

% length_subseqs
tff(fact_7916_length__subseqs,axiom,
    ! [Xs: list_nat] : size_s3023201423986296836st_nat(subseqs_nat(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),size_size_list_nat(Xs)) ).

% length_subseqs
tff(fact_7917_atLeastLessThan__iff,axiom,
    ! [Ia: $o,L: $o,U: $o] :
      ( aa_set_o_o(member_o2((Ia)),set_or7139685690850216873Than_o((L),(U)))
    <=> ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(L)),(Ia))
        & aa_o_o(aa_o_fun_o_o(ord_less_o,(Ia)),(U)) ) ) ).

% atLeastLessThan_iff
tff(fact_7918_atLeastLessThan__iff,axiom,
    ! [Ia: extended_enat,L: extended_enat,U: extended_enat] :
      ( aa_set3337841544853333262enat_o(member_Extended_enat(Ia),set_or4374356025156299511d_enat(L,U))
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,L),Ia)
        & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Ia),U) ) ) ).

% atLeastLessThan_iff
tff(fact_7919_atLeastLessThan__iff,axiom,
    ! [Ia: filter_nat,L: filter_nat,U: filter_nat] :
      ( member_filter_nat(Ia,set_or1773934645810362255er_nat(L,U))
    <=> ( ord_le2510731241096832064er_nat(L,Ia)
        & ord_less_filter_nat(Ia,U) ) ) ).

% atLeastLessThan_iff
tff(fact_7920_atLeastLessThan__iff,axiom,
    ! [Ia: real,L: real,U: real] :
      ( aa_set_real_o(member_real2(Ia),set_or66887138388493659n_real(L,U))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),Ia)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Ia),U) ) ) ).

% atLeastLessThan_iff
tff(fact_7921_atLeastLessThan__iff,axiom,
    ! [Ia: set_nat,L: set_nat,U: set_nat] :
      ( aa_set_set_nat_o(member_set_nat2(Ia),set_or3540276404033026485et_nat(L,U))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,L),Ia)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Ia),U) ) ) ).

% atLeastLessThan_iff
tff(fact_7922_atLeastLessThan__iff,axiom,
    ! [Ia: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat2(Ia),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,L),U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),Ia)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),U) ) ) ).

% atLeastLessThan_iff
tff(fact_7923_atLeastLessThan__iff,axiom,
    ! [Ia: int,L: int,U: int] :
      ( aa_set_int_o(member_int2(Ia),set_or4662586982721622107an_int(L,U))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),Ia)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Ia),U) ) ) ).

% atLeastLessThan_iff
tff(fact_7924_atLeastLessThan__empty,axiom,
    ! [B: $o,A: $o] :
      ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(B)),(A))
     => ( set_or7139685690850216873Than_o((A),(B)) = bot_bot_set_o ) ) ).

% atLeastLessThan_empty
tff(fact_7925_atLeastLessThan__empty,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( ord_le2510731241096832064er_nat(B,A)
     => ( set_or1773934645810362255er_nat(A,B) = bot_bo498966703094740906er_nat ) ) ).

% atLeastLessThan_empty
tff(fact_7926_atLeastLessThan__empty,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( set_or66887138388493659n_real(A,B) = bot_bot_set_real ) ) ).

% atLeastLessThan_empty
tff(fact_7927_atLeastLessThan__empty,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( set_or3540276404033026485et_nat(A,B) = bot_bot_set_set_nat ) ) ).

% atLeastLessThan_empty
tff(fact_7928_atLeastLessThan__empty,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B) = bot_bot_set_nat ) ) ).

% atLeastLessThan_empty
tff(fact_7929_atLeastLessThan__empty,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( set_or4662586982721622107an_int(A,B) = bot_bot_set_int ) ) ).

% atLeastLessThan_empty
tff(fact_7930_ivl__subset,axiom,
    ! [Ia: real,J: real,Mb: real,Nb: real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_or66887138388493659n_real(Ia,J)),set_or66887138388493659n_real(Mb,Nb))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,J),Ia)
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Mb),Ia)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,J),Nb) ) ) ) ).

% ivl_subset
tff(fact_7931_ivl__subset,axiom,
    ! [Ia: nat,J: nat,Mb: nat,Nb: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Ia),J)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),Ia)
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Ia)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),Nb) ) ) ) ).

% ivl_subset
tff(fact_7932_ivl__subset,axiom,
    ! [Ia: int,J: int,Mb: int,Nb: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_or4662586982721622107an_int(Ia,J)),set_or4662586982721622107an_int(Mb,Nb))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),Ia)
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Mb),Ia)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),Nb) ) ) ) ).

% ivl_subset
tff(fact_7933_atLeastLessThan__empty__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( set_or7139685690850216873Than_o((A),(B)) = bot_bot_set_o )
    <=> ~ aa_o_o(aa_o_fun_o_o(ord_less_o,(A)),(B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_7934_atLeastLessThan__empty__iff,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( set_or4374356025156299511d_enat(A,B) = bot_bo7653980558646680370d_enat )
    <=> ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_7935_atLeastLessThan__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( set_or66887138388493659n_real(A,B) = bot_bot_set_real )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_7936_atLeastLessThan__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_7937_atLeastLessThan__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( set_or4662586982721622107an_int(A,B) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_7938_atLeastLessThan__empty__iff2,axiom,
    ! [A: $o,B: $o] :
      ( ( bot_bot_set_o = set_or7139685690850216873Than_o((A),(B)) )
    <=> ~ aa_o_o(aa_o_fun_o_o(ord_less_o,(A)),(B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_7939_atLeastLessThan__empty__iff2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( bot_bo7653980558646680370d_enat = set_or4374356025156299511d_enat(A,B) )
    <=> ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_7940_atLeastLessThan__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real = set_or66887138388493659n_real(A,B) )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_7941_atLeastLessThan__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_7942_atLeastLessThan__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = set_or4662586982721622107an_int(A,B) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_7943_infinite__Ico__iff,axiom,
    ! [A: real,B: real] :
      ( ~ aa_set_real_o(finite_finite_real,set_or66887138388493659n_real(A,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% infinite_Ico_iff
tff(fact_7944_ivl__diff,axiom,
    ! [Ia: real,Nb: real,Mb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ia),Nb)
     => ( minus_minus_set_real(set_or66887138388493659n_real(Ia,Mb),set_or66887138388493659n_real(Ia,Nb)) = set_or66887138388493659n_real(Nb,Mb) ) ) ).

% ivl_diff
tff(fact_7945_ivl__diff,axiom,
    ! [Ia: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),Nb)
     => ( minus_minus_set_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Ia),Mb),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Ia),Nb)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),Mb) ) ) ).

% ivl_diff
tff(fact_7946_ivl__diff,axiom,
    ! [Ia: int,Nb: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),Nb)
     => ( minus_minus_set_int(set_or4662586982721622107an_int(Ia,Mb),set_or4662586982721622107an_int(Ia,Nb)) = set_or4662586982721622107an_int(Nb,Mb) ) ) ).

% ivl_diff
tff(fact_7947_is__singletonI,axiom,
    ! [Xb: real] : aa_set_real_o(is_singleton_real,insert_real2(Xb,bot_bot_set_real)) ).

% is_singletonI
tff(fact_7948_is__singletonI,axiom,
    ! [Xb: $o] : aa_set_o_o(is_singleton_o,insert_o2((Xb),bot_bot_set_o)) ).

% is_singletonI
tff(fact_7949_is__singletonI,axiom,
    ! [Xb: nat] : aa_set_nat_o(is_singleton_nat,insert_nat2(Xb,bot_bot_set_nat)) ).

% is_singletonI
tff(fact_7950_is__singletonI,axiom,
    ! [Xb: int] : aa_set_int_o(is_singleton_int,insert_int2(Xb,bot_bot_set_int)) ).

% is_singletonI
tff(fact_7951_atLeastLessThan__singleton,axiom,
    ! [Mb: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),aa_nat_nat(suc,Mb)) = insert_nat2(Mb,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_7952_sum_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] :
      aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_nat_int(G,Nb))) ).

% sum.op_ivl_Suc
tff(fact_7953_sum_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_complex,Mb: nat,Nb: nat] :
      groups2073611262835488442omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),zero_zero_complex,aa_complex_complex(plus_plus_complex(groups2073611262835488442omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_nat_complex(G,Nb))) ).

% sum.op_ivl_Suc
tff(fact_7954_sum_Oop__ivl__Suc,axiom,
    ! [G: fun_na8797796813480665199d_enat,Mb: nat,Nb: nat] :
      groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),zero_z5237406670263579293d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_nat_Extended_enat(G,Nb))) ).

% sum.op_ivl_Suc
tff(fact_7955_sum_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_nat_nat(G,Nb))) ).

% sum.op_ivl_Suc
tff(fact_7956_sum_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),zero_zero_real,aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_nat_real(G,Nb))) ).

% sum.op_ivl_Suc
tff(fact_7957_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),one_one_real,aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_nat_real(G,Nb))) ).

% prod.op_ivl_Suc
tff(fact_7958_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_complex,Mb: nat,Nb: nat] :
      groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),one_one_complex,aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_nat_complex(G,Nb))) ).

% prod.op_ivl_Suc
tff(fact_7959_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_na8797796813480665199d_enat,Mb: nat,Nb: nat] :
      groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),one_on7984719198319812577d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_nat_Extended_enat(G,Nb))) ).

% prod.op_ivl_Suc
tff(fact_7960_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] :
      aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_nat_int(G,Nb))) ).

% prod.op_ivl_Suc
tff(fact_7961_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] :
      aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_nat_nat(G,Nb))) ).

% prod.op_ivl_Suc
tff(fact_7962_atLeastLessThan__eq__iff,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),D)
       => ( ( set_or4374356025156299511d_enat(A,B) = set_or4374356025156299511d_enat(C,D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_7963_atLeastLessThan__eq__iff,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)
       => ( ( set_or66887138388493659n_real(A,B) = set_or66887138388493659n_real(C,D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_7964_atLeastLessThan__eq__iff,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_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,C),D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_7965_atLeastLessThan__eq__iff,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)
       => ( ( set_or4662586982721622107an_int(A,B) = set_or4662586982721622107an_int(C,D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_7966_atLeastLessThan__inj_I1_J,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( set_or4374356025156299511d_enat(A,B) = set_or4374356025156299511d_enat(C,D) )
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
       => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),D)
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_7967_atLeastLessThan__inj_I1_J,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( set_or66887138388493659n_real(A,B) = set_or66887138388493659n_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)
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_7968_atLeastLessThan__inj_I1_J,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,C),D) )
     => ( 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)
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_7969_atLeastLessThan__inj_I1_J,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( set_or4662586982721622107an_int(A,B) = set_or4662586982721622107an_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)
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_7970_atLeastLessThan__inj_I2_J,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( set_or4374356025156299511d_enat(A,B) = set_or4374356025156299511d_enat(C,D) )
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
       => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),D)
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_7971_atLeastLessThan__inj_I2_J,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( set_or66887138388493659n_real(A,B) = set_or66887138388493659n_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)
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_7972_atLeastLessThan__inj_I2_J,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,C),D) )
     => ( 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)
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_7973_atLeastLessThan__inj_I2_J,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( set_or4662586982721622107an_int(A,B) = set_or4662586982721622107an_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)
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_7974_atLeastLessThan__subset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_or66887138388493659n_real(A,B)),set_or66887138388493659n_real(C,D))
     => ( 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),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),D) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_7975_atLeastLessThan__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,C),D))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_7976_atLeastLessThan__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_or4662586982721622107an_int(A,B)),set_or4662586982721622107an_int(C,D))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_7977_infinite__Ico,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_or66887138388493659n_real(A,B)) ) ).

% infinite_Ico
tff(fact_7978_all__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),Nb)
         => aa_nat_o(P,M5) )
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))
         => aa_nat_o(P,X3) ) ) ).

% all_nat_less_eq
tff(fact_7979_ex__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),Nb)
          & aa_nat_o(P,M5) )
    <=> ? [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))
          & aa_nat_o(P,X3) ) ) ).

% ex_nat_less_eq
tff(fact_7980_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,L),aa_nat_nat(suc,U)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L),U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_7981_lessThan__atLeast0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,Nb) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb) ).

% lessThan_atLeast0
tff(fact_7982_atLeastLessThan0,axiom,
    ! [Mb: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_7983_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_go(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) ).

% sum.shift_bounds_Suc_ivl
tff(fact_7984_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gp(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) ).

% sum.shift_bounds_Suc_ivl
tff(fact_7985_sum_Oshift__bounds__nat__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,K: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_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,Nb),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gq(G),K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) ).

% sum.shift_bounds_nat_ivl
tff(fact_7986_sum_Oshift__bounds__nat__ivl,axiom,
    ! [G: fun_nat_real,Mb: nat,K: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_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,Nb),K))) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_gr(G),K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) ).

% sum.shift_bounds_nat_ivl
tff(fact_7987_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_hh(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) ).

% prod.shift_bounds_Suc_ivl
tff(fact_7988_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_go(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) ).

% prod.shift_bounds_Suc_ivl
tff(fact_7989_prod_Oshift__bounds__nat__ivl,axiom,
    ! [G: fun_nat_int,Mb: nat,K: nat,Nb: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_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,Nb),K))) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_kf(G),K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) ).

% prod.shift_bounds_nat_ivl
tff(fact_7990_prod_Oshift__bounds__nat__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,K: nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_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,Nb),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gq(G),K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) ).

% prod.shift_bounds_nat_ivl
tff(fact_7991_is__singletonI_H,axiom,
    ! [A2: set_set_nat] :
      ( ( A2 != bot_bot_set_set_nat )
     => ( ! [X4: set_nat,Y3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat2(X4),A2)
           => ( aa_set_set_nat_o(member_set_nat2(Y3),A2)
             => ( X4 = Y3 ) ) )
       => aa_set_set_nat_o(is_singleton_set_nat,A2) ) ) ).

% is_singletonI'
tff(fact_7992_is__singletonI_H,axiom,
    ! [A2: set_real] :
      ( ( A2 != bot_bot_set_real )
     => ( ! [X4: real,Y3: real] :
            ( aa_set_real_o(member_real2(X4),A2)
           => ( aa_set_real_o(member_real2(Y3),A2)
             => ( X4 = Y3 ) ) )
       => aa_set_real_o(is_singleton_real,A2) ) ) ).

% is_singletonI'
tff(fact_7993_is__singletonI_H,axiom,
    ! [A2: set_o] :
      ( ( A2 != bot_bot_set_o )
     => ( ! [X4: $o,Y3: $o] :
            ( aa_set_o_o(member_o2((X4)),A2)
           => ( aa_set_o_o(member_o2((Y3)),A2)
             => ( (X4)
              <=> (Y3) ) ) )
       => aa_set_o_o(is_singleton_o,A2) ) ) ).

% is_singletonI'
tff(fact_7994_is__singletonI_H,axiom,
    ! [A2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ! [X4: nat,Y3: nat] :
            ( aa_set_nat_o(member_nat2(X4),A2)
           => ( aa_set_nat_o(member_nat2(Y3),A2)
             => ( X4 = Y3 ) ) )
       => aa_set_nat_o(is_singleton_nat,A2) ) ) ).

% is_singletonI'
tff(fact_7995_is__singletonI_H,axiom,
    ! [A2: set_int] :
      ( ( A2 != bot_bot_set_int )
     => ( ! [X4: int,Y3: int] :
            ( aa_set_int_o(member_int2(X4),A2)
           => ( aa_set_int_o(member_int2(Y3),A2)
             => ( X4 = Y3 ) ) )
       => aa_set_int_o(is_singleton_int,A2) ) ) ).

% is_singletonI'
tff(fact_7996_sum_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( ( A = C )
     => ( ( B = D )
       => ( ! [X4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X4)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),D)
               => ( aa_nat_nat(G,X4) = aa_nat_nat(H,X4) ) ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B)) = aa_set_nat_nat(groups3542108847815614940at_nat(H),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,C),D)) ) ) ) ) ).

% sum.ivl_cong
tff(fact_7997_sum_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D: nat,G: fun_nat_real,H: fun_nat_real] :
      ( ( A = C )
     => ( ( B = D )
       => ( ! [X4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X4)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),D)
               => ( aa_nat_real(G,X4) = aa_nat_real(H,X4) ) ) )
         => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B)) = aa_set_nat_real(groups6591440286371151544t_real(H),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,C),D)) ) ) ) ) ).

% sum.ivl_cong
tff(fact_7998_prod_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D: nat,G: fun_nat_int,H: fun_nat_int] :
      ( ( A = C )
     => ( ( B = D )
       => ( ! [X4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X4)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),D)
               => ( aa_nat_int(G,X4) = aa_nat_int(H,X4) ) ) )
         => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B)) = aa_set_nat_int(groups705719431365010083at_int(H),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,C),D)) ) ) ) ) ).

% prod.ivl_cong
tff(fact_7999_prod_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( ( A = C )
     => ( ( B = D )
       => ( ! [X4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X4)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),D)
               => ( aa_nat_nat(G,X4) = aa_nat_nat(H,X4) ) ) )
         => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B)) = aa_set_nat_nat(groups708209901874060359at_nat(H),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,C),D)) ) ) ) ) ).

% prod.ivl_cong
tff(fact_8000_prod_Oivl__cong,axiom,
    ! [A: int,C: int,B: int,D: int,G: fun_int_int,H: fun_int_int] :
      ( ( A = C )
     => ( ( B = D )
       => ( ! [X4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),X4)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),D)
               => ( aa_int_int(G,X4) = aa_int_int(H,X4) ) ) )
         => ( groups1705073143266064639nt_int(G,set_or4662586982721622107an_int(A,B)) = groups1705073143266064639nt_int(H,set_or4662586982721622107an_int(C,D)) ) ) ) ) ).

% prod.ivl_cong
tff(fact_8001_sum_OatLeastLessThan__concat,axiom,
    ! [Mb: nat,Nb: nat,P2: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),P2)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),P2))) = aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),P2)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_8002_sum_OatLeastLessThan__concat,axiom,
    ! [Mb: nat,Nb: nat,P2: nat,G: fun_na8797796813480665199d_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),P2)
       => ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),P2))) = groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),P2)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_8003_sum_OatLeastLessThan__concat,axiom,
    ! [Mb: nat,Nb: nat,P2: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),P2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),P2))) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),P2)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_8004_sum_OatLeastLessThan__concat,axiom,
    ! [Mb: nat,Nb: nat,P2: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),P2)
       => ( aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),P2))) = aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),P2)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_8005_sum__diff__nat__ivl,axiom,
    ! [Mb: nat,Nb: nat,P2: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),P2)
       => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),P2))),aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))) = aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),P2)) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_8006_sum__diff__nat__ivl,axiom,
    ! [Mb: nat,Nb: nat,P2: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),P2)
       => ( aa_real_real(minus_minus_real(aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),P2))),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))) = aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),P2)) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_8007_size__list__estimation,axiom,
    ! [Xb: real,Xs: list_real,Y: nat,F: fun_real_nat] :
      ( aa_set_real_o(member_real2(Xb),set_real2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_real_nat(F,Xb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),size_list_real(F,Xs)) ) ) ).

% size_list_estimation
tff(fact_8008_size__list__estimation,axiom,
    ! [Xb: $o,Xs: list_o,Y: nat,F: fun_o_nat] :
      ( aa_set_o_o(member_o2((Xb)),set_o2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_o_nat(F,(Xb)))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),size_list_o(F,Xs)) ) ) ).

% size_list_estimation
tff(fact_8009_size__list__estimation,axiom,
    ! [Xb: set_nat,Xs: list_set_nat,Y: nat,F: fun_set_nat_nat] :
      ( aa_set_set_nat_o(member_set_nat2(Xb),set_set_nat2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_set_nat_nat(F,Xb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),size_list_set_nat(F,Xs)) ) ) ).

% size_list_estimation
tff(fact_8010_size__list__estimation,axiom,
    ! [Xb: int,Xs: list_int,Y: nat,F: fun_int_nat] :
      ( aa_set_int_o(member_int2(Xb),set_int2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_int_nat(F,Xb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),size_list_int(F,Xs)) ) ) ).

% size_list_estimation
tff(fact_8011_size__list__estimation,axiom,
    ! [Xb: nat,Xs: list_nat,Y: nat,F: fun_nat_nat] :
      ( aa_set_nat_o(member_nat2(Xb),set_nat2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(F,Xb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),size_list_nat(F,Xs)) ) ) ).

% size_list_estimation
tff(fact_8012_size__list__estimation,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT,Y: nat,F: fun_VEBT_VEBT_nat] :
      ( member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_VEBT_VEBT_nat(F,Xb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),size_list_VEBT_VEBT(F,Xs)) ) ) ).

% size_list_estimation
tff(fact_8013_size__list__pointwise,axiom,
    ! [Xs: list_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [X4: real] :
          ( aa_set_real_o(member_real2(X4),set_real2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X4)),aa_real_nat(G,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_list_real(F,Xs)),size_list_real(G,Xs)) ) ).

% size_list_pointwise
tff(fact_8014_size__list__pointwise,axiom,
    ! [Xs: list_o,F: fun_o_nat,G: fun_o_nat] :
      ( ! [X4: $o] :
          ( aa_set_o_o(member_o2((X4)),set_o2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(X4))),aa_o_nat(G,(X4))) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_list_o(F,Xs)),size_list_o(G,Xs)) ) ).

% size_list_pointwise
tff(fact_8015_size__list__pointwise,axiom,
    ! [Xs: list_set_nat,F: fun_set_nat_nat,G: fun_set_nat_nat] :
      ( ! [X4: set_nat] :
          ( aa_set_set_nat_o(member_set_nat2(X4),set_set_nat2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(F,X4)),aa_set_nat_nat(G,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_list_set_nat(F,Xs)),size_list_set_nat(G,Xs)) ) ).

% size_list_pointwise
tff(fact_8016_size__list__pointwise,axiom,
    ! [Xs: list_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int2(X4),set_int2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X4)),aa_int_nat(G,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_list_int(F,Xs)),size_list_int(G,Xs)) ) ).

% size_list_pointwise
tff(fact_8017_size__list__pointwise,axiom,
    ! [Xs: list_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat2(X4),set_nat2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X4)),aa_nat_nat(G,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_list_nat(F,Xs)),size_list_nat(G,Xs)) ) ).

% size_list_pointwise
tff(fact_8018_size__list__pointwise,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT2(X4,set_VEBT_VEBT2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,X4)),aa_VEBT_VEBT_nat(G,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_list_VEBT_VEBT(F,Xs)),size_list_VEBT_VEBT(G,Xs)) ) ).

% size_list_pointwise
tff(fact_8019_size__list__estimation_H,axiom,
    ! [Xb: real,Xs: list_real,Y: nat,F: fun_real_nat] :
      ( aa_set_real_o(member_real2(Xb),set_real2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_real_nat(F,Xb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),size_list_real(F,Xs)) ) ) ).

% size_list_estimation'
tff(fact_8020_size__list__estimation_H,axiom,
    ! [Xb: $o,Xs: list_o,Y: nat,F: fun_o_nat] :
      ( aa_set_o_o(member_o2((Xb)),set_o2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_o_nat(F,(Xb)))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),size_list_o(F,Xs)) ) ) ).

% size_list_estimation'
tff(fact_8021_size__list__estimation_H,axiom,
    ! [Xb: set_nat,Xs: list_set_nat,Y: nat,F: fun_set_nat_nat] :
      ( aa_set_set_nat_o(member_set_nat2(Xb),set_set_nat2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_set_nat_nat(F,Xb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),size_list_set_nat(F,Xs)) ) ) ).

% size_list_estimation'
tff(fact_8022_size__list__estimation_H,axiom,
    ! [Xb: int,Xs: list_int,Y: nat,F: fun_int_nat] :
      ( aa_set_int_o(member_int2(Xb),set_int2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_int_nat(F,Xb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),size_list_int(F,Xs)) ) ) ).

% size_list_estimation'
tff(fact_8023_size__list__estimation_H,axiom,
    ! [Xb: nat,Xs: list_nat,Y: nat,F: fun_nat_nat] :
      ( aa_set_nat_o(member_nat2(Xb),set_nat2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_nat_nat(F,Xb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),size_list_nat(F,Xs)) ) ) ).

% size_list_estimation'
tff(fact_8024_size__list__estimation_H,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT,Y: nat,F: fun_VEBT_VEBT_nat] :
      ( member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_VEBT_VEBT_nat(F,Xb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),size_list_VEBT_VEBT(F,Xs)) ) ) ).

% size_list_estimation'
tff(fact_8025_prod_OatLeastLessThan__concat,axiom,
    ! [Mb: nat,Nb: nat,P2: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),P2)
       => ( aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),P2))) = aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),P2)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_8026_prod_OatLeastLessThan__concat,axiom,
    ! [Mb: nat,Nb: nat,P2: nat,G: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),P2)
       => ( aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),P2))) = groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),P2)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_8027_prod_OatLeastLessThan__concat,axiom,
    ! [Mb: nat,Nb: nat,P2: nat,G: fun_na8797796813480665199d_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),P2)
       => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),P2))) = groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),P2)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_8028_prod_OatLeastLessThan__concat,axiom,
    ! [Mb: nat,Nb: nat,P2: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),P2)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),P2))) = aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),P2)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_8029_prod_OatLeastLessThan__concat,axiom,
    ! [Mb: nat,Nb: nat,P2: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),P2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),P2))) = aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),P2)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_8030_atLeast0__lessThan__Suc,axiom,
    ! [Nb: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat2(Nb,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb)) ).

% atLeast0_lessThan_Suc
tff(fact_8031_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N4: set_nat,Nb: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N4),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))
     => aa_set_nat_o(finite_finite_nat,N4) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_8032_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(set_or5403411693681687835d_enat(A,B)),set_or4374356025156299511d_enat(C,D))
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
       => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),A)
          & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),D) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_8033_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat,D: filter_nat] :
      ( ord_le2426478655948331894er_nat(set_or1955772592623580779er_nat(A,B),set_or1773934645810362255er_nat(C,D))
    <=> ( ord_le2510731241096832064er_nat(A,B)
       => ( ord_le2510731241096832064er_nat(C,A)
          & ord_less_filter_nat(B,D) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_8034_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(A,B)),set_or3540276404033026485et_nat(C,D))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A)
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),D) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_8035_atLeastAtMost__subseteq__atLeastLessThan__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_or66887138388493659n_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_real,B),D) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_8036_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_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_nat,B),D) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_8037_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_or1266510415728281911st_int(A,B)),set_or4662586982721622107an_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_int,B),D) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_8038_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_or66887138388493659n_real(A,B)),set_or1222579329274155063t_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_eq_real,C),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),D) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_8039_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_int,K: nat] :
      ( ( aa_nat_int(F,zero_zero_nat) = zero_zero_int )
     => ( aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_8040_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( groups2073611262835488442omplex(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,zero_zero_nat)),K)) = groups2073611262835488442omplex(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_8041_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_na8797796813480665199d_enat,K: nat] :
      ( ( aa_nat_Extended_enat(F,zero_zero_nat) = zero_z5237406670263579293d_enat )
     => ( groups7108830773950497114d_enat(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,zero_zero_nat)),K)) = groups7108830773950497114d_enat(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_8042_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_nat,K: nat] :
      ( ( aa_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_8043_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_8044_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))),aa_nat_int(G,Nb)) ).

% sum.atLeast0_lessThan_Suc
tff(fact_8045_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_na8797796813480665199d_enat,Nb: nat] : groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))),aa_nat_Extended_enat(G,Nb)) ).

% sum.atLeast0_lessThan_Suc
tff(fact_8046_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))),aa_nat_nat(G,Nb)) ).

% sum.atLeast0_lessThan_Suc
tff(fact_8047_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))),aa_nat_real(G,Nb)) ).

% sum.atLeast0_lessThan_Suc
tff(fact_8048_sum_OatLeast__Suc__lessThan,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Mb)),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_8049_sum_OatLeast__Suc__lessThan,axiom,
    ! [Mb: nat,Nb: nat,G: fun_na8797796813480665199d_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(G,Mb)),groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_8050_sum_OatLeast__Suc__lessThan,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_8051_sum_OatLeast__Suc__lessThan,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) = aa_real_real(plus_plus_real(aa_nat_real(G,Mb)),aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_8052_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),aa_nat_nat(suc,B))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B))),aa_nat_int(G,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_8053_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_na8797796813480665199d_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),aa_nat_nat(suc,B))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B))),aa_nat_Extended_enat(G,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_8054_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),aa_nat_nat(suc,B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B))),aa_nat_nat(G,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_8055_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),aa_nat_nat(suc,B))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B))),aa_nat_real(G,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_8056_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $o,B: $o] : set_or7139685690850216873Than_o((A),(B)) = minus_minus_set_o(set_or8904488021354931149Most_o((A),(B)),insert_o2((B),bot_bot_set_o)) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_8057_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: real,B: real] : set_or66887138388493659n_real(A,B) = minus_minus_set_real(set_or1222579329274155063t_real(A,B),insert_real2(B,bot_bot_set_real)) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_8058_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: nat,B: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B) = minus_minus_set_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B),insert_nat2(B,bot_bot_set_nat)) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_8059_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: int,B: int] : set_or4662586982721622107an_int(A,B) = minus_minus_set_int(set_or1266510415728281911st_int(A,B),insert_int2(B,bot_bot_set_int)) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_8060_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))),aa_nat_real(G,Nb)) ).

% prod.atLeast0_lessThan_Suc
tff(fact_8061_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_complex,Nb: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))),aa_nat_complex(G,Nb)) ).

% prod.atLeast0_lessThan_Suc
tff(fact_8062_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_na8797796813480665199d_enat,Nb: nat] : groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))),aa_nat_Extended_enat(G,Nb)) ).

% prod.atLeast0_lessThan_Suc
tff(fact_8063_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))),aa_nat_int(G,Nb)) ).

% prod.atLeast0_lessThan_Suc
tff(fact_8064_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))),aa_nat_nat(G,Nb)) ).

% prod.atLeast0_lessThan_Suc
tff(fact_8065_prod_OatLeast__Suc__lessThan,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) = aa_real_real(times_times_real(aa_nat_real(G,Mb)),aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_8066_prod_OatLeast__Suc__lessThan,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(G,Mb)),groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_8067_prod_OatLeast__Suc__lessThan,axiom,
    ! [Mb: nat,Nb: nat,G: fun_na8797796813480665199d_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(G,Mb)),groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_8068_prod_OatLeast__Suc__lessThan,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,Mb)),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_8069_prod_OatLeast__Suc__lessThan,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_8070_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),aa_nat_nat(suc,B))) = aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B))),aa_nat_real(G,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_8071_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),aa_nat_nat(suc,B))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B))),aa_nat_complex(G,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_8072_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_na8797796813480665199d_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),aa_nat_nat(suc,B))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B))),aa_nat_Extended_enat(G,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_8073_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),aa_nat_nat(suc,B))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B))),aa_nat_int(G,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_8074_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),aa_nat_nat(suc,B))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B))),aa_nat_nat(G,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_8075_sum_Olast__plus,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Nb)),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))) ) ) ).

% sum.last_plus
tff(fact_8076_sum_Olast__plus,axiom,
    ! [Mb: nat,Nb: nat,G: fun_na8797796813480665199d_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(G,Nb)),groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))) ) ) ).

% sum.last_plus
tff(fact_8077_sum_Olast__plus,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Nb)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))) ) ) ).

% sum.last_plus
tff(fact_8078_sum_Olast__plus,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_real_real(plus_plus_real(aa_nat_real(G,Nb)),aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))) ) ) ).

% sum.last_plus
tff(fact_8079_prod_Olast__plus,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_real_real(times_times_real(aa_nat_real(G,Nb)),aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))) ) ) ).

% prod.last_plus
tff(fact_8080_prod_Olast__plus,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(G,Nb)),groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))) ) ) ).

% prod.last_plus
tff(fact_8081_prod_Olast__plus,axiom,
    ! [Mb: nat,Nb: nat,G: fun_na8797796813480665199d_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(G,Nb)),groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))) ) ) ).

% prod.last_plus
tff(fact_8082_prod_Olast__plus,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,Nb)),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))) ) ) ).

% prod.last_plus
tff(fact_8083_prod_Olast__plus,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,Nb)),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))) ) ) ).

% prod.last_plus
tff(fact_8084_sum__Suc__diff_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_hj(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Nb)),aa_nat_int(F,Mb)) ) ) ).

% sum_Suc_diff'
tff(fact_8085_sum__Suc__diff_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hk(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) = aa_real_real(minus_minus_real(aa_nat_real(F,Nb)),aa_nat_real(F,Mb)) ) ) ).

% sum_Suc_diff'
tff(fact_8086_sum_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_nat,Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),Mb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_uh(G),Nb),Mb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),Mb)) ).

% sum.atLeastLessThan_rev
tff(fact_8087_sum_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_real,Nb: nat,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),Mb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_ui(G),Nb),Mb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),Mb)) ).

% sum.atLeastLessThan_rev
tff(fact_8088_atLeastLessThanSuc,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),insert_nat2(Nb,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_8089_sum_Onested__swap,axiom,
    ! [A: fun_nat_fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_uj(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ms(A),Nb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb)) ).

% sum.nested_swap
tff(fact_8090_sum_Onested__swap,axiom,
    ! [A: fun_nat_fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_uk(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_mv(A),Nb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb)) ).

% sum.nested_swap
tff(fact_8091_prod_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_int,Nb: nat,Mb: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),Mb)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_ul(G),Nb),Mb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),Mb)) ).

% prod.atLeastLessThan_rev
tff(fact_8092_prod_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_nat,Nb: nat,Mb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),Mb)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_uh(G),Nb),Mb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),Mb)) ).

% prod.atLeastLessThan_rev
tff(fact_8093_prod_Onested__swap,axiom,
    ! [A: fun_nat_fun_nat_int,Nb: nat] : aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_um(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_my(A),Nb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb)) ).

% prod.nested_swap
tff(fact_8094_prod_Onested__swap,axiom,
    ! [A: fun_nat_fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_un(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_na(A),Nb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb)) ).

% prod.nested_swap
tff(fact_8095_sum_Onat__group,axiom,
    ! [G: fun_nat_nat,K: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uo(G),K)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K))) ).

% sum.nat_group
tff(fact_8096_sum_Onat__group,axiom,
    ! [G: fun_nat_real,K: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_up(G),K)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K))) ).

% sum.nat_group
tff(fact_8097_prod_Onat__group,axiom,
    ! [G: fun_nat_int,K: nat,Nb: nat] : aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_uq(G),K)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K))) ).

% prod.nat_group
tff(fact_8098_prod_Onat__group,axiom,
    ! [G: fun_nat_nat,K: nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ur(G),K)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K))) ).

% prod.nat_group
tff(fact_8099_prod__Suc__Suc__fact,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_Suc_fact
tff(fact_8100_prod__Suc__fact,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_fact
tff(fact_8101_is__singletonE,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(is_singleton_real,A2)
     => ~ ! [X4: real] : A2 != insert_real2(X4,bot_bot_set_real) ) ).

% is_singletonE
tff(fact_8102_is__singletonE,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(is_singleton_o,A2)
     => ~ ! [X4: $o] : A2 != insert_o2((X4),bot_bot_set_o) ) ).

% is_singletonE
tff(fact_8103_is__singletonE,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A2)
     => ~ ! [X4: nat] : A2 != insert_nat2(X4,bot_bot_set_nat) ) ).

% is_singletonE
tff(fact_8104_is__singletonE,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(is_singleton_int,A2)
     => ~ ! [X4: int] : A2 != insert_int2(X4,bot_bot_set_int) ) ).

% is_singletonE
tff(fact_8105_is__singleton__def,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(is_singleton_real,A2)
    <=> ? [X3: real] : A2 = insert_real2(X3,bot_bot_set_real) ) ).

% is_singleton_def
tff(fact_8106_is__singleton__def,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(is_singleton_o,A2)
    <=> ? [X3: $o] : A2 = insert_o2((X3),bot_bot_set_o) ) ).

% is_singleton_def
tff(fact_8107_is__singleton__def,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A2)
    <=> ? [X3: nat] : A2 = insert_nat2(X3,bot_bot_set_nat) ) ).

% is_singleton_def
tff(fact_8108_is__singleton__def,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(is_singleton_int,A2)
    <=> ? [X3: int] : A2 = insert_int2(X3,bot_bot_set_int) ) ).

% is_singleton_def
tff(fact_8109_sum_Ohead__if,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] :
      aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_nat_int(G,Nb))) ).

% sum.head_if
tff(fact_8110_sum_Ohead__if,axiom,
    ! [G: fun_nat_complex,Mb: nat,Nb: nat] :
      groups2073611262835488442omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),zero_zero_complex,aa_complex_complex(plus_plus_complex(groups2073611262835488442omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_nat_complex(G,Nb))) ).

% sum.head_if
tff(fact_8111_sum_Ohead__if,axiom,
    ! [G: fun_na8797796813480665199d_enat,Mb: nat,Nb: nat] :
      groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),zero_z5237406670263579293d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(groups7108830773950497114d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_nat_Extended_enat(G,Nb))) ).

% sum.head_if
tff(fact_8112_sum_Ohead__if,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_nat_nat(G,Nb))) ).

% sum.head_if
tff(fact_8113_sum_Ohead__if,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),zero_zero_real,aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_nat_real(G,Nb))) ).

% sum.head_if
tff(fact_8114_prod_Ohead__if,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),one_one_real,aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_nat_real(G,Nb))) ).

% prod.head_if
tff(fact_8115_prod_Ohead__if,axiom,
    ! [G: fun_nat_complex,Mb: nat,Nb: nat] :
      groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),one_one_complex,aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_nat_complex(G,Nb))) ).

% prod.head_if
tff(fact_8116_prod_Ohead__if,axiom,
    ! [G: fun_na8797796813480665199d_enat,Mb: nat,Nb: nat] :
      groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),one_on7984719198319812577d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(groups7961826882256487087d_enat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_nat_Extended_enat(G,Nb))) ).

% prod.head_if
tff(fact_8117_prod_Ohead__if,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] :
      aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_nat_int(G,Nb))) ).

% prod.head_if
tff(fact_8118_prod_Ohead__if,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] :
      aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb))),aa_nat_nat(G,Nb))) ).

% prod.head_if
tff(fact_8119_fact__prod__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))) ).

% fact_prod_Suc
tff(fact_8120_fact__prod__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))) ).

% fact_prod_Suc
tff(fact_8121_fact__prod__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))) ).

% fact_prod_Suc
tff(fact_8122_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: fun_nat_nat,Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),Mb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_hf(G),Nb),Mb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Nb)),Mb)) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_8123_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: fun_nat_real,Nb: nat,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),Mb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_hg(G),Nb),Mb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Nb)),Mb)) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_8124_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: fun_nat_int,Nb: nat,Mb: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),Mb)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_kh(G),Nb),Mb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Nb)),Mb)) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_8125_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: fun_nat_nat,Nb: nat,Mb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Nb),Mb)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_hf(G),Nb),Mb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Nb)),Mb)) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_8126_pochhammer__prod,axiom,
    ! [A: extended_enat,Nb: nat] : aa_nat_Extended_enat(comm_s3181272606743183617d_enat(A),Nb) = groups7961826882256487087d_enat(aTP_Lamm_km(A),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb)) ).

% pochhammer_prod
tff(fact_8127_pochhammer__prod,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),Nb) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_kn(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb)) ).

% pochhammer_prod
tff(fact_8128_pochhammer__prod,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),Nb) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_ko(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb)) ).

% pochhammer_prod
tff(fact_8129_pochhammer__prod,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kp,A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb)) ).

% pochhammer_prod
tff(fact_8130_atLeastLessThan__nat__numeral,axiom,
    ! [Mb: nat,K: num] :
      aa_nat_set_nat(aa_nat1818925681583257441et_nat(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_nat2(aa_num_nat(pred_numeral,K),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),aa_num_nat(pred_numeral,K))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_8131_fact__prod__rev,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))) ).

% fact_prod_rev
tff(fact_8132_fact__prod__rev,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))) ).

% fact_prod_rev
tff(fact_8133_fact__prod__rev,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))) ).

% fact_prod_rev
tff(fact_8134_summable__Cauchy,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
    <=> ! [E3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E3)
         => ? [N8: nat] :
            ! [M5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),M5)
             => ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M5),N3)))),E3) ) ) ) ).

% summable_Cauchy
tff(fact_8135_summable__Cauchy,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
    <=> ! [E3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E3)
         => ? [N8: nat] :
            ! [M5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),M5)
             => ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M5),N3)))),E3) ) ) ) ).

% summable_Cauchy
tff(fact_8136_CauchyD,axiom,
    ! [X5: fun_nat_complex,E2: real] :
      ( topolo6517432010174082258omplex(X5)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
       => ? [M7: nat] :
          ! [M2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),M2)
           => ! [N5: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),N5)
               => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(X5,M2)),aa_nat_complex(X5,N5)))),E2) ) ) ) ) ).

% CauchyD
tff(fact_8137_CauchyD,axiom,
    ! [X5: fun_nat_real,E2: real] :
      ( topolo4055970368930404560y_real(X5)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
       => ? [M7: nat] :
          ! [M2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),M2)
           => ! [N5: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),N5)
               => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_nat_real(X5,M2)),aa_nat_real(X5,N5)))),E2) ) ) ) ) ).

% CauchyD
tff(fact_8138_CauchyI,axiom,
    ! [X5: fun_nat_complex] :
      ( ! [E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => ? [M9: nat] :
            ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),M)
             => ! [N: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),N)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(X5,M)),aa_nat_complex(X5,N)))),E) ) ) )
     => topolo6517432010174082258omplex(X5) ) ).

% CauchyI
tff(fact_8139_CauchyI,axiom,
    ! [X5: fun_nat_real] :
      ( ! [E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => ? [M9: nat] :
            ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),M)
             => ! [N: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),N)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_nat_real(X5,M)),aa_nat_real(X5,N)))),E) ) ) )
     => topolo4055970368930404560y_real(X5) ) ).

% CauchyI
tff(fact_8140_Cauchy__iff,axiom,
    ! [X5: fun_nat_complex] :
      ( topolo6517432010174082258omplex(X5)
    <=> ! [E3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E3)
         => ? [M8: nat] :
            ! [M5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),M5)
             => ! [N3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),N3)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(X5,M5)),aa_nat_complex(X5,N3)))),E3) ) ) ) ) ).

% Cauchy_iff
tff(fact_8141_Cauchy__iff,axiom,
    ! [X5: fun_nat_real] :
      ( topolo4055970368930404560y_real(X5)
    <=> ! [E3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E3)
         => ? [M8: nat] :
            ! [M5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),M5)
             => ! [N3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),N3)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_nat_real(X5,M5)),aa_nat_real(X5,N3)))),E3) ) ) ) ) ).

% Cauchy_iff
tff(fact_8142_sums__group,axiom,
    ! [F: fun_nat_nat,S: nat,K: nat] :
      ( aa_nat_o(sums_nat(F),S)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(sums_nat(aa_nat_fun_nat_nat(aTP_Lamm_uo(F),K)),S) ) ) ).

% sums_group
tff(fact_8143_sums__group,axiom,
    ! [F: fun_nat_real,S: real,K: nat] :
      ( aa_real_o(sums_real(F),S)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_up(F),K)),S) ) ) ).

% sums_group
tff(fact_8144_atLeast1__lessThan__eq__remove0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Nb),insert_nat2(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_8145_fact__split,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_complex(semiri5044797733671781792omplex,Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),Nb)))),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))) ) ) ).

% fact_split
tff(fact_8146_fact__split,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_Extended_enat(semiri4449623510593786356d_enat,Nb) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),Nb)))),aa_nat_Extended_enat(semiri4449623510593786356d_enat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))) ) ) ).

% fact_split
tff(fact_8147_fact__split,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),Nb)))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))) ) ) ).

% fact_split
tff(fact_8148_fact__split,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),Nb)))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))) ) ) ).

% fact_split
tff(fact_8149_fact__split,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),Nb)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))) ) ) ).

% fact_split
tff(fact_8150_binomial__altdef__of__nat,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_us(K),Nb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K)) ) ) ).

% binomial_altdef_of_nat
tff(fact_8151_gbinomial__altdef__of__nat,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),K) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_ut(A),K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K)) ).

% gbinomial_altdef_of_nat
tff(fact_8152_gbinomial__mult__fact,axiom,
    ! [K: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(gbinomial_complex(A),K)) = groups6464643781859351333omplex(aTP_Lamm_uu(A),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K)) ).

% gbinomial_mult_fact
tff(fact_8153_gbinomial__mult__fact,axiom,
    ! [K: nat,A: real] : aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(gbinomial_real(A),K)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_qo(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K)) ).

% gbinomial_mult_fact
tff(fact_8154_gbinomial__mult__fact_H,axiom,
    ! [A: complex,K: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_nat_complex(semiri5044797733671781792omplex,K)) = groups6464643781859351333omplex(aTP_Lamm_uu(A),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K)) ).

% gbinomial_mult_fact'
tff(fact_8155_gbinomial__mult__fact_H,axiom,
    ! [A: real,K: nat] : aa_real_real(times_times_real(aa_nat_real(gbinomial_real(A),K)),aa_nat_real(semiri2265585572941072030t_real,K)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_qo(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K)) ).

% gbinomial_mult_fact'
tff(fact_8156_gbinomial__prod__rev,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),K) = aa_real_real(divide_divide_real(aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_qo(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K))),aa_nat_real(semiri2265585572941072030t_real,K)) ).

% gbinomial_prod_rev
tff(fact_8157_gbinomial__prod__rev,axiom,
    ! [A: int,K: nat] : aa_nat_int(gbinomial_int(A),K) = aa_int_int(divide_divide_int(aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_qp(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K))),aa_nat_int(semiri1406184849735516958ct_int,K)) ).

% gbinomial_prod_rev
tff(fact_8158_gbinomial__prod__rev,axiom,
    ! [A: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qq,A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K))),aa_nat_nat(semiri1408675320244567234ct_nat,K)) ).

% gbinomial_prod_rev
tff(fact_8159_horner__sum__eq__sum,axiom,
    ! [F: fun_VEBT_VEBT_int,A: int,Xs: list_VEBT_VEBT] : groups6842663049115397189BT_int(F,A,Xs) = aa_set_nat_int(groups3539618377306564664at_int(aa_lis3633651700486656887at_int(aa_int4104624148011801308at_int(aTP_Lamm_uv(F),A),Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))) ).

% horner_sum_eq_sum
tff(fact_8160_horner__sum__eq__sum,axiom,
    ! [F: fun_int_int,A: int,Xs: list_int] : groups8485231416243008693nt_int(F,A,Xs) = aa_set_nat_int(groups3539618377306564664at_int(aa_lis1073296200319854579at_int(aa_int2572695527019912092at_int(aTP_Lamm_uw(F),A),Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_size_list_int(Xs))) ).

% horner_sum_eq_sum
tff(fact_8161_horner__sum__eq__sum,axiom,
    ! [F: fun_nat_int,A: int,Xs: list_nat] : groups7485877704341954137at_int(F,A,Xs) = aa_set_nat_int(groups3539618377306564664at_int(aa_lis6345223608758921111at_int(aa_int7844622935458978624at_int(aTP_Lamm_ux(F),A),Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_size_list_nat(Xs))) ).

% horner_sum_eq_sum
tff(fact_8162_horner__sum__eq__sum,axiom,
    ! [F: fun_VE5569070535936029429omplex,A: complex,Xs: list_VEBT_VEBT] : groups1931381680841367751omplex(F,A,Xs) = groups2073611262835488442omplex(aa_lis8713343626058569977omplex(aa_com9025189225015940188omplex(aTP_Lamm_uy(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))) ).

% horner_sum_eq_sum
tff(fact_8163_horner__sum__eq__sum,axiom,
    ! [F: fun_int_complex,A: complex,Xs: list_int] : groups1380173120649922871omplex(F,A,Xs) = groups2073611262835488442omplex(aa_lis226970529952281973omplex(aa_com4579324164465666976omplex(aTP_Lamm_uz(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_size_list_int(Xs))) ).

% horner_sum_eq_sum
tff(fact_8164_horner__sum__eq__sum,axiom,
    ! [F: fun_nat_complex,A: complex,Xs: list_nat] : groups404637655443745499omplex(F,A,Xs) = groups2073611262835488442omplex(aa_lis3451821876164259097omplex(aa_com7804175510677644100omplex(aTP_Lamm_va(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_size_list_nat(Xs))) ).

% horner_sum_eq_sum
tff(fact_8165_horner__sum__eq__sum,axiom,
    ! [F: fun_VE3569950477627547253d_enat,A: extended_enat,Xs: list_VEBT_VEBT] : groups3844460828123498381d_enat(F,A,Xs) = groups7108830773950497114d_enat(aa_lis7361820021430418801d_enat(aa_Ext4914522244092902178d_enat(aTP_Lamm_vb(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))) ).

% horner_sum_eq_sum
tff(fact_8166_horner__sum__eq__sum,axiom,
    ! [F: fun_in5914218760682845459d_enat,A: extended_enat,Xs: list_int] : groups7888997386326813469d_enat(F,A,Xs) = groups7108830773950497114d_enat(aa_lis3069066898809486325d_enat(aa_Ext5403380450472165346d_enat(aTP_Lamm_vc(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_size_list_int(Xs))) ).

% horner_sum_eq_sum
tff(fact_8167_horner__sum__eq__sum,axiom,
    ! [F: fun_na8797796813480665199d_enat,A: extended_enat,Xs: list_nat] : groups1549203402269857401d_enat(F,A,Xs) = groups7108830773950497114d_enat(aa_lis8229176623098646865d_enat(aa_Ext1340118137906550078d_enat(aTP_Lamm_vd(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_size_list_nat(Xs))) ).

% horner_sum_eq_sum
tff(fact_8168_horner__sum__eq__sum,axiom,
    ! [F: fun_VEBT_VEBT_nat,A: nat,Xs: list_VEBT_VEBT] : groups6845153519624447465BT_nat(F,A,Xs) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_lis7811502719995853595at_nat(aa_nat265247640139305436at_nat(aTP_Lamm_ve(F),A),Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))) ).

% horner_sum_eq_sum
tff(fact_8169_Chebyshev__sum__upper,axiom,
    ! [Nb: nat,A: fun_nat_int,B: fun_nat_int] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(A,I2)),aa_nat_int(A,J2)) ) )
     => ( ! [I2: nat,J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(B,J2)),aa_nat_int(B,I2)) ) )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_vf(A),B)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups3539618377306564664at_int(A),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))),aa_set_nat_int(groups3539618377306564664at_int(B),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb)))) ) ) ).

% Chebyshev_sum_upper
tff(fact_8170_Chebyshev__sum__upper,axiom,
    ! [Nb: nat,A: fun_nat_real,B: fun_nat_real] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,I2)),aa_nat_real(A,J2)) ) )
     => ( ! [I2: nat,J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(B,J2)),aa_nat_real(B,I2)) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_vg(A),B)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb)))),aa_real_real(times_times_real(aa_set_nat_real(groups6591440286371151544t_real(A),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))),aa_set_nat_real(groups6591440286371151544t_real(B),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb)))) ) ) ).

% Chebyshev_sum_upper
tff(fact_8171_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(A,I2)),aa_nat_nat(A,J2)) ) )
     => ( ! [I2: nat,J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(B,J2)),aa_nat_nat(B,I2)) ) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_vh(A),B)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(B),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_8172_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_8173_length__mul__elem,axiom,
    ! [Xs: list_list_VEBT_VEBT,Nb: nat] :
      ( ! [X4: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X4,set_list_VEBT_VEBT2(Xs))
         => ( size_s6755466524823107622T_VEBT(X4) = Nb ) )
     => ( size_s6755466524823107622T_VEBT(concat_VEBT_VEBT(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s8217280938318005548T_VEBT(Xs)),Nb) ) ) ).

% length_mul_elem
tff(fact_8174_length__mul__elem,axiom,
    ! [Xs: list_list_int,Nb: nat] :
      ( ! [X4: list_int] :
          ( member_list_int(X4,set_list_int2(Xs))
         => ( size_size_list_int(X4) = Nb ) )
     => ( size_size_list_int(concat_int(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s533118279054570080st_int(Xs)),Nb) ) ) ).

% length_mul_elem
tff(fact_8175_length__mul__elem,axiom,
    ! [Xs: list_list_nat,Nb: nat] :
      ( ! [X4: list_nat] :
          ( member_list_nat(X4,set_list_nat2(Xs))
         => ( size_size_list_nat(X4) = Nb ) )
     => ( size_size_list_nat(concat_nat(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s3023201423986296836st_nat(Xs)),Nb) ) ) ).

% length_mul_elem
tff(fact_8176_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_8177_set__n__lists,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT] : set_list_VEBT_VEBT2(n_lists_VEBT_VEBT(Nb,Xs)) = collec5608196760682091941T_VEBT(aa_lis764731566332388511VEBT_o(aTP_Lamm_vi(Nb),Xs)) ).

% set_n_lists
tff(fact_8178_set__n__lists,axiom,
    ! [Nb: nat,Xs: list_int] : set_list_int2(n_lists_int(Nb,Xs)) = collect_list_int(aa_lis7773489400162621755_int_o(aTP_Lamm_vj(Nb),Xs)) ).

% set_n_lists
tff(fact_8179_set__n__lists,axiom,
    ! [Nb: nat,Xs: list_nat] : set_list_nat2(n_lists_nat(Nb,Xs)) = collect_list_nat(aa_lis557685189327716851_nat_o(aTP_Lamm_vk(Nb),Xs)) ).

% set_n_lists
tff(fact_8180_pair__leqI2,axiom,
    ! [A: nat,B: nat,S: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,S),Ta)
       => member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),S)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B),Ta)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_8181_pair__leqI1,axiom,
    ! [A: nat,B: nat,S: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),S)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B),Ta)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_8182_case__nat__add__eq__if,axiom,
    ! [A: $o,F: fun_nat_o,V: num,Nb: nat] :
      ( aa_nat_o(case_nat_o((A),F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),Nb))
    <=> aa_nat_o(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(pred_numeral,V)),Nb)) ) ).

% case_nat_add_eq_if
tff(fact_8183_case__nat__add__eq__if,axiom,
    ! [A: nat,F: fun_nat_nat,V: num,Nb: nat] : aa_nat_nat(case_nat_nat(A,F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),Nb)) = aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(pred_numeral,V)),Nb)) ).

% case_nat_add_eq_if
tff(fact_8184_nat_Ocase__distrib,axiom,
    ! [H: fun_o_o,F1: $o,F22: fun_nat_o,Nat: nat] :
      ( aa_o_o(H,aa_nat_o(case_nat_o((F1),F22),Nat))
    <=> aa_nat_o(case_nat_o(aa_o_o(H,(F1)),aa_fun5937193402907943164_nat_o(aTP_Lamm_vl(H),F22)),Nat) ) ).

% nat.case_distrib
tff(fact_8185_nat_Ocase__distrib,axiom,
    ! [H: fun_o_nat,F1: $o,F22: fun_nat_o,Nat: nat] : aa_o_nat(H,aa_nat_o(case_nat_o((F1),F22),Nat)) = aa_nat_nat(case_nat_nat(aa_o_nat(H,(F1)),aa_fun4204860950123276802at_nat(aTP_Lamm_vm(H),F22)),Nat) ).

% nat.case_distrib
tff(fact_8186_nat_Ocase__distrib,axiom,
    ! [H: fun_nat_o,F1: nat,F22: fun_nat_nat,Nat: nat] :
      ( aa_nat_o(H,aa_nat_nat(case_nat_nat(F1,F22),Nat))
    <=> aa_nat_o(case_nat_o(aa_nat_o(H,F1),aa_fun8673969331427483654_nat_o(aTP_Lamm_vn(H),F22)),Nat) ) ).

% nat.case_distrib
tff(fact_8187_nat_Ocase__distrib,axiom,
    ! [H: fun_nat_nat,F1: nat,F22: fun_nat_nat,Nat: nat] : aa_nat_nat(H,aa_nat_nat(case_nat_nat(F1,F22),Nat)) = aa_nat_nat(case_nat_nat(aa_nat_nat(H,F1),aa_fun5791449513213591928at_nat(aTP_Lamm_vo(H),F22)),Nat) ).

% nat.case_distrib
tff(fact_8188_old_Onat_Osimps_I5_J,axiom,
    ! [F1: $o,F22: fun_nat_o,X22: nat] :
      ( aa_nat_o(case_nat_o((F1),F22),aa_nat_nat(suc,X22))
    <=> aa_nat_o(F22,X22) ) ).

% old.nat.simps(5)
tff(fact_8189_old_Onat_Osimps_I5_J,axiom,
    ! [F1: nat,F22: fun_nat_nat,X22: nat] : aa_nat_nat(case_nat_nat(F1,F22),aa_nat_nat(suc,X22)) = aa_nat_nat(F22,X22) ).

% old.nat.simps(5)
tff(fact_8190_old_Onat_Osimps_I4_J,axiom,
    ! [F1: $o,F22: fun_nat_o] :
      ( aa_nat_o(case_nat_o((F1),F22),zero_zero_nat)
    <=> (F1) ) ).

% old.nat.simps(4)
tff(fact_8191_old_Onat_Osimps_I4_J,axiom,
    ! [F1: nat,F22: fun_nat_nat] : aa_nat_nat(case_nat_nat(F1,F22),zero_zero_nat) = F1 ).

% old.nat.simps(4)
tff(fact_8192_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_vp),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_8193_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_av),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_8194_less__eq__nat_Osimps_I2_J,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,Mb)),Nb) ) ).

% less_eq_nat.simps(2)
tff(fact_8195_max__Suc1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),Mb) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_vq,Nb)),Mb) ).

% max_Suc1
tff(fact_8196_max__Suc2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_vr,Nb)),Mb) ).

% max_Suc2
tff(fact_8197_length__n__lists__elem,axiom,
    ! [Ys: list_VEBT_VEBT,Nb: nat,Xs: list_VEBT_VEBT] :
      ( member2936631157270082147T_VEBT(Ys,set_list_VEBT_VEBT2(n_lists_VEBT_VEBT(Nb,Xs)))
     => ( size_s6755466524823107622T_VEBT(Ys) = Nb ) ) ).

% length_n_lists_elem
tff(fact_8198_length__n__lists__elem,axiom,
    ! [Ys: list_int,Nb: nat,Xs: list_int] :
      ( member_list_int(Ys,set_list_int2(n_lists_int(Nb,Xs)))
     => ( size_size_list_int(Ys) = Nb ) ) ).

% length_n_lists_elem
tff(fact_8199_length__n__lists__elem,axiom,
    ! [Ys: list_nat,Nb: nat,Xs: list_nat] :
      ( member_list_nat(Ys,set_list_nat2(n_lists_nat(Nb,Xs)))
     => ( size_size_list_nat(Ys) = Nb ) ) ).

% length_n_lists_elem
tff(fact_8200_diff__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_bc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ).

% diff_Suc
tff(fact_8201_Nitpick_Ocase__nat__unfold,axiom,
    ! [Xb: $o,F: fun_nat_o,Nb: nat] :
      ( aa_nat_o(case_nat_o((Xb),F),Nb)
    <=> $ite(Nb = zero_zero_nat,(Xb),aa_nat_o(F,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ).

% Nitpick.case_nat_unfold
tff(fact_8202_Nitpick_Ocase__nat__unfold,axiom,
    ! [Xb: nat,F: fun_nat_nat,Nb: nat] :
      aa_nat_nat(case_nat_nat(Xb,F),Nb) = $ite(Nb = zero_zero_nat,Xb,aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ).

% Nitpick.case_nat_unfold
tff(fact_8203_nat_Osplit__sels_I1_J,axiom,
    ! [P: fun_o_o,F1: $o,F22: fun_nat_o,Nat: nat] :
      ( aa_o_o(P,aa_nat_o(case_nat_o((F1),F22),Nat))
    <=> ( ( ( Nat = zero_zero_nat )
         => aa_o_o(P,(F1)) )
        & ( ( Nat = aa_nat_nat(suc,aa_nat_nat(pred,Nat)) )
         => aa_o_o(P,aa_nat_o(F22,aa_nat_nat(pred,Nat))) ) ) ) ).

% nat.split_sels(1)
tff(fact_8204_nat_Osplit__sels_I1_J,axiom,
    ! [P: fun_nat_o,F1: nat,F22: fun_nat_nat,Nat: nat] :
      ( aa_nat_o(P,aa_nat_nat(case_nat_nat(F1,F22),Nat))
    <=> ( ( ( Nat = zero_zero_nat )
         => aa_nat_o(P,F1) )
        & ( ( Nat = aa_nat_nat(suc,aa_nat_nat(pred,Nat)) )
         => aa_nat_o(P,aa_nat_nat(F22,aa_nat_nat(pred,Nat))) ) ) ) ).

% nat.split_sels(1)
tff(fact_8205_nat_Osplit__sels_I2_J,axiom,
    ! [P: fun_o_o,F1: $o,F22: fun_nat_o,Nat: nat] :
      ( aa_o_o(P,aa_nat_o(case_nat_o((F1),F22),Nat))
    <=> ~ ( ( ( Nat = zero_zero_nat )
            & ~ aa_o_o(P,(F1)) )
          | ( ( Nat = aa_nat_nat(suc,aa_nat_nat(pred,Nat)) )
            & ~ aa_o_o(P,aa_nat_o(F22,aa_nat_nat(pred,Nat))) ) ) ) ).

% nat.split_sels(2)
tff(fact_8206_nat_Osplit__sels_I2_J,axiom,
    ! [P: fun_nat_o,F1: nat,F22: fun_nat_nat,Nat: nat] :
      ( aa_nat_o(P,aa_nat_nat(case_nat_nat(F1,F22),Nat))
    <=> ~ ( ( ( Nat = zero_zero_nat )
            & ~ aa_nat_o(P,F1) )
          | ( ( Nat = aa_nat_nat(suc,aa_nat_nat(pred,Nat)) )
            & ~ aa_nat_o(P,aa_nat_nat(F22,aa_nat_nat(pred,Nat))) ) ) ) ).

% nat.split_sels(2)
tff(fact_8207_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_bc),Nat) ).

% pred_def
tff(fact_8208_subset__CollectI,axiom,
    ! [B2: set_o,A2: set_o,Q: fun_o_o,P: fun_o_o] :
      ( aa_set_o_o(ord_less_eq_set_o(B2),A2)
     => ( ! [X4: $o] :
            ( aa_set_o_o(member_o2((X4)),B2)
           => ( aa_o_o(Q,(X4))
             => aa_o_o(P,(X4)) ) )
       => aa_set_o_o(ord_less_eq_set_o(collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_al(B2),Q))),collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_al(A2),P))) ) ) ).

% subset_CollectI
tff(fact_8209_subset__CollectI,axiom,
    ! [B2: set_real,A2: set_real,Q: fun_real_o,P: fun_real_o] :
      ( aa_set_real_o(ord_less_eq_set_real(B2),A2)
     => ( ! [X4: real] :
            ( aa_set_real_o(member_real2(X4),B2)
           => ( aa_real_o(Q,X4)
             => aa_real_o(P,X4) ) )
       => aa_set_real_o(ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_am(B2),Q))),collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_am(A2),P))) ) ) ).

% subset_CollectI
tff(fact_8210_subset__CollectI,axiom,
    ! [B2: set_list_nat,A2: set_list_nat,Q: fun_list_nat_o,P: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(B2,A2)
     => ( ! [X4: list_nat] :
            ( member_list_nat(X4,B2)
           => ( aa_list_nat_o(Q,X4)
             => aa_list_nat_o(P,X4) ) )
       => ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_an(B2),Q)),collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_an(A2),P))) ) ) ).

% subset_CollectI
tff(fact_8211_subset__CollectI,axiom,
    ! [B2: set_set_nat,A2: set_set_nat,Q: fun_set_nat_o,P: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B2),A2)
     => ( ! [X4: set_nat] :
            ( aa_set_set_nat_o(member_set_nat2(X4),B2)
           => ( aa_set_nat_o(Q,X4)
             => aa_set_nat_o(P,X4) ) )
       => aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ao(B2),Q))),collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ao(A2),P))) ) ) ).

% subset_CollectI
tff(fact_8212_subset__CollectI,axiom,
    ! [B2: set_int,A2: set_int,Q: fun_int_o,P: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A2)
     => ( ! [X4: int] :
            ( aa_set_int_o(member_int2(X4),B2)
           => ( aa_int_o(Q,X4)
             => aa_int_o(P,X4) ) )
       => aa_set_int_o(ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ap(B2),Q))),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ap(A2),P))) ) ) ).

% subset_CollectI
tff(fact_8213_subset__CollectI,axiom,
    ! [B2: set_nat,A2: set_nat,Q: fun_nat_o,P: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat2(X4),B2)
           => ( aa_nat_o(Q,X4)
             => aa_nat_o(P,X4) ) )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aq(B2),Q))),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aq(A2),P))) ) ) ).

% subset_CollectI
tff(fact_8214_subset__Collect__iff,axiom,
    ! [B2: set_o,A2: set_o,P: fun_o_o] :
      ( aa_set_o_o(ord_less_eq_set_o(B2),A2)
     => ( aa_set_o_o(ord_less_eq_set_o(B2),collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_al(A2),P)))
      <=> ! [X3: $o] :
            ( aa_set_o_o(member_o2((X3)),B2)
           => aa_o_o(P,(X3)) ) ) ) ).

% subset_Collect_iff
tff(fact_8215_subset__Collect__iff,axiom,
    ! [B2: set_real,A2: set_real,P: fun_real_o] :
      ( aa_set_real_o(ord_less_eq_set_real(B2),A2)
     => ( aa_set_real_o(ord_less_eq_set_real(B2),collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_am(A2),P)))
      <=> ! [X3: real] :
            ( aa_set_real_o(member_real2(X3),B2)
           => aa_real_o(P,X3) ) ) ) ).

% subset_Collect_iff
tff(fact_8216_subset__Collect__iff,axiom,
    ! [B2: set_list_nat,A2: set_list_nat,P: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(B2,A2)
     => ( ord_le6045566169113846134st_nat(B2,collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_an(A2),P)))
      <=> ! [X3: list_nat] :
            ( member_list_nat(X3,B2)
           => aa_list_nat_o(P,X3) ) ) ) ).

% subset_Collect_iff
tff(fact_8217_subset__Collect__iff,axiom,
    ! [B2: set_set_nat,A2: set_set_nat,P: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B2),A2)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B2),collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ao(A2),P)))
      <=> ! [X3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat2(X3),B2)
           => aa_set_nat_o(P,X3) ) ) ) ).

% subset_Collect_iff
tff(fact_8218_subset__Collect__iff,axiom,
    ! [B2: set_int,A2: set_int,P: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A2)
     => ( aa_set_int_o(ord_less_eq_set_int(B2),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ap(A2),P)))
      <=> ! [X3: int] :
            ( aa_set_int_o(member_int2(X3),B2)
           => aa_int_o(P,X3) ) ) ) ).

% subset_Collect_iff
tff(fact_8219_subset__Collect__iff,axiom,
    ! [B2: set_nat,A2: set_nat,P: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aq(A2),P)))
      <=> ! [X3: nat] :
            ( aa_set_nat_o(member_nat2(X3),B2)
           => aa_nat_o(P,X3) ) ) ) ).

% subset_Collect_iff
tff(fact_8220_signed__take__bit__code,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(Nb),A) = $let(
        l: int,
        l:= aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),A),
        $ite(aa_nat_o(bit_se1146084159140164899it_int(l),Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,l),aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,one_one_int))),l) ) ).

% signed_take_bit_code
tff(fact_8221_bezw__0,axiom,
    ! [Xb: nat] : bezw(Xb,zero_zero_nat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ).

% bezw_0
tff(fact_8222_prod__decode__aux_Osimps,axiom,
    ! [K: nat,Mb: nat] :
      aa_nat7603221163087113075at_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(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),Mb)),aa_nat7603221163087113075at_nat(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_8223_prod__decode__aux_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(Xb),Xaa) = Y )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xb),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Xaa)),aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xb)))) ) ) ).

% prod_decode_aux.elims
tff(fact_8224_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_8225_push__bit__of__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),zero_zero_nat) = zero_zero_nat ).

% push_bit_of_0
tff(fact_8226_push__bit__of__0,axiom,
    ! [Nb: nat] : aa_int_int(bit_se545348938243370406it_int(Nb),zero_zero_int) = zero_zero_int ).

% push_bit_of_0
tff(fact_8227_push__bit__eq__0__iff,axiom,
    ! [Nb: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),A) = zero_zero_nat )
    <=> ( A = zero_zero_nat ) ) ).

% push_bit_eq_0_iff
tff(fact_8228_push__bit__eq__0__iff,axiom,
    ! [Nb: nat,A: int] :
      ( ( aa_int_int(bit_se545348938243370406it_int(Nb),A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% push_bit_eq_0_iff
tff(fact_8229_push__bit__push__bit,axiom,
    ! [Mb: nat,Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),A) ).

% push_bit_push_bit
tff(fact_8230_push__bit__push__bit,axiom,
    ! [Mb: nat,Nb: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(Mb),aa_int_int(bit_se545348938243370406it_int(Nb),A)) = aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),A) ).

% push_bit_push_bit
tff(fact_8231_push__bit__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_num_nat(numeral_numeral_nat,bit0(K))) ).

% push_bit_Suc_numeral
tff(fact_8232_push__bit__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(suc,Nb)),numeral_numeral_int(K)) = aa_int_int(bit_se545348938243370406it_int(Nb),numeral_numeral_int(bit0(K))) ).

% push_bit_Suc_numeral
tff(fact_8233_push__bit__Suc__minus__numeral,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,numeral_numeral_int(K))) = aa_int_int(bit_se545348938243370406it_int(Nb),aa_int_int(uminus_uminus_int,numeral_numeral_int(bit0(K)))) ).

% push_bit_Suc_minus_numeral
tff(fact_8234_push__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% push_bit_Suc
tff(fact_8235_push__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(bit_se545348938243370406it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,A),numeral_numeral_int(bit0(one)))) ).

% push_bit_Suc
tff(fact_8236_push__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb) ).

% push_bit_of_Suc_0
tff(fact_8237_even__push__bit__iff,axiom,
    ! [Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),A))
    <=> ( ( Nb != zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A) ) ) ).

% even_push_bit_iff
tff(fact_8238_even__push__bit__iff,axiom,
    ! [Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se545348938243370406it_int(Nb),A))
    <=> ( ( Nb != zero_zero_nat )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A) ) ) ).

% even_push_bit_iff
tff(fact_8239_push__bit__add,axiom,
    ! [Nb: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),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(bit_se547839408752420682it_nat,Nb),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),B)) ).

% push_bit_add
tff(fact_8240_push__bit__add,axiom,
    ! [Nb: nat,A: int,B: int] : aa_int_int(bit_se545348938243370406it_int(Nb),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(bit_se545348938243370406it_int(Nb),A)),aa_int_int(bit_se545348938243370406it_int(Nb),B)) ).

% push_bit_add
tff(fact_8241_fst__conv,axiom,
    ! [X1: product_prod_nat_nat,X22: product_prod_nat_nat] : produc3213797794245857475at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,X1),X22)) = X1 ).

% fst_conv
tff(fact_8242_fst__conv,axiom,
    ! [X1: vEBT_VEBT,X22: nat] : produc8713918199166443969BT_nat(aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,X1),X22)) = X1 ).

% fst_conv
tff(fact_8243_fst__conv,axiom,
    ! [X1: vEBT_VEBT,X22: extended_enat] : produc967593531271825845d_enat(aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,X1),X22)) = X1 ).

% fst_conv
tff(fact_8244_fst__conv,axiom,
    ! [X1: nat,X22: nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X22)) = X1 ).

% fst_conv
tff(fact_8245_fst__conv,axiom,
    ! [X1: int,X22: int] : product_fst_int_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22)) = X1 ).

% fst_conv
tff(fact_8246_fst__eqD,axiom,
    ! [Xb: product_prod_nat_nat,Y: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( produc3213797794245857475at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,Xb),Y)) = A )
     => ( Xb = A ) ) ).

% fst_eqD
tff(fact_8247_fst__eqD,axiom,
    ! [Xb: vEBT_VEBT,Y: nat,A: vEBT_VEBT] :
      ( ( produc8713918199166443969BT_nat(aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Y)) = A )
     => ( Xb = A ) ) ).

% fst_eqD
tff(fact_8248_fst__eqD,axiom,
    ! [Xb: vEBT_VEBT,Y: extended_enat,A: vEBT_VEBT] :
      ( ( produc967593531271825845d_enat(aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,Xb),Y)) = A )
     => ( Xb = A ) ) ).

% fst_eqD
tff(fact_8249_fst__eqD,axiom,
    ! [Xb: nat,Y: nat,A: nat] :
      ( ( aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Y)) = A )
     => ( Xb = A ) ) ).

% fst_eqD
tff(fact_8250_fst__eqD,axiom,
    ! [Xb: int,Y: int,A: int] :
      ( ( product_fst_int_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Y)) = A )
     => ( Xb = A ) ) ).

% fst_eqD
tff(fact_8251_push__bit__take__bit,axiom,
    ! [Mb: nat,Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),A)) ).

% push_bit_take_bit
tff(fact_8252_push__bit__take__bit,axiom,
    ! [Mb: nat,Nb: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A)) = aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),aa_int_int(bit_se545348938243370406it_int(Mb),A)) ).

% push_bit_take_bit
tff(fact_8253_bit__push__bit__iff__int,axiom,
    ! [Mb: nat,K: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se545348938243370406it_int(Mb),K)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & aa_nat_o(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% bit_push_bit_iff_int
tff(fact_8254_bit__push__bit__iff__nat,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),Q2)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_8255_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),one_one_nat)) != zero_zero_nat ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_8256_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
    <=> ( aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int)) != zero_zero_int ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_8257_take__bit__sum,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vs,A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb)) ).

% take_bit_sum
tff(fact_8258_take__bit__sum,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),A) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_vt(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb)) ).

% take_bit_sum
tff(fact_8259_in__set__enumerate__eq,axiom,
    ! [P2: produc8025551001238799321T_VEBT,Nb: nat,Xs: list_VEBT_VEBT] :
      ( member8549952807677709168T_VEBT(P2,set_Pr5984661752051438084T_VEBT(enumerate_VEBT_VEBT(Nb,Xs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),produc8575180428842422559T_VEBT(P2))
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,produc8575180428842422559T_VEBT(P2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_s6755466524823107622T_VEBT(Xs)),Nb))
        & ( nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,produc8575180428842422559T_VEBT(P2)),Nb)) = produc8172668247895388509T_VEBT(P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_8260_in__set__enumerate__eq,axiom,
    ! [P2: product_prod_nat_int,Nb: nat,Xs: list_int] :
      ( member4262671552274231302at_int(P2,set_Pr1470767568048878706at_int(enumerate_int(Nb,Xs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),product_fst_nat_int(P2))
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,product_fst_nat_int(P2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_int(Xs)),Nb))
        & ( aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,product_fst_nat_int(P2)),Nb)) = product_snd_nat_int(P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_8261_in__set__enumerate__eq,axiom,
    ! [P2: product_prod_nat_nat,Nb: nat,Xs: list_nat] :
      ( member8440522571783428010at_nat(P2,set_Pr5648618587558075414at_nat(enumerate_nat(Nb,Xs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2))
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_nat(Xs)),Nb))
        & ( aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2)),Nb)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_8262_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_8263_nth__rotate1,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(rotate1_VEBT_VEBT(Xs),Nb) = nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Nb)),size_s6755466524823107622T_VEBT(Xs))) ) ) ).

% nth_rotate1
tff(fact_8264_nth__rotate1,axiom,
    ! [Nb: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(rotate1_int(Xs)),Nb) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Nb)),size_size_list_int(Xs))) ) ) ).

% nth_rotate1
tff(fact_8265_nth__rotate1,axiom,
    ! [Nb: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(rotate1_nat(Xs)),Nb) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Nb)),size_size_list_nat(Xs))) ) ) ).

% nth_rotate1
tff(fact_8266_set__update__distinct,axiom,
    ! [Xs: list_VEBT_VEBT,Nb: nat,Xb: vEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
       => ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,Nb,Xb)) = insert_VEBT_VEBT2(Xb,minus_5127226145743854075T_VEBT(set_VEBT_VEBT2(Xs),insert_VEBT_VEBT2(nth_VEBT_VEBT(Xs,Nb),bot_bo8194388402131092736T_VEBT))) ) ) ) ).

% set_update_distinct
tff(fact_8267_set__update__distinct,axiom,
    ! [Xs: list_real,Nb: nat,Xb: real] :
      ( distinct_real(Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
       => ( set_real2(list_update_real(Xs,Nb,Xb)) = insert_real2(Xb,minus_minus_set_real(set_real2(Xs),insert_real2(aa_nat_real(nth_real(Xs),Nb),bot_bot_set_real))) ) ) ) ).

% set_update_distinct
tff(fact_8268_set__update__distinct,axiom,
    ! [Xs: list_o,Nb: nat,Xb: $o] :
      ( distinct_o(Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
       => ( set_o2(list_update_o(Xs,Nb,(Xb))) = insert_o2((Xb),minus_minus_set_o(set_o2(Xs),insert_o2(aa_nat_o(nth_o(Xs),Nb),bot_bot_set_o))) ) ) ) ).

% set_update_distinct
tff(fact_8269_set__update__distinct,axiom,
    ! [Xs: list_int,Nb: nat,Xb: int] :
      ( aa_list_int_o(distinct_int,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
       => ( set_int2(list_update_int(Xs,Nb,Xb)) = insert_int2(Xb,minus_minus_set_int(set_int2(Xs),insert_int2(aa_nat_int(nth_int(Xs),Nb),bot_bot_set_int))) ) ) ) ).

% set_update_distinct
tff(fact_8270_set__update__distinct,axiom,
    ! [Xs: list_nat,Nb: nat,Xb: nat] :
      ( aa_list_nat_o(distinct_nat,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
       => ( set_nat2(list_update_nat(Xs,Nb,Xb)) = insert_nat2(Xb,minus_minus_set_nat(set_nat2(Xs),insert_nat2(aa_nat_nat(nth_nat(Xs),Nb),bot_bot_set_nat))) ) ) ) ).

% set_update_distinct
tff(fact_8271_drop__bit__rec,axiom,
    ! [Nb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),A) = $ite(Nb = zero_zero_nat,A,aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% drop_bit_rec
tff(fact_8272_drop__bit__rec,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(bit_se8568078237143864401it_int(Nb),A) = $ite(Nb = zero_zero_nat,A,aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one))))) ).

% drop_bit_rec
tff(fact_8273_drop__bit__of__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),zero_zero_nat) = zero_zero_nat ).

% drop_bit_of_0
tff(fact_8274_drop__bit__of__0,axiom,
    ! [Nb: nat] : aa_int_int(bit_se8568078237143864401it_int(Nb),zero_zero_int) = zero_zero_int ).

% drop_bit_of_0
tff(fact_8275_drop__bit__drop__bit,axiom,
    ! [Mb: nat,Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),A) ).

% drop_bit_drop_bit
tff(fact_8276_drop__bit__drop__bit,axiom,
    ! [Mb: nat,Nb: nat,A: int] : aa_int_int(bit_se8568078237143864401it_int(Mb),aa_int_int(bit_se8568078237143864401it_int(Nb),A)) = aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),A) ).

% drop_bit_drop_bit
tff(fact_8277_set__rotate1,axiom,
    ! [Xs: list_VEBT_VEBT] : set_VEBT_VEBT2(rotate1_VEBT_VEBT(Xs)) = set_VEBT_VEBT2(Xs) ).

% set_rotate1
tff(fact_8278_set__rotate1,axiom,
    ! [Xs: list_int] : set_int2(rotate1_int(Xs)) = set_int2(Xs) ).

% set_rotate1
tff(fact_8279_set__rotate1,axiom,
    ! [Xs: list_nat] : set_nat2(rotate1_nat(Xs)) = set_nat2(Xs) ).

% set_rotate1
tff(fact_8280_length__rotate1,axiom,
    ! [Xs: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(rotate1_VEBT_VEBT(Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_rotate1
tff(fact_8281_length__rotate1,axiom,
    ! [Xs: list_int] : size_size_list_int(rotate1_int(Xs)) = size_size_list_int(Xs) ).

% length_rotate1
tff(fact_8282_length__rotate1,axiom,
    ! [Xs: list_nat] : size_size_list_nat(rotate1_nat(Xs)) = size_size_list_nat(Xs) ).

% length_rotate1
tff(fact_8283_distinct1__rotate,axiom,
    ! [Xs: list_int] :
      ( aa_list_int_o(distinct_int,rotate1_int(Xs))
    <=> aa_list_int_o(distinct_int,Xs) ) ).

% distinct1_rotate
tff(fact_8284_distinct1__rotate,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(distinct_nat,rotate1_nat(Xs))
    <=> aa_list_nat_o(distinct_nat,Xs) ) ).

% distinct1_rotate
tff(fact_8285_distinct__insert,axiom,
    ! [Xb: int,Xs: list_int] :
      ( aa_list_int_o(distinct_int,insert_int(Xb,Xs))
    <=> aa_list_int_o(distinct_int,Xs) ) ).

% distinct_insert
tff(fact_8286_distinct__insert,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( aa_list_nat_o(distinct_nat,insert_nat(Xb,Xs))
    <=> aa_list_nat_o(distinct_nat,Xs) ) ).

% distinct_insert
tff(fact_8287_prod_Ocollapse,axiom,
    ! [Prod: produc859450856879609959at_nat] : aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,produc3213797794245857475at_nat(Prod)),produc6408287024330202629at_nat(Prod)) = Prod ).

% prod.collapse
tff(fact_8288_prod_Ocollapse,axiom,
    ! [Prod: produc9072475918466114483BT_nat] : aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,produc8713918199166443969BT_nat(Prod)),produc8311406018219409919BT_nat(Prod)) = Prod ).

% prod.collapse
tff(fact_8289_prod_Ocollapse,axiom,
    ! [Prod: produc7272778201969148633d_enat] : aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,produc967593531271825845d_enat(Prod)),produc522021263368680183d_enat(Prod)) = Prod ).

% prod.collapse
tff(fact_8290_prod_Ocollapse,axiom,
    ! [Prod: product_prod_nat_nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) = Prod ).

% prod.collapse
tff(fact_8291_prod_Ocollapse,axiom,
    ! [Prod: product_prod_int_int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_fst_int_int(Prod)),product_snd_int_int(Prod)) = Prod ).

% prod.collapse
tff(fact_8292_drop__bit__of__bool,axiom,
    ! [Nb: nat,B: $o] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,(B))) = aa_o_nat(zero_n2687167440665602831ol_nat,
        ( ( Nb = zero_zero_nat )
        & (B) )) ).

% drop_bit_of_bool
tff(fact_8293_drop__bit__of__bool,axiom,
    ! [Nb: nat,B: $o] :
      aa_int_int(bit_se8568078237143864401it_int(Nb),aa_o_int(zero_n2684676970156552555ol_int,(B))) = aa_o_int(zero_n2684676970156552555ol_int,
        ( ( Nb = zero_zero_nat )
        & (B) )) ).

% drop_bit_of_bool
tff(fact_8294_drop__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% drop_bit_of_Suc_0
tff(fact_8295_drop__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,numeral_numeral_int(bit0(K)))) = aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,numeral_numeral_int(K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_8296_drop__bit__Suc__bit0,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,bit0(K))) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_num_nat(numeral_numeral_nat,K)) ).

% drop_bit_Suc_bit0
tff(fact_8297_drop__bit__Suc__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),numeral_numeral_int(bit0(K))) = aa_int_int(bit_se8568078237143864401it_int(Nb),numeral_numeral_int(K)) ).

% drop_bit_Suc_bit0
tff(fact_8298_drop__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_num_nat(numeral_numeral_nat,K)) ).

% drop_bit_Suc_bit1
tff(fact_8299_drop__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),numeral_numeral_int(aa_num_num(bit1,K))) = aa_int_int(bit_se8568078237143864401it_int(Nb),numeral_numeral_int(K)) ).

% drop_bit_Suc_bit1
tff(fact_8300_drop__bit__of__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),one_one_nat) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% drop_bit_of_1
tff(fact_8301_drop__bit__of__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_se8568078237143864401it_int(Nb),one_one_int) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% drop_bit_of_1
tff(fact_8302_rotate1__length01,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),one_one_nat)
     => ( rotate1_VEBT_VEBT(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_8303_rotate1__length01,axiom,
    ! [Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),one_one_nat)
     => ( rotate1_int(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_8304_rotate1__length01,axiom,
    ! [Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),one_one_nat)
     => ( rotate1_nat(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_8305_distinct__swap,axiom,
    ! [Ia: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,Ia,nth_VEBT_VEBT(Xs,J)),J,nth_VEBT_VEBT(Xs,Ia)))
        <=> aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Xs) ) ) ) ).

% distinct_swap
tff(fact_8306_distinct__swap,axiom,
    ! [Ia: nat,Xs: list_int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(Xs))
       => ( aa_list_int_o(distinct_int,list_update_int(list_update_int(Xs,Ia,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),Ia)))
        <=> aa_list_int_o(distinct_int,Xs) ) ) ) ).

% distinct_swap
tff(fact_8307_distinct__swap,axiom,
    ! [Ia: nat,Xs: list_nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_nat(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_nat(Xs))
       => ( aa_list_nat_o(distinct_nat,list_update_nat(list_update_nat(Xs,Ia,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),Ia)))
        <=> aa_list_nat_o(distinct_nat,Xs) ) ) ) ).

% distinct_swap
tff(fact_8308_drop__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,numeral_numeral_int(aa_num_num(bit1,K)))) = aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,numeral_numeral_int(inc(K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_8309_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_vu(A2),Nb))) ) ).

% finite_lists_distinct_length_eq
tff(fact_8310_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_Extended_enat,Nb: nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => finite1862508098717546133d_enat(collec8433460942617342167d_enat(aa_nat4163650157883470461enat_o(aTP_Lamm_vv(A2),Nb))) ) ).

% finite_lists_distinct_length_eq
tff(fact_8311_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,Nb: nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_vw(A2),Nb))) ) ).

% finite_lists_distinct_length_eq
tff(fact_8312_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_int,Nb: nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_vx(A2),Nb))) ) ).

% finite_lists_distinct_length_eq
tff(fact_8313_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_vy(A2),Nb))) ) ).

% finite_lists_distinct_length_eq
tff(fact_8314_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [Xs: list_int] :
      ( aa_list_int_o(distinct_int,Xs)
     => aa_list_int_o(distinct_int,Xs) ) ).

% sorted_list_of_set.distinct_if_distinct_map
tff(fact_8315_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(distinct_nat,Xs)
     => aa_list_nat_o(distinct_nat,Xs) ) ).

% sorted_list_of_set.distinct_if_distinct_map
tff(fact_8316_distinct__product,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( aa_list_int_o(distinct_int,Xs)
     => ( aa_list_int_o(distinct_int,Ys)
       => distin3744728255968310194nt_int(product_int_int(Xs,Ys)) ) ) ).

% distinct_product
tff(fact_8317_distinct__product,axiom,
    ! [Xs: list_int,Ys: list_nat] :
      ( aa_list_int_o(distinct_int,Xs)
     => ( aa_list_nat_o(distinct_nat,Ys)
       => distin7922579275477506902nt_nat(product_int_nat(Xs,Ys)) ) ) ).

% distinct_product
tff(fact_8318_distinct__product,axiom,
    ! [Xs: list_nat,Ys: list_int] :
      ( aa_list_nat_o(distinct_nat,Xs)
     => ( aa_list_int_o(distinct_int,Ys)
       => distin2745374544067255638at_int(product_nat_int(Xs,Ys)) ) ) ).

% distinct_product
tff(fact_8319_distinct__product,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( aa_list_nat_o(distinct_nat,Xs)
     => ( aa_list_nat_o(distinct_nat,Ys)
       => distin6923225563576452346at_nat(product_nat_nat(Xs,Ys)) ) ) ).

% distinct_product
tff(fact_8320_snd__eqD,axiom,
    ! [Xb: product_prod_nat_nat,Y: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( produc6408287024330202629at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,Xb),Y)) = A )
     => ( Y = A ) ) ).

% snd_eqD
tff(fact_8321_snd__eqD,axiom,
    ! [Xb: vEBT_VEBT,Y: nat,A: nat] :
      ( ( produc8311406018219409919BT_nat(aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Y)) = A )
     => ( Y = A ) ) ).

% snd_eqD
tff(fact_8322_snd__eqD,axiom,
    ! [Xb: vEBT_VEBT,Y: extended_enat,A: extended_enat] :
      ( ( produc522021263368680183d_enat(aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,Xb),Y)) = A )
     => ( Y = A ) ) ).

% snd_eqD
tff(fact_8323_snd__eqD,axiom,
    ! [Xb: nat,Y: nat,A: nat] :
      ( ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Y)) = A )
     => ( Y = A ) ) ).

% snd_eqD
tff(fact_8324_snd__eqD,axiom,
    ! [Xb: int,Y: int,A: int] :
      ( ( product_snd_int_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Y)) = A )
     => ( Y = A ) ) ).

% snd_eqD
tff(fact_8325_snd__conv,axiom,
    ! [X1: product_prod_nat_nat,X22: product_prod_nat_nat] : produc6408287024330202629at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,X1),X22)) = X22 ).

% snd_conv
tff(fact_8326_snd__conv,axiom,
    ! [X1: vEBT_VEBT,X22: nat] : produc8311406018219409919BT_nat(aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,X1),X22)) = X22 ).

% snd_conv
tff(fact_8327_snd__conv,axiom,
    ! [X1: vEBT_VEBT,X22: extended_enat] : produc522021263368680183d_enat(aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,X1),X22)) = X22 ).

% snd_conv
tff(fact_8328_snd__conv,axiom,
    ! [X1: nat,X22: nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X22)) = X22 ).

% snd_conv
tff(fact_8329_snd__conv,axiom,
    ! [X1: int,X22: int] : product_snd_int_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22)) = X22 ).

% snd_conv
tff(fact_8330_finite__distinct__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A2)
     => ? [Xs2: list_VEBT_VEBT] :
          ( ( set_VEBT_VEBT2(Xs2) = A2 )
          & aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Xs2) ) ) ).

% finite_distinct_list
tff(fact_8331_finite__distinct__list,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ? [Xs2: list_nat] :
          ( ( set_nat2(Xs2) = A2 )
          & aa_list_nat_o(distinct_nat,Xs2) ) ) ).

% finite_distinct_list
tff(fact_8332_finite__distinct__list,axiom,
    ! [A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ? [Xs2: list_complex] :
          ( ( set_complex2(Xs2) = A2 )
          & distinct_complex(Xs2) ) ) ).

% finite_distinct_list
tff(fact_8333_finite__distinct__list,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ? [Xs2: list_int] :
          ( ( set_int2(Xs2) = A2 )
          & aa_list_int_o(distinct_int,Xs2) ) ) ).

% finite_distinct_list
tff(fact_8334_finite__distinct__list,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ? [Xs2: list_Extended_enat] :
          ( ( set_Extended_enat2(Xs2) = A2 )
          & distin4523846830085650399d_enat(Xs2) ) ) ).

% finite_distinct_list
tff(fact_8335_prod_Oexhaust__sel,axiom,
    ! [Prod: produc859450856879609959at_nat] : Prod = aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,produc3213797794245857475at_nat(Prod)),produc6408287024330202629at_nat(Prod)) ).

% prod.exhaust_sel
tff(fact_8336_prod_Oexhaust__sel,axiom,
    ! [Prod: produc9072475918466114483BT_nat] : Prod = aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,produc8713918199166443969BT_nat(Prod)),produc8311406018219409919BT_nat(Prod)) ).

% prod.exhaust_sel
tff(fact_8337_prod_Oexhaust__sel,axiom,
    ! [Prod: produc7272778201969148633d_enat] : Prod = aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,produc967593531271825845d_enat(Prod)),produc522021263368680183d_enat(Prod)) ).

% prod.exhaust_sel
tff(fact_8338_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_nat_nat] : Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) ).

% prod.exhaust_sel
tff(fact_8339_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_int_int] : Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_fst_int_int(Prod)),product_snd_int_int(Prod)) ).

% prod.exhaust_sel
tff(fact_8340_surjective__pairing,axiom,
    ! [Ta: produc859450856879609959at_nat] : Ta = aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,produc3213797794245857475at_nat(Ta)),produc6408287024330202629at_nat(Ta)) ).

% surjective_pairing
tff(fact_8341_surjective__pairing,axiom,
    ! [Ta: produc9072475918466114483BT_nat] : Ta = aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,produc8713918199166443969BT_nat(Ta)),produc8311406018219409919BT_nat(Ta)) ).

% surjective_pairing
tff(fact_8342_surjective__pairing,axiom,
    ! [Ta: produc7272778201969148633d_enat] : Ta = aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,produc967593531271825845d_enat(Ta)),produc522021263368680183d_enat(Ta)) ).

% surjective_pairing
tff(fact_8343_surjective__pairing,axiom,
    ! [Ta: product_prod_nat_nat] : Ta = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Ta)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Ta)) ).

% surjective_pairing
tff(fact_8344_surjective__pairing,axiom,
    ! [Ta: product_prod_int_int] : Ta = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_fst_int_int(Ta)),product_snd_int_int(Ta)) ).

% surjective_pairing
tff(fact_8345_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: fun_Pr2028404973800837934_nat_o,Xb: product_prod_nat_nat,Y: product_prod_nat_nat,A: produc859450856879609959at_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(P,Xb),Y)
     => ( ( A = aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,Xb),Y) )
       => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(P,produc3213797794245857475at_nat(A)),produc6408287024330202629at_nat(A)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_8346_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: fun_VE6480751511521678896_nat_o,Xb: vEBT_VEBT,Y: nat,A: produc9072475918466114483BT_nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(P,Xb),Y)
     => ( ( A = aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Y) )
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(P,produc8713918199166443969BT_nat(A)),produc8311406018219409919BT_nat(A)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_8347_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: fun_VE3387248342596292540enat_o,Xb: vEBT_VEBT,Y: extended_enat,A: produc7272778201969148633d_enat] :
      ( aa_Extended_enat_o(aa_VEB2705759856535948141enat_o(P,Xb),Y)
     => ( ( A = aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,Xb),Y) )
       => aa_Extended_enat_o(aa_VEB2705759856535948141enat_o(P,produc967593531271825845d_enat(A)),produc522021263368680183d_enat(A)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_8348_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: fun_nat_fun_nat_o,Xb: nat,Y: nat,A: product_prod_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(P,Xb),Y)
     => ( ( A = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Y) )
       => aa_nat_o(aa_nat_fun_nat_o(P,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,A)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,A)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_8349_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: fun_int_fun_int_o,Xb: int,Y: int,A: product_prod_int_int] :
      ( aa_int_o(aa_int_fun_int_o(P,Xb),Y)
     => ( ( A = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Y) )
       => aa_int_o(aa_int_fun_int_o(P,product_fst_int_int(A)),product_snd_int_int(A)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_8350_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [Nb: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = A )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),A) = zero_zero_nat ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_8351_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [Nb: nat,A: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),A) = A )
    <=> ( aa_int_int(bit_se8568078237143864401it_int(Nb),A) = zero_zero_int ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_8352_take__bit__drop__bit,axiom,
    ! [Mb: nat,Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),A)) ).

% take_bit_drop_bit
tff(fact_8353_take__bit__drop__bit,axiom,
    ! [Mb: nat,Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Mb),aa_int_int(bit_se8568078237143864401it_int(Nb),A)) = aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),A)) ).

% take_bit_drop_bit
tff(fact_8354_subseqs__distinctD,axiom,
    ! [Ys: list_int,Xs: list_int] :
      ( member_list_int(Ys,set_list_int2(subseqs_int(Xs)))
     => ( aa_list_int_o(distinct_int,Xs)
       => aa_list_int_o(distinct_int,Ys) ) ) ).

% subseqs_distinctD
tff(fact_8355_subseqs__distinctD,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( member_list_nat(Ys,set_list_nat2(subseqs_nat(Xs)))
     => ( aa_list_nat_o(distinct_nat,Xs)
       => aa_list_nat_o(distinct_nat,Ys) ) ) ).

% subseqs_distinctD
tff(fact_8356_divides__aux__def,axiom,
    ! [Qr: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,Qr)
    <=> ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Qr) = zero_zero_nat ) ) ).

% divides_aux_def
tff(fact_8357_divides__aux__def,axiom,
    ! [Qr: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,Qr)
    <=> ( product_snd_int_int(Qr) = zero_zero_int ) ) ).

% divides_aux_def
tff(fact_8358_distinct__conv__nth,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Xs)
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs))
         => ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_s6755466524823107622T_VEBT(Xs))
             => ( ( I4 != J3 )
               => ( nth_VEBT_VEBT(Xs,I4) != nth_VEBT_VEBT(Xs,J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_8359_distinct__conv__nth,axiom,
    ! [Xs: list_int] :
      ( aa_list_int_o(distinct_int,Xs)
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_int(Xs))
         => ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_int(Xs))
             => ( ( I4 != J3 )
               => ( aa_nat_int(nth_int(Xs),I4) != aa_nat_int(nth_int(Xs),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_8360_distinct__conv__nth,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(distinct_nat,Xs)
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_nat(Xs))
         => ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_nat(Xs))
             => ( ( I4 != J3 )
               => ( aa_nat_nat(nth_nat(Xs),I4) != aa_nat_nat(nth_nat(Xs),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_8361_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ia: nat,J: nat] :
      ( aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs))
         => ( ( nth_VEBT_VEBT(Xs,Ia) = nth_VEBT_VEBT(Xs,J) )
          <=> ( Ia = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_8362_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_int,Ia: nat,J: nat] :
      ( aa_list_int_o(distinct_int,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_int(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(Xs))
         => ( ( aa_nat_int(nth_int(Xs),Ia) = aa_nat_int(nth_int(Xs),J) )
          <=> ( Ia = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_8363_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_nat,Ia: nat,J: nat] :
      ( aa_list_nat_o(distinct_nat,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_nat(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_nat(Xs))
         => ( ( aa_nat_nat(nth_nat(Xs),Ia) = aa_nat_nat(nth_nat(Xs),J) )
          <=> ( Ia = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_8364_prod_Osplit__sel__asm,axiom,
    ! [P: fun_o_o,F: fun_nat_fun_nat_o,Prod: product_prod_nat_nat] :
      ( aa_o_o(P,aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),Prod))
    <=> ~ ( ( Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
          & ~ aa_o_o(P,aa_nat_o(aa_nat_fun_nat_o(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_8365_prod_Osplit__sel__asm,axiom,
    ! [P: fun_nat_o,F: fun_nat_fun_nat_nat,Prod: product_prod_nat_nat] :
      ( aa_nat_o(P,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),Prod))
    <=> ~ ( ( Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
          & ~ aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_8366_prod_Osplit__sel__asm,axiom,
    ! [P: fun_fu8391177546919232046_nat_o,F: fun_na1668100129766719005at_nat,Prod: product_prod_nat_nat] :
      ( aa_fun3538214411680374423_nat_o(P,aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),Prod))
    <=> ~ ( ( Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
          & ~ aa_fun3538214411680374423_nat_o(P,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_8367_prod_Osplit__sel__asm,axiom,
    ! [P: fun_fu6891447142836604759at_o_o,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] :
      ( aa_fun8408151826566693824at_o_o(P,aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),Prod))
    <=> ~ ( ( Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
          & ~ aa_fun8408151826566693824at_o_o(P,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_8368_prod_Osplit__sel__asm,axiom,
    ! [P: fun_o_o,F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(P,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod))
    <=> ~ ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_fst_int_int(Prod)),product_snd_int_int(Prod)) )
          & ~ aa_o_o(P,aa_int_o(aa_int_fun_int_o(F,product_fst_int_int(Prod)),product_snd_int_int(Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_8369_prod_Osplit__sel,axiom,
    ! [P: fun_o_o,F: fun_nat_fun_nat_o,Prod: product_prod_nat_nat] :
      ( aa_o_o(P,aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),Prod))
    <=> ( ( Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
       => aa_o_o(P,aa_nat_o(aa_nat_fun_nat_o(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel
tff(fact_8370_prod_Osplit__sel,axiom,
    ! [P: fun_nat_o,F: fun_nat_fun_nat_nat,Prod: product_prod_nat_nat] :
      ( aa_nat_o(P,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),Prod))
    <=> ( ( Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
       => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel
tff(fact_8371_prod_Osplit__sel,axiom,
    ! [P: fun_fu8391177546919232046_nat_o,F: fun_na1668100129766719005at_nat,Prod: product_prod_nat_nat] :
      ( aa_fun3538214411680374423_nat_o(P,aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),Prod))
    <=> ( ( Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
       => aa_fun3538214411680374423_nat_o(P,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel
tff(fact_8372_prod_Osplit__sel,axiom,
    ! [P: fun_fu6891447142836604759at_o_o,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] :
      ( aa_fun8408151826566693824at_o_o(P,aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),Prod))
    <=> ( ( Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
       => aa_fun8408151826566693824at_o_o(P,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel
tff(fact_8373_prod_Osplit__sel,axiom,
    ! [P: fun_o_o,F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(P,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod))
    <=> ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_fst_int_int(Prod)),product_snd_int_int(Prod)) )
       => aa_o_o(P,aa_int_o(aa_int_fun_int_o(F,product_fst_int_int(Prod)),product_snd_int_int(Prod))) ) ) ).

% prod.split_sel
tff(fact_8374_bits__ident,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),A))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A)) = A ).

% bits_ident
tff(fact_8375_bits__ident,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se545348938243370406it_int(Nb),aa_int_int(bit_se8568078237143864401it_int(Nb),A))),aa_int_int(bit_se2923211474154528505it_int(Nb),A)) = A ).

% bits_ident
tff(fact_8376_distinct__Ex1,axiom,
    ! [Xs: list_real,Xb: real] :
      ( distinct_real(Xs)
     => ( aa_set_real_o(member_real2(Xb),set_real2(Xs))
       => ? [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),size_size_list_real(Xs))
            & ( aa_nat_real(nth_real(Xs),X4) = Xb )
            & ! [Y4: nat] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),size_size_list_real(Xs))
                  & ( aa_nat_real(nth_real(Xs),Y4) = Xb ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_8377_distinct__Ex1,axiom,
    ! [Xs: list_o,Xb: $o] :
      ( distinct_o(Xs)
     => ( aa_set_o_o(member_o2((Xb)),set_o2(Xs))
       => ? [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),size_size_list_o(Xs))
            & ( aa_nat_o(nth_o(Xs),X4)
            <=> (Xb) )
            & ! [Y4: nat] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),size_size_list_o(Xs))
                  & ( aa_nat_o(nth_o(Xs),Y4)
                  <=> (Xb) ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_8378_distinct__Ex1,axiom,
    ! [Xs: list_set_nat,Xb: set_nat] :
      ( distinct_set_nat(Xs)
     => ( aa_set_set_nat_o(member_set_nat2(Xb),set_set_nat2(Xs))
       => ? [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),size_s3254054031482475050et_nat(Xs))
            & ( aa_nat_set_nat(nth_set_nat(Xs),X4) = Xb )
            & ! [Y4: nat] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),size_s3254054031482475050et_nat(Xs))
                  & ( aa_nat_set_nat(nth_set_nat(Xs),Y4) = Xb ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_8379_distinct__Ex1,axiom,
    ! [Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Xs)
     => ( member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(Xs))
       => ? [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),size_s6755466524823107622T_VEBT(Xs))
            & ( nth_VEBT_VEBT(Xs,X4) = Xb )
            & ! [Y4: nat] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),size_s6755466524823107622T_VEBT(Xs))
                  & ( nth_VEBT_VEBT(Xs,Y4) = Xb ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_8380_distinct__Ex1,axiom,
    ! [Xs: list_int,Xb: int] :
      ( aa_list_int_o(distinct_int,Xs)
     => ( aa_set_int_o(member_int2(Xb),set_int2(Xs))
       => ? [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),size_size_list_int(Xs))
            & ( aa_nat_int(nth_int(Xs),X4) = Xb )
            & ! [Y4: nat] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),size_size_list_int(Xs))
                  & ( aa_nat_int(nth_int(Xs),Y4) = Xb ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_8381_distinct__Ex1,axiom,
    ! [Xs: list_nat,Xb: nat] :
      ( aa_list_nat_o(distinct_nat,Xs)
     => ( aa_set_nat_o(member_nat2(Xb),set_nat2(Xs))
       => ? [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),size_size_list_nat(Xs))
            & ( aa_nat_nat(nth_nat(Xs),X4) = Xb )
            & ! [Y4: nat] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),size_size_list_nat(Xs))
                  & ( aa_nat_nat(nth_nat(Xs),Y4) = Xb ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_8382_update__zip,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Ia: nat,Xy: produc8243902056947475879T_VEBT] : list_u6961636818849549845T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys),Ia,Xy) = zip_VE537291747668921783T_VEBT(list_u1324408373059187874T_VEBT(Xs,Ia,produc6118266651365575811T_VEBT(Xy)),list_u1324408373059187874T_VEBT(Ys,Ia,produc351232539757196229T_VEBT(Xy))) ).

% update_zip
tff(fact_8383_update__zip,axiom,
    ! [Xs: list_nat,Ys: list_nat,Ia: nat,Xy: product_prod_nat_nat] : list_u6180841689913720943at_nat(zip_nat_nat(Xs,Ys),Ia,Xy) = zip_nat_nat(list_update_nat(Xs,Ia,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xy)),list_update_nat(Ys,Ia,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xy))) ).

% update_zip
tff(fact_8384_update__zip,axiom,
    ! [Xs: list_int,Ys: list_int,Ia: nat,Xy: product_prod_int_int] : list_u3002344382305578791nt_int(zip_int_int(Xs,Ys),Ia,Xy) = zip_int_int(list_update_int(Xs,Ia,product_fst_int_int(Xy)),list_update_int(Ys,Ia,product_snd_int_int(Xy))) ).

% update_zip
tff(fact_8385_drop__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% drop_bit_Suc
tff(fact_8386_drop__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(divide_divide_int(A),numeral_numeral_int(bit0(one)))) ).

% drop_bit_Suc
tff(fact_8387_distinct__list__update,axiom,
    ! [Xs: list_set_nat,A: set_nat,Ia: nat] :
      ( distinct_set_nat(Xs)
     => ( ~ aa_set_set_nat_o(member_set_nat2(A),minus_2163939370556025621et_nat(set_set_nat2(Xs),insert_set_nat2(aa_nat_set_nat(nth_set_nat(Xs),Ia),bot_bot_set_set_nat)))
       => distinct_set_nat(list_update_set_nat(Xs,Ia,A)) ) ) ).

% distinct_list_update
tff(fact_8388_distinct__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,A: vEBT_VEBT,Ia: nat] :
      ( aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Xs)
     => ( ~ member_VEBT_VEBT2(A,minus_5127226145743854075T_VEBT(set_VEBT_VEBT2(Xs),insert_VEBT_VEBT2(nth_VEBT_VEBT(Xs,Ia),bot_bo8194388402131092736T_VEBT)))
       => aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,list_u1324408373059187874T_VEBT(Xs,Ia,A)) ) ) ).

% distinct_list_update
tff(fact_8389_distinct__list__update,axiom,
    ! [Xs: list_real,A: real,Ia: nat] :
      ( distinct_real(Xs)
     => ( ~ aa_set_real_o(member_real2(A),minus_minus_set_real(set_real2(Xs),insert_real2(aa_nat_real(nth_real(Xs),Ia),bot_bot_set_real)))
       => distinct_real(list_update_real(Xs,Ia,A)) ) ) ).

% distinct_list_update
tff(fact_8390_distinct__list__update,axiom,
    ! [Xs: list_o,A: $o,Ia: nat] :
      ( distinct_o(Xs)
     => ( ~ aa_set_o_o(member_o2((A)),minus_minus_set_o(set_o2(Xs),insert_o2(aa_nat_o(nth_o(Xs),Ia),bot_bot_set_o)))
       => distinct_o(list_update_o(Xs,Ia,(A))) ) ) ).

% distinct_list_update
tff(fact_8391_distinct__list__update,axiom,
    ! [Xs: list_int,A: int,Ia: nat] :
      ( aa_list_int_o(distinct_int,Xs)
     => ( ~ aa_set_int_o(member_int2(A),minus_minus_set_int(set_int2(Xs),insert_int2(aa_nat_int(nth_int(Xs),Ia),bot_bot_set_int)))
       => aa_list_int_o(distinct_int,list_update_int(Xs,Ia,A)) ) ) ).

% distinct_list_update
tff(fact_8392_distinct__list__update,axiom,
    ! [Xs: list_nat,A: nat,Ia: nat] :
      ( aa_list_nat_o(distinct_nat,Xs)
     => ( ~ aa_set_nat_o(member_nat2(A),minus_minus_set_nat(set_nat2(Xs),insert_nat2(aa_nat_nat(nth_nat(Xs),Ia),bot_bot_set_nat)))
       => aa_list_nat_o(distinct_nat,list_update_nat(Xs,Ia,A)) ) ) ).

% distinct_list_update
tff(fact_8393_in__set__zip,axiom,
    ! [P2: produc8243902056947475879T_VEBT,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( member568628332442017744T_VEBT(P2,set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys)))
    <=> ? [N3: nat] :
          ( ( nth_VEBT_VEBT(Xs,N3) = produc6118266651365575811T_VEBT(P2) )
          & ( nth_VEBT_VEBT(Ys,N3) = produc351232539757196229T_VEBT(P2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_s6755466524823107622T_VEBT(Xs))
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_s6755466524823107622T_VEBT(Ys)) ) ) ).

% in_set_zip
tff(fact_8394_in__set__zip,axiom,
    ! [P2: produc4894624898956917775BT_int,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( member5419026705395827622BT_int(P2,set_Pr2853735649769556538BT_int(zip_VEBT_VEBT_int(Xs,Ys)))
    <=> ? [N3: nat] :
          ( ( nth_VEBT_VEBT(Xs,N3) = produc8711427728657393693BT_int(P2) )
          & ( aa_nat_int(nth_int(Ys),N3) = produc8308915547710359643BT_int(P2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_s6755466524823107622T_VEBT(Xs))
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_size_list_int(Ys)) ) ) ).

% in_set_zip
tff(fact_8395_in__set__zip,axiom,
    ! [P2: produc9072475918466114483BT_nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( member373505688050248522BT_nat(P2,set_Pr7031586669278753246BT_nat(zip_VEBT_VEBT_nat(Xs,Ys)))
    <=> ? [N3: nat] :
          ( ( nth_VEBT_VEBT(Xs,N3) = produc8713918199166443969BT_nat(P2) )
          & ( aa_nat_nat(nth_nat(Ys),N3) = produc8311406018219409919BT_nat(P2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_s6755466524823107622T_VEBT(Xs))
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_size_list_nat(Ys)) ) ) ).

% in_set_zip
tff(fact_8396_in__set__zip,axiom,
    ! [P2: produc1531783533982839933T_VEBT,Xs: list_int,Ys: list_VEBT_VEBT] :
      ( member2056185340421749780T_VEBT(P2,set_Pr8714266321650254504T_VEBT(zip_int_VEBT_VEBT(Xs,Ys)))
    <=> ? [N3: nat] :
          ( ( aa_nat_int(nth_int(Xs),N3) = produc2081412961586463171T_VEBT(P2) )
          & ( nth_VEBT_VEBT(Ys,N3) = produc1678900780639429121T_VEBT(P2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_size_list_int(Xs))
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_s6755466524823107622T_VEBT(Ys)) ) ) ).

% in_set_zip
tff(fact_8397_in__set__zip,axiom,
    ! [P2: product_prod_int_nat,Xs: list_int,Ys: list_nat] :
      ( member216504246829706758nt_nat(P2,set_Pr6647972299459129970nt_nat(zip_int_nat(Xs,Ys)))
    <=> ? [N3: nat] :
          ( ( aa_nat_int(nth_int(Xs),N3) = product_fst_int_nat(P2) )
          & ( aa_nat_nat(nth_nat(Ys),N3) = product_snd_int_nat(P2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_size_list_int(Xs))
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_size_list_nat(Ys)) ) ) ).

% in_set_zip
tff(fact_8398_in__set__zip,axiom,
    ! [P2: produc8025551001238799321T_VEBT,Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( member8549952807677709168T_VEBT(P2,set_Pr5984661752051438084T_VEBT(zip_nat_VEBT_VEBT(Xs,Ys)))
    <=> ? [N3: nat] :
          ( ( aa_nat_nat(nth_nat(Xs),N3) = produc8575180428842422559T_VEBT(P2) )
          & ( nth_VEBT_VEBT(Ys,N3) = produc8172668247895388509T_VEBT(P2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_size_list_nat(Xs))
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_s6755466524823107622T_VEBT(Ys)) ) ) ).

% in_set_zip
tff(fact_8399_in__set__zip,axiom,
    ! [P2: product_prod_nat_int,Xs: list_nat,Ys: list_int] :
      ( member4262671552274231302at_int(P2,set_Pr1470767568048878706at_int(zip_nat_int(Xs,Ys)))
    <=> ? [N3: nat] :
          ( ( aa_nat_nat(nth_nat(Xs),N3) = product_fst_nat_int(P2) )
          & ( aa_nat_int(nth_int(Ys),N3) = product_snd_nat_int(P2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_size_list_nat(Xs))
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_size_list_int(Ys)) ) ) ).

% in_set_zip
tff(fact_8400_in__set__zip,axiom,
    ! [P2: product_prod_nat_nat,Xs: list_nat,Ys: list_nat] :
      ( member8440522571783428010at_nat(P2,set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Ys)))
    <=> ? [N3: nat] :
          ( ( aa_nat_nat(nth_nat(Xs),N3) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2) )
          & ( aa_nat_nat(nth_nat(Ys),N3) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_size_list_nat(Xs))
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_size_list_nat(Ys)) ) ) ).

% in_set_zip
tff(fact_8401_in__set__zip,axiom,
    ! [P2: product_prod_int_int,Xs: list_int,Ys: list_int] :
      ( member5262025264175285858nt_int(P2,set_Pr2470121279949933262nt_int(zip_int_int(Xs,Ys)))
    <=> ? [N3: nat] :
          ( ( aa_nat_int(nth_int(Xs),N3) = product_fst_int_int(P2) )
          & ( aa_nat_int(nth_int(Ys),N3) = product_snd_int_int(P2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_size_list_int(Xs))
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),size_size_list_int(Ys)) ) ) ).

% in_set_zip
tff(fact_8402_size__prod__simp,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,P2: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(basic_876126793109182934at_nat(F,G),P2) = 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(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2))),aa_nat_nat(G,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P2)))),aa_nat_nat(suc,zero_zero_nat)) ).

% size_prod_simp
tff(fact_8403_size__prod__simp,axiom,
    ! [F: fun_int_nat,G: fun_int_nat,P2: product_prod_int_int] : basic_1872990034501187214nt_int(F,G,P2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(F,product_fst_int_int(P2))),aa_int_nat(G,product_snd_int_int(P2)))),aa_nat_nat(suc,zero_zero_nat)) ).

% size_prod_simp
tff(fact_8404_exI__realizer,axiom,
    ! [P: fun_Pr2028404973800837934_nat_o,Y: product_prod_nat_nat,Xb: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(P,Y),Xb)
     => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(P,produc6408287024330202629at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,Xb),Y))),produc3213797794245857475at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,Xb),Y))) ) ).

% exI_realizer
tff(fact_8405_exI__realizer,axiom,
    ! [P: fun_na6887940067898170868VEBT_o,Y: nat,Xb: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,Y),Xb)
     => aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,produc8311406018219409919BT_nat(aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Y))),produc8713918199166443969BT_nat(aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Y))) ) ).

% exI_realizer
tff(fact_8406_exI__realizer,axiom,
    ! [P: fun_Ex5372025725550730656VEBT_o,Y: extended_enat,Xb: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_Ext6823463560544661073VEBT_o(P,Y),Xb)
     => aa_VEBT_VEBT_o(aa_Ext6823463560544661073VEBT_o(P,produc522021263368680183d_enat(aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,Xb),Y))),produc967593531271825845d_enat(aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,Xb),Y))) ) ).

% exI_realizer
tff(fact_8407_exI__realizer,axiom,
    ! [P: fun_nat_fun_nat_o,Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(P,Y),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(P,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Y))),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Y))) ) ).

% exI_realizer
tff(fact_8408_exI__realizer,axiom,
    ! [P: fun_int_fun_int_o,Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(P,Y),Xb)
     => aa_int_o(aa_int_fun_int_o(P,product_snd_int_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Y))),product_fst_int_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Y))) ) ).

% exI_realizer
tff(fact_8409_conjI__realizer,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,P2: product_prod_nat_nat,Q: fun_Pr4796943543331071514_nat_o,Q2: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(P,P2)
     => ( aa_Pro6150637604445964947_nat_o(Q,Q2)
       => ( aa_Pro6150637604445964947_nat_o(P,produc3213797794245857475at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,P2),Q2)))
          & aa_Pro6150637604445964947_nat_o(Q,produc6408287024330202629at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_8410_conjI__realizer,axiom,
    ! [P: fun_VEBT_VEBT_o,P2: vEBT_VEBT,Q: fun_nat_o,Q2: nat] :
      ( aa_VEBT_VEBT_o(P,P2)
     => ( aa_nat_o(Q,Q2)
       => ( aa_VEBT_VEBT_o(P,produc8713918199166443969BT_nat(aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,P2),Q2)))
          & aa_nat_o(Q,produc8311406018219409919BT_nat(aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_8411_conjI__realizer,axiom,
    ! [P: fun_VEBT_VEBT_o,P2: vEBT_VEBT,Q: fun_Extended_enat_o,Q2: extended_enat] :
      ( aa_VEBT_VEBT_o(P,P2)
     => ( aa_Extended_enat_o(Q,Q2)
       => ( aa_VEBT_VEBT_o(P,produc967593531271825845d_enat(aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,P2),Q2)))
          & aa_Extended_enat_o(Q,produc522021263368680183d_enat(aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_8412_conjI__realizer,axiom,
    ! [P: fun_nat_o,P2: nat,Q: fun_nat_o,Q2: nat] :
      ( aa_nat_o(P,P2)
     => ( aa_nat_o(Q,Q2)
       => ( aa_nat_o(P,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,P2),Q2)))
          & aa_nat_o(Q,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_8413_conjI__realizer,axiom,
    ! [P: fun_int_o,P2: int,Q: fun_int_o,Q2: int] :
      ( aa_int_o(P,P2)
     => ( aa_int_o(Q,Q2)
       => ( aa_int_o(P,product_fst_int_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2)))
          & aa_int_o(Q,product_snd_int_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_8414_bezw_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_int_int] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( Y = $ite(Xaa = 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(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Xaa)))))) ) ) ).

% bezw.elims
tff(fact_8415_bezw_Osimps,axiom,
    ! [Xb: nat,Y: nat] :
      bezw(Xb,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,Xb),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,Xb),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,Xb),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Y)))))) ).

% bezw.simps
tff(fact_8416_bezw__non__0,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Y)
     => ( bezw(Xb,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,Xb),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,Xb),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,Xb),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Y))))) ) ) ).

% bezw_non_0
tff(fact_8417_bezw_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_int_int] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xaa))
       => ~ ( ( Y = $ite(Xaa = 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(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Xaa)))))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xaa)) ) ) ) ).

% bezw.pelims
tff(fact_8418_distinct__union,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( aa_list_int_o(distinct_int,union_int(Xs,Ys))
    <=> aa_list_int_o(distinct_int,Ys) ) ).

% distinct_union
tff(fact_8419_distinct__union,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( aa_list_nat_o(distinct_nat,union_nat(Xs,Ys))
    <=> aa_list_nat_o(distinct_nat,Ys) ) ).

% distinct_union
tff(fact_8420_set__remove1__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Xs)
     => ( set_VEBT_VEBT2(remove1_VEBT_VEBT(Xb,Xs)) = minus_5127226145743854075T_VEBT(set_VEBT_VEBT2(Xs),insert_VEBT_VEBT2(Xb,bot_bo8194388402131092736T_VEBT)) ) ) ).

% set_remove1_eq
tff(fact_8421_set__remove1__eq,axiom,
    ! [Xs: list_real,Xb: real] :
      ( distinct_real(Xs)
     => ( set_real2(remove1_real(Xb,Xs)) = minus_minus_set_real(set_real2(Xs),insert_real2(Xb,bot_bot_set_real)) ) ) ).

% set_remove1_eq
tff(fact_8422_set__remove1__eq,axiom,
    ! [Xs: list_o,Xb: $o] :
      ( distinct_o(Xs)
     => ( set_o2(remove1_o((Xb),Xs)) = minus_minus_set_o(set_o2(Xs),insert_o2((Xb),bot_bot_set_o)) ) ) ).

% set_remove1_eq
tff(fact_8423_set__remove1__eq,axiom,
    ! [Xs: list_int,Xb: int] :
      ( aa_list_int_o(distinct_int,Xs)
     => ( set_int2(remove1_int(Xb,Xs)) = minus_minus_set_int(set_int2(Xs),insert_int2(Xb,bot_bot_set_int)) ) ) ).

% set_remove1_eq
tff(fact_8424_set__remove1__eq,axiom,
    ! [Xs: list_nat,Xb: nat] :
      ( aa_list_nat_o(distinct_nat,Xs)
     => ( set_nat2(remove1_nat(Xb,Xs)) = minus_minus_set_nat(set_nat2(Xs),insert_nat2(Xb,bot_bot_set_nat)) ) ) ).

% set_remove1_eq
tff(fact_8425_root__powr__inverse,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = aa_real_real(powr_real(Xb),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ) ).

% root_powr_inverse
tff(fact_8426_horner__sum__eq__sum__funpow,axiom,
    ! [F: fun_VEBT_VEBT_int,A: int,Xs: list_VEBT_VEBT] : groups6842663049115397189BT_int(F,A,Xs) = aa_set_nat_int(groups3539618377306564664at_int(aa_lis3633651700486656887at_int(aa_int4104624148011801308at_int(aTP_Lamm_vz(F),A),Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))) ).

% horner_sum_eq_sum_funpow
tff(fact_8427_horner__sum__eq__sum__funpow,axiom,
    ! [F: fun_int_int,A: int,Xs: list_int] : groups8485231416243008693nt_int(F,A,Xs) = aa_set_nat_int(groups3539618377306564664at_int(aa_lis1073296200319854579at_int(aa_int2572695527019912092at_int(aTP_Lamm_wa(F),A),Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_size_list_int(Xs))) ).

% horner_sum_eq_sum_funpow
tff(fact_8428_horner__sum__eq__sum__funpow,axiom,
    ! [F: fun_nat_int,A: int,Xs: list_nat] : groups7485877704341954137at_int(F,A,Xs) = aa_set_nat_int(groups3539618377306564664at_int(aa_lis6345223608758921111at_int(aa_int7844622935458978624at_int(aTP_Lamm_wb(F),A),Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_size_list_nat(Xs))) ).

% horner_sum_eq_sum_funpow
tff(fact_8429_horner__sum__eq__sum__funpow,axiom,
    ! [F: fun_VE5569070535936029429omplex,A: complex,Xs: list_VEBT_VEBT] : groups1931381680841367751omplex(F,A,Xs) = groups2073611262835488442omplex(aa_lis8713343626058569977omplex(aa_com9025189225015940188omplex(aTP_Lamm_wc(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))) ).

% horner_sum_eq_sum_funpow
tff(fact_8430_horner__sum__eq__sum__funpow,axiom,
    ! [F: fun_int_complex,A: complex,Xs: list_int] : groups1380173120649922871omplex(F,A,Xs) = groups2073611262835488442omplex(aa_lis226970529952281973omplex(aa_com4579324164465666976omplex(aTP_Lamm_wd(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_size_list_int(Xs))) ).

% horner_sum_eq_sum_funpow
tff(fact_8431_horner__sum__eq__sum__funpow,axiom,
    ! [F: fun_nat_complex,A: complex,Xs: list_nat] : groups404637655443745499omplex(F,A,Xs) = groups2073611262835488442omplex(aa_lis3451821876164259097omplex(aa_com7804175510677644100omplex(aTP_Lamm_we(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_size_list_nat(Xs))) ).

% horner_sum_eq_sum_funpow
tff(fact_8432_horner__sum__eq__sum__funpow,axiom,
    ! [F: fun_VE3569950477627547253d_enat,A: extended_enat,Xs: list_VEBT_VEBT] : groups3844460828123498381d_enat(F,A,Xs) = groups7108830773950497114d_enat(aa_lis7361820021430418801d_enat(aa_Ext4914522244092902178d_enat(aTP_Lamm_wf(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))) ).

% horner_sum_eq_sum_funpow
tff(fact_8433_horner__sum__eq__sum__funpow,axiom,
    ! [F: fun_in5914218760682845459d_enat,A: extended_enat,Xs: list_int] : groups7888997386326813469d_enat(F,A,Xs) = groups7108830773950497114d_enat(aa_lis3069066898809486325d_enat(aa_Ext5403380450472165346d_enat(aTP_Lamm_wg(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_size_list_int(Xs))) ).

% horner_sum_eq_sum_funpow
tff(fact_8434_horner__sum__eq__sum__funpow,axiom,
    ! [F: fun_na8797796813480665199d_enat,A: extended_enat,Xs: list_nat] : groups1549203402269857401d_enat(F,A,Xs) = groups7108830773950497114d_enat(aa_lis8229176623098646865d_enat(aa_Ext1340118137906550078d_enat(aTP_Lamm_wh(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_size_list_nat(Xs))) ).

% horner_sum_eq_sum_funpow
tff(fact_8435_horner__sum__eq__sum__funpow,axiom,
    ! [F: fun_VEBT_VEBT_nat,A: nat,Xs: list_VEBT_VEBT] : groups6845153519624447465BT_nat(F,A,Xs) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_lis7811502719995853595at_nat(aa_nat265247640139305436at_nat(aTP_Lamm_wi(F),A),Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))) ).

% horner_sum_eq_sum_funpow
tff(fact_8436_Suc__funpow,axiom,
    ! [Nb: nat] : aa_fun5791449513213591928at_nat(aa_nat2689442564459254213at_nat(compow_fun_nat_nat,Nb),suc) = aa_nat_fun_nat_nat(plus_plus_nat,Nb) ).

% Suc_funpow
tff(fact_8437_funpow__0,axiom,
    ! [F: fun_nat_nat,Xb: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_nat2689442564459254213at_nat(compow_fun_nat_nat,zero_zero_nat),F),Xb) = Xb ).

% funpow_0
tff(fact_8438_in__set__remove1,axiom,
    ! [A: real,B: real,Xs: list_real] :
      ( ( A != B )
     => ( aa_set_real_o(member_real2(A),set_real2(remove1_real(B,Xs)))
      <=> aa_set_real_o(member_real2(A),set_real2(Xs)) ) ) ).

% in_set_remove1
tff(fact_8439_in__set__remove1,axiom,
    ! [A: $o,B: $o,Xs: list_o] :
      ( ~ ( (A)
        <=> (B) )
     => ( aa_set_o_o(member_o2((A)),set_o2(remove1_o((B),Xs)))
      <=> aa_set_o_o(member_o2((A)),set_o2(Xs)) ) ) ).

% in_set_remove1
tff(fact_8440_in__set__remove1,axiom,
    ! [A: set_nat,B: set_nat,Xs: list_set_nat] :
      ( ( A != B )
     => ( aa_set_set_nat_o(member_set_nat2(A),set_set_nat2(remove1_set_nat(B,Xs)))
      <=> aa_set_set_nat_o(member_set_nat2(A),set_set_nat2(Xs)) ) ) ).

% in_set_remove1
tff(fact_8441_in__set__remove1,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( A != B )
     => ( member_VEBT_VEBT2(A,set_VEBT_VEBT2(remove1_VEBT_VEBT(B,Xs)))
      <=> member_VEBT_VEBT2(A,set_VEBT_VEBT2(Xs)) ) ) ).

% in_set_remove1
tff(fact_8442_in__set__remove1,axiom,
    ! [A: int,B: int,Xs: list_int] :
      ( ( A != B )
     => ( aa_set_int_o(member_int2(A),set_int2(remove1_int(B,Xs)))
      <=> aa_set_int_o(member_int2(A),set_int2(Xs)) ) ) ).

% in_set_remove1
tff(fact_8443_in__set__remove1,axiom,
    ! [A: nat,B: nat,Xs: list_nat] :
      ( ( A != B )
     => ( aa_set_nat_o(member_nat2(A),set_nat2(remove1_nat(B,Xs)))
      <=> aa_set_nat_o(member_nat2(A),set_nat2(Xs)) ) ) ).

% in_set_remove1
tff(fact_8444_real__root__Suc__0,axiom,
    ! [Xb: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),Xb) = Xb ).

% real_root_Suc_0
tff(fact_8445_root__0,axiom,
    ! [Xb: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),Xb) = zero_zero_real ).

% root_0
tff(fact_8446_real__root__eq__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = aa_real_real(aa_nat_fun_real_real(root,Nb),Y) )
      <=> ( Xb = Y ) ) ) ).

% real_root_eq_iff
tff(fact_8447_real__root__eq__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = zero_zero_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_8448_real__root__less__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).

% real_root_less_iff
tff(fact_8449_real__root__le__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% real_root_le_iff
tff(fact_8450_real__root__eq__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = one_one_real )
      <=> ( Xb = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_8451_real__root__one,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_8452_real__root__lt__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ) ).

% real_root_lt_0_iff
tff(fact_8453_real__root__gt__0__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ) ).

% real_root_gt_0_iff
tff(fact_8454_real__root__le__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ) ).

% real_root_le_0_iff
tff(fact_8455_real__root__ge__0__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_8456_real__root__lt__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% real_root_lt_1_iff
tff(fact_8457_real__root__gt__1__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ) ).

% real_root_gt_1_iff
tff(fact_8458_real__root__le__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).

% real_root_le_1_iff
tff(fact_8459_real__root__ge__1__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_8460_real__root__pow__pos2,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos2
tff(fact_8461_funpow__swap1,axiom,
    ! [F: fun_nat_nat,Nb: nat,Xb: nat] : aa_nat_nat(F,aa_nat_nat(aa_fun5791449513213591928at_nat(aa_nat2689442564459254213at_nat(compow_fun_nat_nat,Nb),F),Xb)) = aa_nat_nat(aa_fun5791449513213591928at_nat(aa_nat2689442564459254213at_nat(compow_fun_nat_nat,Nb),F),aa_nat_nat(F,Xb)) ).

% funpow_swap1
tff(fact_8462_funpow__mult,axiom,
    ! [Nb: nat,Mb: nat,F: fun_nat_nat] : aa_fun5791449513213591928at_nat(aa_nat2689442564459254213at_nat(compow_fun_nat_nat,Nb),aa_fun5791449513213591928at_nat(aa_nat2689442564459254213at_nat(compow_fun_nat_nat,Mb),F)) = aa_fun5791449513213591928at_nat(aa_nat2689442564459254213at_nat(compow_fun_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),F) ).

% funpow_mult
tff(fact_8463_notin__set__remove1,axiom,
    ! [Xb: real,Xs: list_real,Y: real] :
      ( ~ aa_set_real_o(member_real2(Xb),set_real2(Xs))
     => ~ aa_set_real_o(member_real2(Xb),set_real2(remove1_real(Y,Xs))) ) ).

% notin_set_remove1
tff(fact_8464_notin__set__remove1,axiom,
    ! [Xb: $o,Xs: list_o,Y: $o] :
      ( ~ aa_set_o_o(member_o2((Xb)),set_o2(Xs))
     => ~ aa_set_o_o(member_o2((Xb)),set_o2(remove1_o((Y),Xs))) ) ).

% notin_set_remove1
tff(fact_8465_notin__set__remove1,axiom,
    ! [Xb: set_nat,Xs: list_set_nat,Y: set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat2(Xb),set_set_nat2(Xs))
     => ~ aa_set_set_nat_o(member_set_nat2(Xb),set_set_nat2(remove1_set_nat(Y,Xs))) ) ).

% notin_set_remove1
tff(fact_8466_notin__set__remove1,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ~ member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(Xs))
     => ~ member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(remove1_VEBT_VEBT(Y,Xs))) ) ).

% notin_set_remove1
tff(fact_8467_notin__set__remove1,axiom,
    ! [Xb: int,Xs: list_int,Y: int] :
      ( ~ aa_set_int_o(member_int2(Xb),set_int2(Xs))
     => ~ aa_set_int_o(member_int2(Xb),set_int2(remove1_int(Y,Xs))) ) ).

% notin_set_remove1
tff(fact_8468_notin__set__remove1,axiom,
    ! [Xb: nat,Xs: list_nat,Y: nat] :
      ( ~ aa_set_nat_o(member_nat2(Xb),set_nat2(Xs))
     => ~ aa_set_nat_o(member_nat2(Xb),set_nat2(remove1_nat(Y,Xs))) ) ).

% notin_set_remove1
tff(fact_8469_remove1__idem,axiom,
    ! [Xb: real,Xs: list_real] :
      ( ~ aa_set_real_o(member_real2(Xb),set_real2(Xs))
     => ( remove1_real(Xb,Xs) = Xs ) ) ).

% remove1_idem
tff(fact_8470_remove1__idem,axiom,
    ! [Xb: $o,Xs: list_o] :
      ( ~ aa_set_o_o(member_o2((Xb)),set_o2(Xs))
     => ( remove1_o((Xb),Xs) = Xs ) ) ).

% remove1_idem
tff(fact_8471_remove1__idem,axiom,
    ! [Xb: set_nat,Xs: list_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat2(Xb),set_set_nat2(Xs))
     => ( remove1_set_nat(Xb,Xs) = Xs ) ) ).

% remove1_idem
tff(fact_8472_remove1__idem,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ~ member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(Xs))
     => ( remove1_VEBT_VEBT(Xb,Xs) = Xs ) ) ).

% remove1_idem
tff(fact_8473_remove1__idem,axiom,
    ! [Xb: int,Xs: list_int] :
      ( ~ aa_set_int_o(member_int2(Xb),set_int2(Xs))
     => ( remove1_int(Xb,Xs) = Xs ) ) ).

% remove1_idem
tff(fact_8474_remove1__idem,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( ~ aa_set_nat_o(member_nat2(Xb),set_nat2(Xs))
     => ( remove1_nat(Xb,Xs) = Xs ) ) ).

% remove1_idem
tff(fact_8475_distinct__remove1,axiom,
    ! [Xs: list_int,Xb: int] :
      ( aa_list_int_o(distinct_int,Xs)
     => aa_list_int_o(distinct_int,remove1_int(Xb,Xs)) ) ).

% distinct_remove1
tff(fact_8476_distinct__remove1,axiom,
    ! [Xs: list_nat,Xb: nat] :
      ( aa_list_nat_o(distinct_nat,Xs)
     => aa_list_nat_o(distinct_nat,remove1_nat(Xb,Xs)) ) ).

% distinct_remove1
tff(fact_8477_set__remove1__subset,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] : ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(remove1_VEBT_VEBT(Xb,Xs)),set_VEBT_VEBT2(Xs)) ).

% set_remove1_subset
tff(fact_8478_set__remove1__subset,axiom,
    ! [Xb: int,Xs: list_int] : aa_set_int_o(ord_less_eq_set_int(set_int2(remove1_int(Xb,Xs))),set_int2(Xs)) ).

% set_remove1_subset
tff(fact_8479_set__remove1__subset,axiom,
    ! [Xb: nat,Xs: list_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(remove1_nat(Xb,Xs))),set_nat2(Xs)) ).

% set_remove1_subset
tff(fact_8480_real__root__less__mono,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ) ) ).

% real_root_less_mono
tff(fact_8481_real__root__le__mono,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ) ) ).

% real_root_le_mono
tff(fact_8482_real__root__power,axiom,
    ! [Nb: nat,Xb: real,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(power_power_real(Xb),K)) = aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),K) ) ) ).

% real_root_power
tff(fact_8483_real__root__abs,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(abs_abs_real,Xb)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_abs
tff(fact_8484_sgn__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) = aa_real_real(sgn_sgn_real,Xb) ) ) ).

% sgn_root
tff(fact_8485_of__nat__def,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,Nb) = aa_complex_complex(compow7134260122181328870omplex(Nb,plus_plus_complex(one_one_complex)),zero_zero_complex) ).

% of_nat_def
tff(fact_8486_of__nat__def,axiom,
    ! [Nb: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb) = aa_Ext7969882492699152700d_enat(compow2216990816977393616d_enat(Nb,plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat)),zero_z5237406670263579293d_enat) ).

% of_nat_def
tff(fact_8487_of__nat__def,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(compow_fun_real_real(Nb,plus_plus_real(one_one_real)),zero_zero_real) ).

% of_nat_def
tff(fact_8488_of__nat__def,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,Nb) = aa_int_int(compow_fun_int_int(Nb,aa_int_fun_int_int(plus_plus_int,one_one_int)),zero_zero_int) ).

% of_nat_def
tff(fact_8489_of__nat__def,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = aa_nat_nat(aa_fun5791449513213591928at_nat(aa_nat2689442564459254213at_nat(compow_fun_nat_nat,Nb),aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat)),zero_zero_nat) ).

% of_nat_def
tff(fact_8490_real__root__gt__zero,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_gt_zero
tff(fact_8491_numeral__add__unfold__funpow,axiom,
    ! [K: num,A: complex] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(K)),A) = aa_complex_complex(compow7134260122181328870omplex(aa_num_nat(numeral_numeral_nat,K),plus_plus_complex(one_one_complex)),A) ).

% numeral_add_unfold_funpow
tff(fact_8492_numeral__add__unfold__funpow,axiom,
    ! [K: num,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,K)),A) = aa_nat_nat(aa_fun5791449513213591928at_nat(aa_nat2689442564459254213at_nat(compow_fun_nat_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat)),A) ).

% numeral_add_unfold_funpow
tff(fact_8493_numeral__add__unfold__funpow,axiom,
    ! [K: num,A: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(K)),A) = aa_Ext7969882492699152700d_enat(compow2216990816977393616d_enat(aa_num_nat(numeral_numeral_nat,K),plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat)),A) ).

% numeral_add_unfold_funpow
tff(fact_8494_numeral__add__unfold__funpow,axiom,
    ! [K: num,A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(K)),A) = aa_int_int(compow_fun_int_int(aa_num_nat(numeral_numeral_nat,K),aa_int_fun_int_int(plus_plus_int,one_one_int)),A) ).

% numeral_add_unfold_funpow
tff(fact_8495_numeral__add__unfold__funpow,axiom,
    ! [K: num,A: real] : aa_real_real(plus_plus_real(numeral_numeral_real(K)),A) = aa_real_real(compow_fun_real_real(aa_num_nat(numeral_numeral_nat,K),plus_plus_real(one_one_real)),A) ).

% numeral_add_unfold_funpow
tff(fact_8496_real__root__strict__decreasing,axiom,
    ! [Nb: nat,N4: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N4),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_8497_root__abs__power,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(power_power_real(Y),Nb))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_8498_numeral__unfold__funpow,axiom,
    ! [K: num] : numera6690914467698888265omplex(K) = aa_complex_complex(compow7134260122181328870omplex(aa_num_nat(numeral_numeral_nat,K),plus_plus_complex(one_one_complex)),zero_zero_complex) ).

% numeral_unfold_funpow
tff(fact_8499_numeral__unfold__funpow,axiom,
    ! [K: num] : aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(aa_fun5791449513213591928at_nat(aa_nat2689442564459254213at_nat(compow_fun_nat_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat)),zero_zero_nat) ).

% numeral_unfold_funpow
tff(fact_8500_numeral__unfold__funpow,axiom,
    ! [K: num] : numera1916890842035813515d_enat(K) = aa_Ext7969882492699152700d_enat(compow2216990816977393616d_enat(aa_num_nat(numeral_numeral_nat,K),plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat)),zero_z5237406670263579293d_enat) ).

% numeral_unfold_funpow
tff(fact_8501_numeral__unfold__funpow,axiom,
    ! [K: num] : numeral_numeral_int(K) = aa_int_int(compow_fun_int_int(aa_num_nat(numeral_numeral_nat,K),aa_int_fun_int_int(plus_plus_int,one_one_int)),zero_zero_int) ).

% numeral_unfold_funpow
tff(fact_8502_numeral__unfold__funpow,axiom,
    ! [K: num] : numeral_numeral_real(K) = aa_real_real(compow_fun_real_real(aa_num_nat(numeral_numeral_nat,K),plus_plus_real(one_one_real)),zero_zero_real) ).

% numeral_unfold_funpow
tff(fact_8503_real__root__pos__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_pos_pos
tff(fact_8504_real__root__strict__increasing,axiom,
    ! [Nb: nat,N4: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,N4),Xb)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_8505_real__root__decreasing,axiom,
    ! [Nb: nat,N4: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N4),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ) ).

% real_root_decreasing
tff(fact_8506_real__root__pow__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos
tff(fact_8507_real__root__pos__unique,axiom,
    ! [Nb: nat,Y: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(power_power_real(Y),Nb) = Xb )
         => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_8508_real__root__power__cancel,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(power_power_real(Xb),Nb)) = Xb ) ) ) ).

% real_root_power_cancel
tff(fact_8509_length__remove1,axiom,
    ! [Xb: real,Xs: list_real] :
      size_size_list_real(remove1_real(Xb,Xs)) = $ite(aa_set_real_o(member_real2(Xb),set_real2(Xs)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_real(Xs)),one_one_nat),size_size_list_real(Xs)) ).

% length_remove1
tff(fact_8510_length__remove1,axiom,
    ! [Xb: $o,Xs: list_o] :
      size_size_list_o(remove1_o((Xb),Xs)) = $ite(aa_set_o_o(member_o2((Xb)),set_o2(Xs)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_o(Xs)),one_one_nat),size_size_list_o(Xs)) ).

% length_remove1
tff(fact_8511_length__remove1,axiom,
    ! [Xb: set_nat,Xs: list_set_nat] :
      size_s3254054031482475050et_nat(remove1_set_nat(Xb,Xs)) = $ite(aa_set_set_nat_o(member_set_nat2(Xb),set_set_nat2(Xs)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_s3254054031482475050et_nat(Xs)),one_one_nat),size_s3254054031482475050et_nat(Xs)) ).

% length_remove1
tff(fact_8512_length__remove1,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      size_s6755466524823107622T_VEBT(remove1_VEBT_VEBT(Xb,Xs)) = $ite(member_VEBT_VEBT2(Xb,set_VEBT_VEBT2(Xs)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_s6755466524823107622T_VEBT(Xs)),one_one_nat),size_s6755466524823107622T_VEBT(Xs)) ).

% length_remove1
tff(fact_8513_length__remove1,axiom,
    ! [Xb: int,Xs: list_int] :
      size_size_list_int(remove1_int(Xb,Xs)) = $ite(aa_set_int_o(member_int2(Xb),set_int2(Xs)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_int(Xs)),one_one_nat),size_size_list_int(Xs)) ).

% length_remove1
tff(fact_8514_length__remove1,axiom,
    ! [Xb: nat,Xs: list_nat] :
      size_size_list_nat(remove1_nat(Xb,Xs)) = $ite(aa_set_nat_o(member_nat2(Xb),set_nat2(Xs)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_nat(Xs)),one_one_nat),size_size_list_nat(Xs)) ).

% length_remove1
tff(fact_8515_real__root__increasing,axiom,
    ! [Nb: nat,N4: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,N4),Xb)) ) ) ) ) ).

% real_root_increasing
tff(fact_8516_sgn__power__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))),Nb)) = Xb ) ) ).

% sgn_power_root
tff(fact_8517_root__sgn__power,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Y)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,Y)),Nb))) = Y ) ) ).

% root_sgn_power
tff(fact_8518_ln__root,axiom,
    ! [Nb: nat,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,Nb),B)) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% ln_root
tff(fact_8519_log__root,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_real(log(B),aa_real_real(aa_nat_fun_real_real(root,Nb),A)) = aa_real_real(divide_divide_real(aa_real_real(log(B),A)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_root
tff(fact_8520_log__base__root,axiom,
    ! [Nb: nat,B: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(log(aa_real_real(aa_nat_fun_real_real(root,Nb),B)),Xb) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(B),Xb)) ) ) ) ).

% log_base_root
tff(fact_8521_split__root,axiom,
    ! [P: fun_real_o,Nb: nat,Xb: real] :
      ( aa_real_o(P,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_real_o(P,zero_zero_real) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ! [Y2: real] :
              ( ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Y2)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,Y2)),Nb)) = Xb )
             => aa_real_o(P,Y2) ) ) ) ) ).

% split_root
tff(fact_8522_prod__decode__aux_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(Xb),Xaa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xaa))
       => ~ ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xb),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Xaa)),aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xb)))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xaa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_8523_Nat_Ofunpow__code__def,axiom,
    funpow_nat = compow_fun_nat_nat ).

% Nat.funpow_code_def
tff(fact_8524_polyfun__rootbound,axiom,
    ! [C: fun_nat_complex,K: nat,Nb: nat] :
      ( ( aa_nat_complex(C,K) != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_nl(C),Nb)))
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_nl(C),Nb)))),Nb) ) ) ) ).

% polyfun_rootbound
tff(fact_8525_polyfun__rootbound,axiom,
    ! [C: fun_nat_real,K: nat,Nb: nat] :
      ( ( aa_nat_real(C,K) != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( aa_set_real_o(finite_finite_real,collect_real(aa_nat_fun_real_o(aTP_Lamm_nm(C),Nb)))
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_real(collect_real(aa_nat_fun_real_o(aTP_Lamm_nm(C),Nb)))),Nb) ) ) ) ).

% polyfun_rootbound
tff(fact_8526_card__Collect__less__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aj,Nb))) = Nb ).

% card_Collect_less_nat
tff(fact_8527_card__atMost,axiom,
    ! [U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_ord_atMost_nat,U)) = aa_nat_nat(suc,U) ).

% card_atMost
tff(fact_8528_card__Collect__le__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ak,Nb))) = aa_nat_nat(suc,Nb) ).

% card_Collect_le_nat
tff(fact_8529_card_Oempty,axiom,
    finite_card_complex(bot_bot_set_complex) = zero_zero_nat ).

% card.empty
tff(fact_8530_card_Oempty,axiom,
    finite_card_set_nat(bot_bot_set_set_nat) = zero_zero_nat ).

% card.empty
tff(fact_8531_card_Oempty,axiom,
    finite_card_list_nat(bot_bot_set_list_nat) = zero_zero_nat ).

% card.empty
tff(fact_8532_card_Oempty,axiom,
    finite_card_real(bot_bot_set_real) = zero_zero_nat ).

% card.empty
tff(fact_8533_card_Oempty,axiom,
    finite_card_o(bot_bot_set_o) = zero_zero_nat ).

% card.empty
tff(fact_8534_card_Oempty,axiom,
    aa_set_nat_nat(finite_card_nat,bot_bot_set_nat) = zero_zero_nat ).

% card.empty
tff(fact_8535_card_Oempty,axiom,
    finite_card_int(bot_bot_set_int) = zero_zero_nat ).

% card.empty
tff(fact_8536_card_Oinfinite,axiom,
    ! [A2: set_set_nat] :
      ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( finite_card_set_nat(A2) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_8537_card_Oinfinite,axiom,
    ! [A2: set_list_nat] :
      ( ~ finite8100373058378681591st_nat(A2)
     => ( finite_card_list_nat(A2) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_8538_card_Oinfinite,axiom,
    ! [A2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(finite_card_nat,A2) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_8539_card_Oinfinite,axiom,
    ! [A2: set_complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( finite_card_complex(A2) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_8540_card_Oinfinite,axiom,
    ! [A2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( finite_card_int(A2) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_8541_card_Oinfinite,axiom,
    ! [A2: set_Extended_enat] :
      ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( finite121521170596916366d_enat(A2) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_8542_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(aa_nat1818925681583257441et_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_8543_card__0__eq,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( finite_card_set_nat(A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_set_nat ) ) ) ).

% card_0_eq
tff(fact_8544_card__0__eq,axiom,
    ! [A2: set_list_nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( ( finite_card_list_nat(A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_list_nat ) ) ) ).

% card_0_eq
tff(fact_8545_card__0__eq,axiom,
    ! [A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( finite_card_complex(A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_complex ) ) ) ).

% card_0_eq
tff(fact_8546_card__0__eq,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( finite121521170596916366d_enat(A2) = zero_zero_nat )
      <=> ( A2 = bot_bo7653980558646680370d_enat ) ) ) ).

% card_0_eq
tff(fact_8547_card__0__eq,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( finite_card_real(A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_real ) ) ) ).

% card_0_eq
tff(fact_8548_card__0__eq,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( finite_card_o(A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_o ) ) ) ).

% card_0_eq
tff(fact_8549_card__0__eq,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( aa_set_nat_nat(finite_card_nat,A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_nat ) ) ) ).

% card_0_eq
tff(fact_8550_card__0__eq,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( finite_card_int(A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_int ) ) ) ).

% card_0_eq
tff(fact_8551_card__insert__disjoint,axiom,
    ! [A2: set_real,Xb: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ~ aa_set_real_o(member_real2(Xb),A2)
       => ( finite_card_real(insert_real2(Xb,A2)) = aa_nat_nat(suc,finite_card_real(A2)) ) ) ) ).

% card_insert_disjoint
tff(fact_8552_card__insert__disjoint,axiom,
    ! [A2: set_o,Xb: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ~ aa_set_o_o(member_o2((Xb)),A2)
       => ( finite_card_o(insert_o2((Xb),A2)) = aa_nat_nat(suc,finite_card_o(A2)) ) ) ) ).

% card_insert_disjoint
tff(fact_8553_card__insert__disjoint,axiom,
    ! [A2: set_set_nat,Xb: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ~ aa_set_set_nat_o(member_set_nat2(Xb),A2)
       => ( finite_card_set_nat(insert_set_nat2(Xb,A2)) = aa_nat_nat(suc,finite_card_set_nat(A2)) ) ) ) ).

% card_insert_disjoint
tff(fact_8554_card__insert__disjoint,axiom,
    ! [A2: set_list_nat,Xb: list_nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( ~ member_list_nat(Xb,A2)
       => ( finite_card_list_nat(insert_list_nat(Xb,A2)) = aa_nat_nat(suc,finite_card_list_nat(A2)) ) ) ) ).

% card_insert_disjoint
tff(fact_8555_card__insert__disjoint,axiom,
    ! [A2: set_nat,Xb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat2(Xb),A2)
       => ( aa_set_nat_nat(finite_card_nat,insert_nat2(Xb,A2)) = aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,A2)) ) ) ) ).

% card_insert_disjoint
tff(fact_8556_card__insert__disjoint,axiom,
    ! [A2: set_complex,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ~ aa_set_complex_o(member_complex(Xb),A2)
       => ( finite_card_complex(insert_complex(Xb,A2)) = aa_nat_nat(suc,finite_card_complex(A2)) ) ) ) ).

% card_insert_disjoint
tff(fact_8557_card__insert__disjoint,axiom,
    ! [A2: set_int,Xb: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int2(Xb),A2)
       => ( finite_card_int(insert_int2(Xb,A2)) = aa_nat_nat(suc,finite_card_int(A2)) ) ) ) ).

% card_insert_disjoint
tff(fact_8558_card__insert__disjoint,axiom,
    ! [A2: set_Extended_enat,Xb: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ~ aa_set3337841544853333262enat_o(member_Extended_enat(Xb),A2)
       => ( finite121521170596916366d_enat(insert_Extended_enat(Xb,A2)) = aa_nat_nat(suc,finite121521170596916366d_enat(A2)) ) ) ) ).

% card_insert_disjoint
tff(fact_8559_n__subsets,axiom,
    ! [A2: set_set_nat,K: nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( finite1149291290879098388et_nat(collect_set_set_nat(aa_nat5160323483289836711_nat_o(aTP_Lamm_wj(A2),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite_card_set_nat(A2)),K) ) ) ).

% n_subsets
tff(fact_8560_n__subsets,axiom,
    ! [A2: set_list_nat,K: nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( finite2364142230527598318st_nat(collect_set_list_nat(aa_nat7721107121239158477_nat_o(aTP_Lamm_wk(A2),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite_card_list_nat(A2)),K) ) ) ).

% n_subsets
tff(fact_8561_n__subsets,axiom,
    ! [A2: set_complex,K: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( finite903997441450111292omplex(collect_set_complex(aa_nat5471291301243885951plex_o(aTP_Lamm_wl(A2),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite_card_complex(A2)),K) ) ) ).

% n_subsets
tff(fact_8562_n__subsets,axiom,
    ! [A2: set_int,K: nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( finite_card_set_int(collect_set_int(aa_nat_fun_set_int_o(aTP_Lamm_wm(A2),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite_card_int(A2)),K) ) ) ).

% n_subsets
tff(fact_8563_n__subsets,axiom,
    ! [A2: set_Extended_enat,K: nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( finite3719263829065406702d_enat(collec2260605976452661553d_enat(aa_nat9166527414068455075enat_o(aTP_Lamm_wn(A2),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite121521170596916366d_enat(A2)),K) ) ) ).

% n_subsets
tff(fact_8564_n__subsets,axiom,
    ! [A2: set_nat,K: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_wo(A2),K))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_set_nat_nat(finite_card_nat,A2)),K) ) ) ).

% n_subsets
tff(fact_8565_infinite__arbitrarily__large,axiom,
    ! [A2: set_set_nat,Nb: nat] :
      ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ? [B6: set_set_nat] :
          ( aa_set_set_nat_o(finite1152437895449049373et_nat,B6)
          & ( finite_card_set_nat(B6) = Nb )
          & aa_set_set_nat_o(ord_le6893508408891458716et_nat(B6),A2) ) ) ).

% infinite_arbitrarily_large
tff(fact_8566_infinite__arbitrarily__large,axiom,
    ! [A2: set_list_nat,Nb: nat] :
      ( ~ finite8100373058378681591st_nat(A2)
     => ? [B6: set_list_nat] :
          ( finite8100373058378681591st_nat(B6)
          & ( finite_card_list_nat(B6) = Nb )
          & ord_le6045566169113846134st_nat(B6,A2) ) ) ).

% infinite_arbitrarily_large
tff(fact_8567_infinite__arbitrarily__large,axiom,
    ! [A2: set_complex,Nb: nat] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ? [B6: set_complex] :
          ( aa_set_complex_o(finite3207457112153483333omplex,B6)
          & ( finite_card_complex(B6) = Nb )
          & aa_set_complex_o(ord_le211207098394363844omplex(B6),A2) ) ) ).

% infinite_arbitrarily_large
tff(fact_8568_infinite__arbitrarily__large,axiom,
    ! [A2: set_int,Nb: nat] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ? [B6: set_int] :
          ( aa_set_int_o(finite_finite_int,B6)
          & ( finite_card_int(B6) = Nb )
          & aa_set_int_o(ord_less_eq_set_int(B6),A2) ) ) ).

% infinite_arbitrarily_large
tff(fact_8569_infinite__arbitrarily__large,axiom,
    ! [A2: set_Extended_enat,Nb: nat] :
      ( ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ? [B6: set_Extended_enat] :
          ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B6)
          & ( finite121521170596916366d_enat(B6) = Nb )
          & aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(B6),A2) ) ) ).

% infinite_arbitrarily_large
tff(fact_8570_infinite__arbitrarily__large,axiom,
    ! [A2: set_nat,Nb: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ? [B6: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B6)
          & ( aa_set_nat_nat(finite_card_nat,B6) = Nb )
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),A2) ) ) ).

% infinite_arbitrarily_large
tff(fact_8571_card__subset__eq,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B2)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
       => ( ( finite_card_set_nat(A2) = finite_card_set_nat(B2) )
         => ( A2 = B2 ) ) ) ) ).

% card_subset_eq
tff(fact_8572_card__subset__eq,axiom,
    ! [B2: set_list_nat,A2: set_list_nat] :
      ( finite8100373058378681591st_nat(B2)
     => ( ord_le6045566169113846134st_nat(A2,B2)
       => ( ( finite_card_list_nat(A2) = finite_card_list_nat(B2) )
         => ( A2 = B2 ) ) ) ) ).

% card_subset_eq
tff(fact_8573_card__subset__eq,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
       => ( ( finite_card_complex(A2) = finite_card_complex(B2) )
         => ( A2 = B2 ) ) ) ) ).

% card_subset_eq
tff(fact_8574_card__subset__eq,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)
       => ( ( finite_card_int(A2) = finite_card_int(B2) )
         => ( A2 = B2 ) ) ) ) ).

% card_subset_eq
tff(fact_8575_card__subset__eq,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B2)
       => ( ( finite121521170596916366d_enat(A2) = finite121521170596916366d_enat(B2) )
         => ( A2 = B2 ) ) ) ) ).

% card_subset_eq
tff(fact_8576_card__subset__eq,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_nat(finite_card_nat,A2) = aa_set_nat_nat(finite_card_nat,B2) )
         => ( A2 = B2 ) ) ) ) ).

% card_subset_eq
tff(fact_8577_card__le__if__inj__on__rel,axiom,
    ! [B2: set_real,A2: set_real,R2: fun_real_fun_real_o] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( ! [A3: real] :
            ( aa_set_real_o(member_real2(A3),A2)
           => ? [B7: real] :
                ( aa_set_real_o(member_real2(B7),B2)
                & aa_real_o(aa_real_fun_real_o(R2,A3),B7) ) )
       => ( ! [A12: real,A23: real,B3: real] :
              ( aa_set_real_o(member_real2(A12),A2)
             => ( aa_set_real_o(member_real2(A23),A2)
               => ( aa_set_real_o(member_real2(B3),B2)
                 => ( aa_real_o(aa_real_fun_real_o(R2,A12),B3)
                   => ( aa_real_o(aa_real_fun_real_o(R2,A23),B3)
                     => ( A12 = A23 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_real(A2)),finite_card_real(B2)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_8578_card__le__if__inj__on__rel,axiom,
    ! [B2: set_o,A2: set_real,R2: fun_real_fun_o_o] :
      ( aa_set_o_o(finite_finite_o,B2)
     => ( ! [A3: real] :
            ( aa_set_real_o(member_real2(A3),A2)
           => ? [B7: $o] :
                ( aa_set_o_o(member_o2((B7)),B2)
                & aa_o_o(aa_real_fun_o_o(R2,A3),(B7)) ) )
       => ( ! [A12: real,A23: real,B3: $o] :
              ( aa_set_real_o(member_real2(A12),A2)
             => ( aa_set_real_o(member_real2(A23),A2)
               => ( aa_set_o_o(member_o2((B3)),B2)
                 => ( aa_o_o(aa_real_fun_o_o(R2,A12),(B3))
                   => ( aa_o_o(aa_real_fun_o_o(R2,A23),(B3))
                     => ( A12 = A23 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_real(A2)),finite_card_o(B2)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_8579_card__le__if__inj__on__rel,axiom,
    ! [B2: set_real,A2: set_o,R2: fun_o_fun_real_o] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( ! [A3: $o] :
            ( aa_set_o_o(member_o2((A3)),A2)
           => ? [B7: real] :
                ( aa_set_real_o(member_real2(B7),B2)
                & aa_real_o(aa_o_fun_real_o(R2,(A3)),B7) ) )
       => ( ! [A12: $o,A23: $o,B3: real] :
              ( aa_set_o_o(member_o2((A12)),A2)
             => ( aa_set_o_o(member_o2((A23)),A2)
               => ( aa_set_real_o(member_real2(B3),B2)
                 => ( aa_real_o(aa_o_fun_real_o(R2,(A12)),B3)
                   => ( aa_real_o(aa_o_fun_real_o(R2,(A23)),B3)
                     => ( (A12)
                      <=> (A23) ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(A2)),finite_card_real(B2)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_8580_card__le__if__inj__on__rel,axiom,
    ! [B2: set_o,A2: set_o,R2: fun_o_fun_o_o] :
      ( aa_set_o_o(finite_finite_o,B2)
     => ( ! [A3: $o] :
            ( aa_set_o_o(member_o2((A3)),A2)
           => ? [B7: $o] :
                ( aa_set_o_o(member_o2((B7)),B2)
                & aa_o_o(aa_o_fun_o_o(R2,(A3)),(B7)) ) )
       => ( ! [A12: $o,A23: $o,B3: $o] :
              ( aa_set_o_o(member_o2((A12)),A2)
             => ( aa_set_o_o(member_o2((A23)),A2)
               => ( aa_set_o_o(member_o2((B3)),B2)
                 => ( aa_o_o(aa_o_fun_o_o(R2,(A12)),(B3))
                   => ( aa_o_o(aa_o_fun_o_o(R2,(A23)),(B3))
                     => ( (A12)
                      <=> (A23) ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(A2)),finite_card_o(B2)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_8581_card__le__if__inj__on__rel,axiom,
    ! [B2: set_real,A2: set_int,R2: fun_int_fun_real_o] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( ! [A3: int] :
            ( aa_set_int_o(member_int2(A3),A2)
           => ? [B7: real] :
                ( aa_set_real_o(member_real2(B7),B2)
                & aa_real_o(aa_int_fun_real_o(R2,A3),B7) ) )
       => ( ! [A12: int,A23: int,B3: real] :
              ( aa_set_int_o(member_int2(A12),A2)
             => ( aa_set_int_o(member_int2(A23),A2)
               => ( aa_set_real_o(member_real2(B3),B2)
                 => ( aa_real_o(aa_int_fun_real_o(R2,A12),B3)
                   => ( aa_real_o(aa_int_fun_real_o(R2,A23),B3)
                     => ( A12 = A23 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A2)),finite_card_real(B2)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_8582_card__le__if__inj__on__rel,axiom,
    ! [B2: set_o,A2: set_int,R2: fun_int_fun_o_o] :
      ( aa_set_o_o(finite_finite_o,B2)
     => ( ! [A3: int] :
            ( aa_set_int_o(member_int2(A3),A2)
           => ? [B7: $o] :
                ( aa_set_o_o(member_o2((B7)),B2)
                & aa_o_o(aa_int_fun_o_o(R2,A3),(B7)) ) )
       => ( ! [A12: int,A23: int,B3: $o] :
              ( aa_set_int_o(member_int2(A12),A2)
             => ( aa_set_int_o(member_int2(A23),A2)
               => ( aa_set_o_o(member_o2((B3)),B2)
                 => ( aa_o_o(aa_int_fun_o_o(R2,A12),(B3))
                   => ( aa_o_o(aa_int_fun_o_o(R2,A23),(B3))
                     => ( A12 = A23 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A2)),finite_card_o(B2)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_8583_card__le__if__inj__on__rel,axiom,
    ! [B2: set_real,A2: set_nat,R2: fun_nat_fun_real_o] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( ! [A3: nat] :
            ( aa_set_nat_o(member_nat2(A3),A2)
           => ? [B7: real] :
                ( aa_set_real_o(member_real2(B7),B2)
                & aa_real_o(aa_nat_fun_real_o(R2,A3),B7) ) )
       => ( ! [A12: nat,A23: nat,B3: real] :
              ( aa_set_nat_o(member_nat2(A12),A2)
             => ( aa_set_nat_o(member_nat2(A23),A2)
               => ( aa_set_real_o(member_real2(B3),B2)
                 => ( aa_real_o(aa_nat_fun_real_o(R2,A12),B3)
                   => ( aa_real_o(aa_nat_fun_real_o(R2,A23),B3)
                     => ( A12 = A23 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A2)),finite_card_real(B2)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_8584_card__le__if__inj__on__rel,axiom,
    ! [B2: set_o,A2: set_nat,R2: fun_nat_fun_o_o] :
      ( aa_set_o_o(finite_finite_o,B2)
     => ( ! [A3: nat] :
            ( aa_set_nat_o(member_nat2(A3),A2)
           => ? [B7: $o] :
                ( aa_set_o_o(member_o2((B7)),B2)
                & aa_o_o(aa_nat_fun_o_o(R2,A3),(B7)) ) )
       => ( ! [A12: nat,A23: nat,B3: $o] :
              ( aa_set_nat_o(member_nat2(A12),A2)
             => ( aa_set_nat_o(member_nat2(A23),A2)
               => ( aa_set_o_o(member_o2((B3)),B2)
                 => ( aa_o_o(aa_nat_fun_o_o(R2,A12),(B3))
                   => ( aa_o_o(aa_nat_fun_o_o(R2,A23),(B3))
                     => ( A12 = A23 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A2)),finite_card_o(B2)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_8585_card__le__if__inj__on__rel,axiom,
    ! [B2: set_real,A2: set_complex,R2: fun_co4998825143550818160real_o] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( ! [A3: complex] :
            ( aa_set_complex_o(member_complex(A3),A2)
           => ? [B7: real] :
                ( aa_set_real_o(member_real2(B7),B2)
                & aa_real_o(aa_com1951324360330529881real_o(R2,A3),B7) ) )
       => ( ! [A12: complex,A23: complex,B3: real] :
              ( aa_set_complex_o(member_complex(A12),A2)
             => ( aa_set_complex_o(member_complex(A23),A2)
               => ( aa_set_real_o(member_real2(B3),B2)
                 => ( aa_real_o(aa_com1951324360330529881real_o(R2,A12),B3)
                   => ( aa_real_o(aa_com1951324360330529881real_o(R2,A23),B3)
                     => ( A12 = A23 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(A2)),finite_card_real(B2)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_8586_card__le__if__inj__on__rel,axiom,
    ! [B2: set_o,A2: set_complex,R2: fun_complex_fun_o_o] :
      ( aa_set_o_o(finite_finite_o,B2)
     => ( ! [A3: complex] :
            ( aa_set_complex_o(member_complex(A3),A2)
           => ? [B7: $o] :
                ( aa_set_o_o(member_o2((B7)),B2)
                & aa_o_o(aa_complex_fun_o_o(R2,A3),(B7)) ) )
       => ( ! [A12: complex,A23: complex,B3: $o] :
              ( aa_set_complex_o(member_complex(A12),A2)
             => ( aa_set_complex_o(member_complex(A23),A2)
               => ( aa_set_o_o(member_o2((B3)),B2)
                 => ( aa_o_o(aa_complex_fun_o_o(R2,A12),(B3))
                   => ( aa_o_o(aa_complex_fun_o_o(R2,A23),(B3))
                     => ( A12 = A23 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(A2)),finite_card_o(B2)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_8587_card__insert__le,axiom,
    ! [A2: set_int,Xb: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A2)),finite_card_int(insert_int2(Xb,A2))) ).

% card_insert_le
tff(fact_8588_card__insert__le,axiom,
    ! [A2: set_real,Xb: real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_real(A2)),finite_card_real(insert_real2(Xb,A2))) ).

% card_insert_le
tff(fact_8589_card__insert__le,axiom,
    ! [A2: set_o,Xb: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(A2)),finite_card_o(insert_o2((Xb),A2))) ).

% card_insert_le
tff(fact_8590_card__insert__le,axiom,
    ! [A2: set_nat,Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A2)),aa_set_nat_nat(finite_card_nat,insert_nat2(Xb,A2))) ).

% card_insert_le
tff(fact_8591_card__insert__le,axiom,
    ! [A2: set_complex,Xb: complex] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(A2)),finite_card_complex(insert_complex(Xb,A2))) ).

% card_insert_le
tff(fact_8592_card__insert__le,axiom,
    ! [A2: set_set_nat,Xb: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(A2)),finite_card_set_nat(insert_set_nat2(Xb,A2))) ).

% card_insert_le
tff(fact_8593_card__insert__le,axiom,
    ! [A2: set_list_nat,Xb: list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(A2)),finite_card_list_nat(insert_list_nat(Xb,A2))) ).

% card_insert_le
tff(fact_8594_card__lists__length__eq,axiom,
    ! [A2: set_set_nat,Nb: nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( finite5631907774883551598et_nat(collect_list_set_nat(aa_nat2247512698059773005_nat_o(aTP_Lamm_wp(A2),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_set_nat(A2)),Nb) ) ) ).

% card_lists_length_eq
tff(fact_8595_card__lists__length__eq,axiom,
    ! [A2: set_list_nat,Nb: nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( finite7325466520557071688st_nat(collec5989764272469232197st_nat(aa_nat158606821896287091_nat_o(aTP_Lamm_wq(A2),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_list_nat(A2)),Nb) ) ) ).

% card_lists_length_eq
tff(fact_8596_card__lists__length__eq,axiom,
    ! [A2: set_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( finite5120063068150530198omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_br(A2),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_complex(A2)),Nb) ) ) ).

% card_lists_length_eq
tff(fact_8597_card__lists__length__eq,axiom,
    ! [A2: set_Extended_enat,Nb: nat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( finite7441382602597825044d_enat(collec8433460942617342167d_enat(aa_nat4163650157883470461enat_o(aTP_Lamm_bs(A2),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite121521170596916366d_enat(A2)),Nb) ) ) ).

% card_lists_length_eq
tff(fact_8598_card__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,Nb: nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( finite5915292604075114978T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_bt(A2),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite7802652506058667612T_VEBT(A2)),Nb) ) ) ).

% card_lists_length_eq
tff(fact_8599_card__lists__length__eq,axiom,
    ! [A2: set_int,Nb: nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( finite_card_list_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_bu(A2),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_int(A2)),Nb) ) ) ).

% card_lists_length_eq
tff(fact_8600_card__lists__length__eq,axiom,
    ! [A2: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_bv(A2),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_set_nat_nat(finite_card_nat,A2)),Nb) ) ) ).

% card_lists_length_eq
tff(fact_8601_card__eq__0__iff,axiom,
    ! [A2: set_set_nat] :
      ( ( finite_card_set_nat(A2) = zero_zero_nat )
    <=> ( ( A2 = bot_bot_set_set_nat )
        | ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A2) ) ) ).

% card_eq_0_iff
tff(fact_8602_card__eq__0__iff,axiom,
    ! [A2: set_list_nat] :
      ( ( finite_card_list_nat(A2) = zero_zero_nat )
    <=> ( ( A2 = bot_bot_set_list_nat )
        | ~ finite8100373058378681591st_nat(A2) ) ) ).

% card_eq_0_iff
tff(fact_8603_card__eq__0__iff,axiom,
    ! [A2: set_complex] :
      ( ( finite_card_complex(A2) = zero_zero_nat )
    <=> ( ( A2 = bot_bot_set_complex )
        | ~ aa_set_complex_o(finite3207457112153483333omplex,A2) ) ) ).

% card_eq_0_iff
tff(fact_8604_card__eq__0__iff,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite121521170596916366d_enat(A2) = zero_zero_nat )
    <=> ( ( A2 = bot_bo7653980558646680370d_enat )
        | ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2) ) ) ).

% card_eq_0_iff
tff(fact_8605_card__eq__0__iff,axiom,
    ! [A2: set_real] :
      ( ( finite_card_real(A2) = zero_zero_nat )
    <=> ( ( A2 = bot_bot_set_real )
        | ~ aa_set_real_o(finite_finite_real,A2) ) ) ).

% card_eq_0_iff
tff(fact_8606_card__eq__0__iff,axiom,
    ! [A2: set_o] :
      ( ( finite_card_o(A2) = zero_zero_nat )
    <=> ( ( A2 = bot_bot_set_o )
        | ~ aa_set_o_o(finite_finite_o,A2) ) ) ).

% card_eq_0_iff
tff(fact_8607_card__eq__0__iff,axiom,
    ! [A2: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A2) = zero_zero_nat )
    <=> ( ( A2 = bot_bot_set_nat )
        | ~ aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% card_eq_0_iff
tff(fact_8608_card__eq__0__iff,axiom,
    ! [A2: set_int] :
      ( ( finite_card_int(A2) = zero_zero_nat )
    <=> ( ( A2 = bot_bot_set_int )
        | ~ aa_set_int_o(finite_finite_int,A2) ) ) ).

% card_eq_0_iff
tff(fact_8609_card__ge__0__finite,axiom,
    ! [A2: set_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_set_nat(A2))
     => aa_set_set_nat_o(finite1152437895449049373et_nat,A2) ) ).

% card_ge_0_finite
tff(fact_8610_card__ge__0__finite,axiom,
    ! [A2: set_list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_list_nat(A2))
     => finite8100373058378681591st_nat(A2) ) ).

% card_ge_0_finite
tff(fact_8611_card__ge__0__finite,axiom,
    ! [A2: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A2))
     => aa_set_nat_o(finite_finite_nat,A2) ) ).

% card_ge_0_finite
tff(fact_8612_card__ge__0__finite,axiom,
    ! [A2: set_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_complex(A2))
     => aa_set_complex_o(finite3207457112153483333omplex,A2) ) ).

% card_ge_0_finite
tff(fact_8613_card__ge__0__finite,axiom,
    ! [A2: set_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_int(A2))
     => aa_set_int_o(finite_finite_int,A2) ) ).

% card_ge_0_finite
tff(fact_8614_card__ge__0__finite,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite121521170596916366d_enat(A2))
     => aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2) ) ).

% card_ge_0_finite
tff(fact_8615_card__Suc__eq__finite,axiom,
    ! [A2: set_real,K: nat] :
      ( ( finite_card_real(A2) = aa_nat_nat(suc,K) )
    <=> ? [B5: real,B8: set_real] :
          ( ( A2 = insert_real2(B5,B8) )
          & ~ aa_set_real_o(member_real2(B5),B8)
          & ( finite_card_real(B8) = K )
          & aa_set_real_o(finite_finite_real,B8) ) ) ).

% card_Suc_eq_finite
tff(fact_8616_card__Suc__eq__finite,axiom,
    ! [A2: set_o,K: nat] :
      ( ( finite_card_o(A2) = aa_nat_nat(suc,K) )
    <=> ? [B5: $o,B8: set_o] :
          ( ( A2 = insert_o2((B5),B8) )
          & ~ aa_set_o_o(member_o2((B5)),B8)
          & ( finite_card_o(B8) = K )
          & aa_set_o_o(finite_finite_o,B8) ) ) ).

% card_Suc_eq_finite
tff(fact_8617_card__Suc__eq__finite,axiom,
    ! [A2: set_set_nat,K: nat] :
      ( ( finite_card_set_nat(A2) = aa_nat_nat(suc,K) )
    <=> ? [B5: set_nat,B8: set_set_nat] :
          ( ( A2 = insert_set_nat2(B5,B8) )
          & ~ aa_set_set_nat_o(member_set_nat2(B5),B8)
          & ( finite_card_set_nat(B8) = K )
          & aa_set_set_nat_o(finite1152437895449049373et_nat,B8) ) ) ).

% card_Suc_eq_finite
tff(fact_8618_card__Suc__eq__finite,axiom,
    ! [A2: set_list_nat,K: nat] :
      ( ( finite_card_list_nat(A2) = aa_nat_nat(suc,K) )
    <=> ? [B5: list_nat,B8: set_list_nat] :
          ( ( A2 = insert_list_nat(B5,B8) )
          & ~ member_list_nat(B5,B8)
          & ( finite_card_list_nat(B8) = K )
          & finite8100373058378681591st_nat(B8) ) ) ).

% card_Suc_eq_finite
tff(fact_8619_card__Suc__eq__finite,axiom,
    ! [A2: set_nat,K: nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A2) = aa_nat_nat(suc,K) )
    <=> ? [B5: nat,B8: set_nat] :
          ( ( A2 = insert_nat2(B5,B8) )
          & ~ aa_set_nat_o(member_nat2(B5),B8)
          & ( aa_set_nat_nat(finite_card_nat,B8) = K )
          & aa_set_nat_o(finite_finite_nat,B8) ) ) ).

% card_Suc_eq_finite
tff(fact_8620_card__Suc__eq__finite,axiom,
    ! [A2: set_complex,K: nat] :
      ( ( finite_card_complex(A2) = aa_nat_nat(suc,K) )
    <=> ? [B5: complex,B8: set_complex] :
          ( ( A2 = insert_complex(B5,B8) )
          & ~ aa_set_complex_o(member_complex(B5),B8)
          & ( finite_card_complex(B8) = K )
          & aa_set_complex_o(finite3207457112153483333omplex,B8) ) ) ).

% card_Suc_eq_finite
tff(fact_8621_card__Suc__eq__finite,axiom,
    ! [A2: set_int,K: nat] :
      ( ( finite_card_int(A2) = aa_nat_nat(suc,K) )
    <=> ? [B5: int,B8: set_int] :
          ( ( A2 = insert_int2(B5,B8) )
          & ~ aa_set_int_o(member_int2(B5),B8)
          & ( finite_card_int(B8) = K )
          & aa_set_int_o(finite_finite_int,B8) ) ) ).

% card_Suc_eq_finite
tff(fact_8622_card__Suc__eq__finite,axiom,
    ! [A2: set_Extended_enat,K: nat] :
      ( ( finite121521170596916366d_enat(A2) = aa_nat_nat(suc,K) )
    <=> ? [B5: extended_enat,B8: set_Extended_enat] :
          ( ( A2 = insert_Extended_enat(B5,B8) )
          & ~ aa_set3337841544853333262enat_o(member_Extended_enat(B5),B8)
          & ( finite121521170596916366d_enat(B8) = K )
          & aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B8) ) ) ).

% card_Suc_eq_finite
tff(fact_8623_card__insert__if,axiom,
    ! [A2: set_real,Xb: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( finite_card_real(insert_real2(Xb,A2)) = $ite(aa_set_real_o(member_real2(Xb),A2),finite_card_real(A2),aa_nat_nat(suc,finite_card_real(A2))) ) ) ).

% card_insert_if
tff(fact_8624_card__insert__if,axiom,
    ! [A2: set_o,Xb: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( finite_card_o(insert_o2((Xb),A2)) = $ite(aa_set_o_o(member_o2((Xb)),A2),finite_card_o(A2),aa_nat_nat(suc,finite_card_o(A2))) ) ) ).

% card_insert_if
tff(fact_8625_card__insert__if,axiom,
    ! [A2: set_set_nat,Xb: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( finite_card_set_nat(insert_set_nat2(Xb,A2)) = $ite(aa_set_set_nat_o(member_set_nat2(Xb),A2),finite_card_set_nat(A2),aa_nat_nat(suc,finite_card_set_nat(A2))) ) ) ).

% card_insert_if
tff(fact_8626_card__insert__if,axiom,
    ! [A2: set_list_nat,Xb: list_nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( finite_card_list_nat(insert_list_nat(Xb,A2)) = $ite(member_list_nat(Xb,A2),finite_card_list_nat(A2),aa_nat_nat(suc,finite_card_list_nat(A2))) ) ) ).

% card_insert_if
tff(fact_8627_card__insert__if,axiom,
    ! [A2: set_nat,Xb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(finite_card_nat,insert_nat2(Xb,A2)) = $ite(aa_set_nat_o(member_nat2(Xb),A2),aa_set_nat_nat(finite_card_nat,A2),aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,A2))) ) ) ).

% card_insert_if
tff(fact_8628_card__insert__if,axiom,
    ! [A2: set_complex,Xb: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( finite_card_complex(insert_complex(Xb,A2)) = $ite(aa_set_complex_o(member_complex(Xb),A2),finite_card_complex(A2),aa_nat_nat(suc,finite_card_complex(A2))) ) ) ).

% card_insert_if
tff(fact_8629_card__insert__if,axiom,
    ! [A2: set_int,Xb: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( finite_card_int(insert_int2(Xb,A2)) = $ite(aa_set_int_o(member_int2(Xb),A2),finite_card_int(A2),aa_nat_nat(suc,finite_card_int(A2))) ) ) ).

% card_insert_if
tff(fact_8630_card__insert__if,axiom,
    ! [A2: set_Extended_enat,Xb: extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( finite121521170596916366d_enat(insert_Extended_enat(Xb,A2)) = $ite(aa_set3337841544853333262enat_o(member_Extended_enat(Xb),A2),finite121521170596916366d_enat(A2),aa_nat_nat(suc,finite121521170596916366d_enat(A2))) ) ) ).

% card_insert_if
tff(fact_8631_card__mono,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B2)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(A2)),finite_card_set_nat(B2)) ) ) ).

% card_mono
tff(fact_8632_card__mono,axiom,
    ! [B2: set_list_nat,A2: set_list_nat] :
      ( finite8100373058378681591st_nat(B2)
     => ( ord_le6045566169113846134st_nat(A2,B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(A2)),finite_card_list_nat(B2)) ) ) ).

% card_mono
tff(fact_8633_card__mono,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(A2)),finite_card_complex(B2)) ) ) ).

% card_mono
tff(fact_8634_card__mono,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_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A2)),finite_card_int(B2)) ) ) ).

% card_mono
tff(fact_8635_card__mono,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(A2)),finite121521170596916366d_enat(B2)) ) ) ).

% card_mono
tff(fact_8636_card__mono,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_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A2)),aa_set_nat_nat(finite_card_nat,B2)) ) ) ).

% card_mono
tff(fact_8637_card__seteq,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B2)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(B2)),finite_card_set_nat(A2))
         => ( A2 = B2 ) ) ) ) ).

% card_seteq
tff(fact_8638_card__seteq,axiom,
    ! [B2: set_list_nat,A2: set_list_nat] :
      ( finite8100373058378681591st_nat(B2)
     => ( ord_le6045566169113846134st_nat(A2,B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(B2)),finite_card_list_nat(A2))
         => ( A2 = B2 ) ) ) ) ).

% card_seteq
tff(fact_8639_card__seteq,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(B2)),finite_card_complex(A2))
         => ( A2 = B2 ) ) ) ) ).

% card_seteq
tff(fact_8640_card__seteq,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_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(B2)),finite_card_int(A2))
         => ( A2 = B2 ) ) ) ) ).

% card_seteq
tff(fact_8641_card__seteq,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
     => ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(A2),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(B2)),finite121521170596916366d_enat(A2))
         => ( A2 = B2 ) ) ) ) ).

% card_seteq
tff(fact_8642_card__seteq,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_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,B2)),aa_set_nat_nat(finite_card_nat,A2))
         => ( A2 = B2 ) ) ) ) ).

% card_seteq
tff(fact_8643_obtain__subset__with__card__n,axiom,
    ! [Nb: nat,S3: set_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),finite_card_set_nat(S3))
     => ~ ! [T4: set_set_nat] :
            ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(T4),S3)
           => ( ( finite_card_set_nat(T4) = Nb )
             => ~ aa_set_set_nat_o(finite1152437895449049373et_nat,T4) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_8644_obtain__subset__with__card__n,axiom,
    ! [Nb: nat,S3: set_list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),finite_card_list_nat(S3))
     => ~ ! [T4: set_list_nat] :
            ( ord_le6045566169113846134st_nat(T4,S3)
           => ( ( finite_card_list_nat(T4) = Nb )
             => ~ finite8100373058378681591st_nat(T4) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_8645_obtain__subset__with__card__n,axiom,
    ! [Nb: nat,S3: set_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),finite_card_complex(S3))
     => ~ ! [T4: set_complex] :
            ( aa_set_complex_o(ord_le211207098394363844omplex(T4),S3)
           => ( ( finite_card_complex(T4) = Nb )
             => ~ aa_set_complex_o(finite3207457112153483333omplex,T4) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_8646_obtain__subset__with__card__n,axiom,
    ! [Nb: nat,S3: set_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),finite_card_int(S3))
     => ~ ! [T4: set_int] :
            ( aa_set_int_o(ord_less_eq_set_int(T4),S3)
           => ( ( finite_card_int(T4) = Nb )
             => ~ aa_set_int_o(finite_finite_int,T4) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_8647_obtain__subset__with__card__n,axiom,
    ! [Nb: nat,S3: set_Extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),finite121521170596916366d_enat(S3))
     => ~ ! [T4: set_Extended_enat] :
            ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(T4),S3)
           => ( ( finite121521170596916366d_enat(T4) = Nb )
             => ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,T4) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_8648_obtain__subset__with__card__n,axiom,
    ! [Nb: nat,S3: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_set_nat_nat(finite_card_nat,S3))
     => ~ ! [T4: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,T4),S3)
           => ( ( aa_set_nat_nat(finite_card_nat,T4) = Nb )
             => ~ aa_set_nat_o(finite_finite_nat,T4) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_8649_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_set_nat,C4: nat] :
      ( ! [G2: set_set_nat] :
          ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(G2),F3)
         => ( aa_set_set_nat_o(finite1152437895449049373et_nat,G2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(G2)),C4) ) )
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(F3)),C4) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_8650_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_list_nat,C4: nat] :
      ( ! [G2: set_list_nat] :
          ( ord_le6045566169113846134st_nat(G2,F3)
         => ( finite8100373058378681591st_nat(G2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(G2)),C4) ) )
     => ( finite8100373058378681591st_nat(F3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(F3)),C4) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_8651_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_complex,C4: nat] :
      ( ! [G2: set_complex] :
          ( aa_set_complex_o(ord_le211207098394363844omplex(G2),F3)
         => ( aa_set_complex_o(finite3207457112153483333omplex,G2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(G2)),C4) ) )
     => ( aa_set_complex_o(finite3207457112153483333omplex,F3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(F3)),C4) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_8652_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_int,C4: nat] :
      ( ! [G2: set_int] :
          ( aa_set_int_o(ord_less_eq_set_int(G2),F3)
         => ( aa_set_int_o(finite_finite_int,G2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(G2)),C4) ) )
     => ( aa_set_int_o(finite_finite_int,F3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(F3)),C4) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_8653_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_Extended_enat,C4: nat] :
      ( ! [G2: set_Extended_enat] :
          ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(G2),F3)
         => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,G2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(G2)),C4) ) )
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,F3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(F3)),C4) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_8654_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_nat,C4: nat] :
      ( ! [G2: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,G2),F3)
         => ( aa_set_nat_o(finite_finite_nat,G2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,G2)),C4) ) )
     => ( aa_set_nat_o(finite_finite_nat,F3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,F3)),C4) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_8655_card__less__sym__Diff,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(A2)),finite_card_set_nat(B2))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A2,B2))),finite_card_set_nat(minus_2163939370556025621et_nat(B2,A2))) ) ) ) ).

% card_less_sym_Diff
tff(fact_8656_card__less__sym__Diff,axiom,
    ! [A2: set_list_nat,B2: set_list_nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( finite8100373058378681591st_nat(B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(A2)),finite_card_list_nat(B2))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A2,B2))),finite_card_list_nat(minus_7954133019191499631st_nat(B2,A2))) ) ) ) ).

% card_less_sym_Diff
tff(fact_8657_card__less__sym__Diff,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_complex(A2)),finite_card_complex(B2))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_complex(minus_811609699411566653omplex(A2,B2))),finite_card_complex(minus_811609699411566653omplex(B2,A2))) ) ) ) ).

% card_less_sym_Diff
tff(fact_8658_card__less__sym__Diff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(A2)),finite_card_int(B2))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(minus_minus_set_int(A2,B2))),finite_card_int(minus_minus_set_int(B2,A2))) ) ) ) ).

% card_less_sym_Diff
tff(fact_8659_card__less__sym__Diff,axiom,
    ! [A2: set_Extended_enat,B2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite121521170596916366d_enat(A2)),finite121521170596916366d_enat(B2))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite121521170596916366d_enat(minus_925952699566721837d_enat(A2,B2))),finite121521170596916366d_enat(minus_925952699566721837d_enat(B2,A2))) ) ) ) ).

% card_less_sym_Diff
tff(fact_8660_card__less__sym__Diff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(finite_card_nat,A2)),aa_set_nat_nat(finite_card_nat,B2))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A2,B2))),aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(B2,A2))) ) ) ) ).

% card_less_sym_Diff
tff(fact_8661_card__le__sym__Diff,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(A2)),finite_card_set_nat(B2))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A2,B2))),finite_card_set_nat(minus_2163939370556025621et_nat(B2,A2))) ) ) ) ).

% card_le_sym_Diff
tff(fact_8662_card__le__sym__Diff,axiom,
    ! [A2: set_list_nat,B2: set_list_nat] :
      ( finite8100373058378681591st_nat(A2)
     => ( finite8100373058378681591st_nat(B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(A2)),finite_card_list_nat(B2))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A2,B2))),finite_card_list_nat(minus_7954133019191499631st_nat(B2,A2))) ) ) ) ).

% card_le_sym_Diff
tff(fact_8663_card__le__sym__Diff,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(A2)),finite_card_complex(B2))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(minus_811609699411566653omplex(A2,B2))),finite_card_complex(minus_811609699411566653omplex(B2,A2))) ) ) ) ).

% card_le_sym_Diff
tff(fact_8664_card__le__sym__Diff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A2)),finite_card_int(B2))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(minus_minus_set_int(A2,B2))),finite_card_int(minus_minus_set_int(B2,A2))) ) ) ) ).

% card_le_sym_Diff
tff(fact_8665_card__le__sym__Diff,axiom,
    ! [A2: set_Extended_enat,B2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(A2)),finite121521170596916366d_enat(B2))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite121521170596916366d_enat(minus_925952699566721837d_enat(A2,B2))),finite121521170596916366d_enat(minus_925952699566721837d_enat(B2,A2))) ) ) ) ).

% card_le_sym_Diff
tff(fact_8666_card__le__sym__Diff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A2)),aa_set_nat_nat(finite_card_nat,B2))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A2,B2))),aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(B2,A2))) ) ) ) ).

% card_le_sym_Diff
tff(fact_8667_card__length,axiom,
    ! [Xs: list_complex] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(set_complex2(Xs))),size_s3451745648224563538omplex(Xs)) ).

% card_length
tff(fact_8668_card__length,axiom,
    ! [Xs: list_set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(set_set_nat2(Xs))),size_s3254054031482475050et_nat(Xs)) ).

% card_length
tff(fact_8669_card__length,axiom,
    ! [Xs: list_list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(set_list_nat2(Xs))),size_s3023201423986296836st_nat(Xs)) ).

% card_length
tff(fact_8670_card__length,axiom,
    ! [Xs: list_VEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite7802652506058667612T_VEBT(set_VEBT_VEBT2(Xs))),size_s6755466524823107622T_VEBT(Xs)) ).

% card_length
tff(fact_8671_card__length,axiom,
    ! [Xs: list_int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(set_int2(Xs))),size_size_list_int(Xs)) ).

% card_length
tff(fact_8672_card__length,axiom,
    ! [Xs: list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,set_nat2(Xs))),size_size_list_nat(Xs)) ).

% card_length
tff(fact_8673_card__1__singletonE,axiom,
    ! [A2: set_complex] :
      ( ( finite_card_complex(A2) = one_one_nat )
     => ~ ! [X4: complex] : A2 != insert_complex(X4,bot_bot_set_complex) ) ).

% card_1_singletonE
tff(fact_8674_card__1__singletonE,axiom,
    ! [A2: set_set_nat] :
      ( ( finite_card_set_nat(A2) = one_one_nat )
     => ~ ! [X4: set_nat] : A2 != insert_set_nat2(X4,bot_bot_set_set_nat) ) ).

% card_1_singletonE
tff(fact_8675_card__1__singletonE,axiom,
    ! [A2: set_list_nat] :
      ( ( finite_card_list_nat(A2) = one_one_nat )
     => ~ ! [X4: list_nat] : A2 != insert_list_nat(X4,bot_bot_set_list_nat) ) ).

% card_1_singletonE
tff(fact_8676_card__1__singletonE,axiom,
    ! [A2: set_real] :
      ( ( finite_card_real(A2) = one_one_nat )
     => ~ ! [X4: real] : A2 != insert_real2(X4,bot_bot_set_real) ) ).

% card_1_singletonE
tff(fact_8677_card__1__singletonE,axiom,
    ! [A2: set_o] :
      ( ( finite_card_o(A2) = one_one_nat )
     => ~ ! [X4: $o] : A2 != insert_o2((X4),bot_bot_set_o) ) ).

% card_1_singletonE
tff(fact_8678_card__1__singletonE,axiom,
    ! [A2: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A2) = one_one_nat )
     => ~ ! [X4: nat] : A2 != insert_nat2(X4,bot_bot_set_nat) ) ).

% card_1_singletonE
tff(fact_8679_card__1__singletonE,axiom,
    ! [A2: set_int] :
      ( ( finite_card_int(A2) = one_one_nat )
     => ~ ! [X4: int] : A2 != insert_int2(X4,bot_bot_set_int) ) ).

% card_1_singletonE
tff(fact_8680_psubset__card__mono,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B2)
     => ( aa_set_set_nat_o(ord_less_set_set_nat(A2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(A2)),finite_card_set_nat(B2)) ) ) ).

% psubset_card_mono
tff(fact_8681_psubset__card__mono,axiom,
    ! [B2: set_list_nat,A2: set_list_nat] :
      ( finite8100373058378681591st_nat(B2)
     => ( ord_le1190675801316882794st_nat(A2,B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(A2)),finite_card_list_nat(B2)) ) ) ).

% psubset_card_mono
tff(fact_8682_psubset__card__mono,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_set_nat,A2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(finite_card_nat,A2)),aa_set_nat_nat(finite_card_nat,B2)) ) ) ).

% psubset_card_mono
tff(fact_8683_psubset__card__mono,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_less_set_complex(A2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_complex(A2)),finite_card_complex(B2)) ) ) ).

% psubset_card_mono
tff(fact_8684_psubset__card__mono,axiom,
    ! [B2: set_int,A2: set_int] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(ord_less_set_int(A2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(A2)),finite_card_int(B2)) ) ) ).

% psubset_card_mono
tff(fact_8685_psubset__card__mono,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,B2)
     => ( aa_set3337841544853333262enat_o(ord_le2529575680413868914d_enat(A2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite121521170596916366d_enat(A2)),finite121521170596916366d_enat(B2)) ) ) ).

% psubset_card_mono
tff(fact_8686_card__distinct,axiom,
    ! [Xs: list_complex] :
      ( ( finite_card_complex(set_complex2(Xs)) = size_s3451745648224563538omplex(Xs) )
     => distinct_complex(Xs) ) ).

% card_distinct
tff(fact_8687_card__distinct,axiom,
    ! [Xs: list_set_nat] :
      ( ( finite_card_set_nat(set_set_nat2(Xs)) = size_s3254054031482475050et_nat(Xs) )
     => distinct_set_nat(Xs) ) ).

% card_distinct
tff(fact_8688_card__distinct,axiom,
    ! [Xs: list_list_nat] :
      ( ( finite_card_list_nat(set_list_nat2(Xs)) = size_s3023201423986296836st_nat(Xs) )
     => distinct_list_nat(Xs) ) ).

% card_distinct
tff(fact_8689_card__distinct,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( ( finite7802652506058667612T_VEBT(set_VEBT_VEBT2(Xs)) = size_s6755466524823107622T_VEBT(Xs) )
     => aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Xs) ) ).

% card_distinct
tff(fact_8690_card__distinct,axiom,
    ! [Xs: list_int] :
      ( ( finite_card_int(set_int2(Xs)) = size_size_list_int(Xs) )
     => aa_list_int_o(distinct_int,Xs) ) ).

% card_distinct
tff(fact_8691_card__distinct,axiom,
    ! [Xs: list_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,set_nat2(Xs)) = size_size_list_nat(Xs) )
     => aa_list_nat_o(distinct_nat,Xs) ) ).

% card_distinct
tff(fact_8692_distinct__card,axiom,
    ! [Xs: list_complex] :
      ( distinct_complex(Xs)
     => ( finite_card_complex(set_complex2(Xs)) = size_s3451745648224563538omplex(Xs) ) ) ).

% distinct_card
tff(fact_8693_distinct__card,axiom,
    ! [Xs: list_set_nat] :
      ( distinct_set_nat(Xs)
     => ( finite_card_set_nat(set_set_nat2(Xs)) = size_s3254054031482475050et_nat(Xs) ) ) ).

% distinct_card
tff(fact_8694_distinct__card,axiom,
    ! [Xs: list_list_nat] :
      ( distinct_list_nat(Xs)
     => ( finite_card_list_nat(set_list_nat2(Xs)) = size_s3023201423986296836st_nat(Xs) ) ) ).

% distinct_card
tff(fact_8695_distinct__card,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Xs)
     => ( finite7802652506058667612T_VEBT(set_VEBT_VEBT2(Xs)) = size_s6755466524823107622T_VEBT(Xs) ) ) ).

% distinct_card
tff(fact_8696_distinct__card,axiom,
    ! [Xs: list_int] :
      ( aa_list_int_o(distinct_int,Xs)
     => ( finite_card_int(set_int2(Xs)) = size_size_list_int(Xs) ) ) ).

% distinct_card
tff(fact_8697_distinct__card,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(distinct_nat,Xs)
     => ( aa_set_nat_nat(finite_card_nat,set_nat2(Xs)) = size_size_list_nat(Xs) ) ) ).

% distinct_card
tff(fact_8698_card__less__Suc2,axiom,
    ! [M6: set_nat,Ia: nat] :
      ( ~ aa_set_nat_o(member_nat2(zero_zero_nat),M6)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_wr(M6),Ia))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ws(M6),Ia))) ) ) ).

% card_less_Suc2
tff(fact_8699_card__less__Suc,axiom,
    ! [M6: set_nat,Ia: nat] :
      ( aa_set_nat_o(member_nat2(zero_zero_nat),M6)
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_wr(M6),Ia)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ws(M6),Ia))) ) ) ).

% card_less_Suc
tff(fact_8700_card__less,axiom,
    ! [M6: set_nat,Ia: nat] :
      ( aa_set_nat_o(member_nat2(zero_zero_nat),M6)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ws(M6),Ia))) != zero_zero_nat ) ) ).

% card_less
tff(fact_8701_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),aa_nat_set_nat(aa_nat1818925681583257441et_nat(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 = aa_nat_set_nat(aa_nat1818925681583257441et_nat(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_8702_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N4: set_nat,Nb: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N4),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N4)),Nb) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_8703_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_bc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,S3)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bc),S3)) ).

% card_sum_le_nat_sum
tff(fact_8704_card__nth__roots,axiom,
    ! [C: complex,Nb: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_wt(C),Nb))) = Nb ) ) ) ).

% card_nth_roots
tff(fact_8705_card__roots__unity__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( finite_card_complex(collect_complex(aTP_Lamm_cd(Nb))) = Nb ) ) ).

% card_roots_unity_eq
tff(fact_8706_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_ak,aTP_Lamm_aj) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_8707_times__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_wv),Xaa),Xb)) ).

% times_int.abs_eq
tff(fact_8708_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_nat2(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_8709_int_Oabs__induct,axiom,
    ! [P: fun_int_o,Xb: int] :
      ( ! [Y3: product_prod_nat_nat] : aa_int_o(P,aa_Pro4240683299535347697at_int(abs_Integ,Y3))
     => aa_int_o(P,Xb) ) ).

% int.abs_induct
tff(fact_8710_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X4: nat,Y3: nat] : Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y3)) ).

% eq_Abs_Integ
tff(fact_8711_nat_Oabs__eq,axiom,
    ! [Xb: product_prod_nat_nat] : aa_int_nat(nat2,aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),Xb) ).

% nat.abs_eq
tff(fact_8712_zero__int__def,axiom,
    zero_zero_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),zero_zero_nat)) ).

% zero_int_def
tff(fact_8713_int__def,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,Nb) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Nb),zero_zero_nat)) ).

% int_def
tff(fact_8714_uminus__int_Oabs__eq,axiom,
    ! [Xb: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ww),Xb)) ).

% uminus_int.abs_eq
tff(fact_8715_one__int__def,axiom,
    one_one_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,one_one_nat),zero_zero_nat)) ).

% one_int_def
tff(fact_8716_less__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_wy),Xaa),Xb) ) ).

% less_int.abs_eq
tff(fact_8717_less__eq__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_xa),Xaa),Xb) ) ).

% less_eq_int.abs_eq
tff(fact_8718_plus__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_xc),Xaa),Xb)) ).

% plus_int.abs_eq
tff(fact_8719_minus__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_xe),Xaa),Xb)) ).

% minus_int.abs_eq
tff(fact_8720_finite__enumerate,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ? [R3: fun_nat_nat] :
          ( aa_set_nat_o(strict1292158309912662752at_nat(R3),aa_nat_set_nat(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_nat2(aa_nat_nat(R3,N5)),S3) ) ) ) ).

% finite_enumerate
tff(fact_8721_less__eq__int_Orep__eq,axiom,
    ! [Xb: int,Xaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_xa),aa_int1417691210696471063at_nat(rep_Integ,Xb)),aa_int1417691210696471063at_nat(rep_Integ,Xaa)) ) ).

% less_eq_int.rep_eq
tff(fact_8722_less__int_Orep__eq,axiom,
    ! [Xb: int,Xaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Xaa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_wy),aa_int1417691210696471063at_nat(rep_Integ,Xb)),aa_int1417691210696471063at_nat(rep_Integ,Xaa)) ) ).

% less_int.rep_eq
tff(fact_8723_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_xf) ).

% prod_encode_def
tff(fact_8724_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(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B))) ).

% le_prod_encode_1
tff(fact_8725_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(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B))) ).

% le_prod_encode_2
tff(fact_8726_nat_Orep__eq,axiom,
    ! [Xb: int] : aa_int_nat(nat2,Xb) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),aa_int1417691210696471063at_nat(rep_Integ,Xb)) ).

% nat.rep_eq
tff(fact_8727_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,Mb: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(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_8728_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_ww)) ).

% uminus_int_def
tff(fact_8729_num__of__nat_Osimps_I2_J,axiom,
    ! [Nb: nat] :
      num_of_nat(aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb),inc(num_of_nat(Nb)),one) ).

% num_of_nat.simps(2)
tff(fact_8730_num__of__nat_Osimps_I1_J,axiom,
    num_of_nat(zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_8731_numeral__num__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_num_nat(numeral_numeral_nat,num_of_nat(Nb)) = Nb ) ) ).

% numeral_num_of_nat
tff(fact_8732_num__of__nat__One,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),one_one_nat)
     => ( num_of_nat(Nb) = one ) ) ).

% num_of_nat_One
tff(fact_8733_num__of__nat__double,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( num_of_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Nb)) = bit0(num_of_nat(Nb)) ) ) ).

% num_of_nat_double
tff(fact_8734_num__of__nat__plus__distrib,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( num_of_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = plus_plus_num(num_of_nat(Mb),num_of_nat(Nb)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_8735_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_wv)) ).

% times_int_def
tff(fact_8736_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_xe)) ).

% minus_int_def
tff(fact_8737_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_xc)) ).

% plus_int_def
tff(fact_8738_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Xb: nat,Y: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xg,C),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Xb),Y)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Y),
        aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),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,Xb),Y),insert_nat2(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_8739_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
tff(fact_8740_image__Suc__atLeastAtMost,axiom,
    ! [Ia: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Ia),J)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Ia)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_8741_image__Suc__atLeastLessThan,axiom,
    ! [Ia: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Ia),J)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,Ia)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_8742_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] : ~ aa_set_nat_o(member_nat2(zero_zero_nat),image_nat_nat(suc,A2)) ).

% zero_notin_Suc_image
tff(fact_8743_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_wy)) ).

% less_int_def
tff(fact_8744_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_xa)) ).

% less_eq_int_def
tff(fact_8745_nat__def,axiom,
    nat2 = map_fu2345160673673942751at_nat(rep_Integ,id_nat,produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat_def
tff(fact_8746_image__Suc__lessThan,axiom,
    ! [Nb: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),Nb) ).

% image_Suc_lessThan
tff(fact_8747_image__Suc__atMost,axiom,
    ! [Nb: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),aa_nat_nat(suc,Nb)) ).

% image_Suc_atMost
tff(fact_8748_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat2(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),Nb))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_8749_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat2(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_8750_lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)) = insert_nat2(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% lessThan_Suc_eq_insert_0
tff(fact_8751_atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb)) = insert_nat2(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% atMost_Suc_eq_insert_0
tff(fact_8752_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_8753_Inf__nat__def1,axiom,
    ! [K5: set_nat] :
      ( ( K5 != bot_bot_set_nat )
     => aa_set_nat_o(member_nat2(aa_set_nat_nat(complete_Inf_Inf_nat,K5)),K5) ) ).

% Inf_nat_def1
tff(fact_8754_UNIV__nat__eq,axiom,
    top_top_set_nat = insert_nat2(zero_zero_nat,image_nat_nat(suc,top_top_set_nat)) ).

% UNIV_nat_eq
tff(fact_8755_range__mod,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xh,Nb),top_top_set_nat) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Nb) ) ) ).

% range_mod
tff(fact_8756_range__mult,axiom,
    ! [A: real] :
      image_real_real(times_times_real(A),top_top_set_real) = $ite(A = zero_zero_real,insert_real2(zero_zero_real,bot_bot_set_real),top_top_set_real) ).

% range_mult
tff(fact_8757_infinite__UNIV__int,axiom,
    ~ aa_set_int_o(finite_finite_int,top_top_set_int) ).

% infinite_UNIV_int
tff(fact_8758_int__in__range__abs,axiom,
    ! [Nb: nat] : aa_set_int_o(member_int2(aa_nat_int(semiri1314217659103216013at_int,Nb)),image_int_int(abs_abs_int,top_top_set_int)) ).

% int_in_range_abs
tff(fact_8759_root__def,axiom,
    ! [Nb: nat,Xb: real] :
      aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = $ite(Nb = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_xi,Nb)),Xb)) ).

% root_def
tff(fact_8760_DERIV__even__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_8761_DERIV__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_8762_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xb: real,Nb: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M: nat,X4: 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)),X4),topolo2177554685111907308n_real(X4,top_top_set_real))
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(F,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xj(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(Xb),Nb))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_8763_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xb: real,Nb: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M: nat,X4: 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)),X4),topolo2177554685111907308n_real(X4,top_top_set_real)) )
     => ? [T5: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xb))
          & ( aa_real_real(F,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xj(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(Xb),Nb))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_8764_DERIV__odd__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
     => ( ( Xb != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_8765_Maclaurin,axiom,
    ! [H: real,Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M: nat,T5: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),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)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
           => ? [T5: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),H)
                & ( aa_real_real(F,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_qg(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(H),Nb))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_8766_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T5: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),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)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
         => ? [T5: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T5)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),H)
              & ( aa_real_real(F,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_qg(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(H),Nb))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_8767_Maclaurin__minus,axiom,
    ! [H: real,Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),zero_zero_real)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M: nat,T5: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),T5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),zero_zero_real) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
           => ? [T5: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),zero_zero_real)
                & ( aa_real_real(F,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_qg(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(H),Nb))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_8768_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,Xb: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( Xb != zero_zero_real )
         => ( ! [M: nat,X4: 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)),X4),topolo2177554685111907308n_real(X4,top_top_set_real))
           => ? [T5: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T5))
                & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xb))
                & ( aa_real_real(F,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xj(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(Xb),Nb))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_8769_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,Xb: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M: nat,T5: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xb)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(F,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xj(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(Xb),Nb))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_8770_Taylor__down,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T5: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),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)),T5),topolo2177554685111907308n_real(T5,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)
             => ? [T5: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),T5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),C)
                  & ( aa_real_real(F,A) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_xk(Diff),A),C)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(A),C)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_8771_Taylor__up,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T5: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),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)),T5),topolo2177554685111907308n_real(T5,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)
             => ? [T5: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),B)
                  & ( aa_real_real(F,B) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_xk(Diff),B),C)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(B),C)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_8772_Taylor,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T5: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),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)),T5),topolo2177554685111907308n_real(T5,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),Xb)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),B)
                 => ( ( Xb != C )
                   => ? [T5: real] :
                        ( $ite(
                            aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),C),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),T5)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),C) ),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T5)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),Xb) ) )
                        & ( aa_real_real(F,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_xl(Diff),C),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(Xb),C)),Nb))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_8773_Maclaurin__lemma2,axiom,
    ! [Nb: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B2: real] :
      ( ! [M: nat,T5: real] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),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)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
     => ( ( Nb = aa_nat_nat(suc,K) )
       => ! [M2: nat,T6: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T6)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),H) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_xn(Nb),Diff),B2),M2),aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T6)),aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_xo(Diff),M2),T6)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M2))))),aa_real_real(times_times_real(B2),aa_real_real(divide_divide_real(aa_nat_real(power_power_real(T6),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M2)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M2))))))),topolo2177554685111907308n_real(T6,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_8774_DERIV__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => has_fi5821293074295781190e_real(aTP_Lamm_xp,suminf_real(aTP_Lamm_xq(Xb)),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_8775_DERIV__real__root__generic,axiom,
    ! [Nb: nat,Xb: real,D5: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( Xb != zero_zero_real )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
             => ( D5 = aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
               => ( D5 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)
               => ( D5 = aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),D5,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_8776_DERIV__pow,axiom,
    ! [Nb: nat,Xb: real,S: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_xr,Nb),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(power_power_real(Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(Xb,S)) ).

% DERIV_pow
tff(fact_8777_DERIV__power__series_H,axiom,
    ! [R: real,F: fun_nat_real,X0: real] :
      ( ! [X4: real] :
          ( aa_set_real_o(member_real2(X4),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_xs(F),X4)) )
     => ( aa_set_real_o(member_real2(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_xu(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_xs(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_8778_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(aa_nat1818925681583257441et_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_8779_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,L)),U) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or5834768355832116004an_nat,L),U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_8780_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Nb: nat,J: nat,Ia: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,Ia)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or5834768355832116004an_nat,Ia),J))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_8781_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_real2(suminf_real(aTP_Lamm_xv(A))),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xv(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N5)),one_one_nat))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xv(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N5))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_8782_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_real2(suminf_real(aTP_Lamm_xv(A))),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xv(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N5))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xv(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N5)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_8783_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_xw(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_xv(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_8784_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(aTP_Lamm_xv(A))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xv(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)),one_one_nat)))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_8785_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
tff(fact_8786_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_8787_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_8788_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_xx,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_8789_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_xy(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_8790_LIMSEQ__inverse__zero,axiom,
    ! [X5: fun_nat_real] :
      ( ! [R3: 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,R3),aa_nat_real(X5,N)) )
     => filterlim_nat_real(aTP_Lamm_xz(X5),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_8791_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_ya,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_8792_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim_nat_real(aTP_Lamm_yb(R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_8793_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(plus_plus_real(aa_nat_real(F,N5)),E)) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_8794_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim_nat_real(aTP_Lamm_yc(R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_8795_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim_nat_real(aTP_Lamm_yd(R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_8796_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_xv(A)) ) ) ) ).

% summable
tff(fact_8797_zeroseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => filterlim_nat_real(aTP_Lamm_si(Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_8798_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_ye(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_xv(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_8799_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xv(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb)))),suminf_real(aTP_Lamm_xv(A))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_8800_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,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xv(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N5)))),L3)
              & filterlim_nat_real(aTP_Lamm_ye(A),topolo2815343760600316023s_real(L3),at_top_nat)
              & ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xv(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N5)),one_one_nat))))
              & filterlim_nat_real(aTP_Lamm_xw(A),topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_8801_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_xw(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_xv(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_8802_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_yf(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_8803_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_yg(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_8804_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_8805_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X4)
         => aa_nat_o(P,X4) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_8806_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_8807_sequentially__offset,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_yg(P),K),at_top_nat) ) ).

% sequentially_offset
tff(fact_8808_filterlim__pow__at__bot__even,axiom,
    ! [Nb: nat,F: fun_real_real,F3: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( 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,bit0(one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yh(Nb),F),at_top_real,F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_8809_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,L)),U) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or6659071591806873216st_nat,L),U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_8810_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Nb: nat,J: nat,Ia: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Ia))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or6659071591806873216st_nat,Ia),J))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_8811_filterlim__pow__at__bot__odd,axiom,
    ! [Nb: nat,F: fun_real_real,F3: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( 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,bit0(one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yh(Nb),F),at_bot_real,F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_8812_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_8813_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_8814_mono__times__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb)) ) ).

% mono_times_nat
tff(fact_8815_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_8816_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_8817_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_nat2(aa_nat_nat(suc,K),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_8818_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,bit0(one))),K)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yi,K)) ) ).

% mono_ge2_power_minus_self
tff(fact_8819_atLeast__0,axiom,
    aa_nat_set_nat(set_ord_atLeast_nat,zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_8820_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_or1210151606488870762an_nat,K) ).

% atLeast_Suc_greaterThan
tff(fact_8821_atLeast__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atLeast_nat,K),insert_nat2(K,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_8822_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_nat2(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_yj,M6))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_8823_Max__divisors__self__nat,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bj,Nb))) = Nb ) ) ).

% Max_divisors_self_nat
tff(fact_8824_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_8825_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_8826_divide__nat__def,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_yk(Mb),Nb)))) ).

% divide_nat_def
tff(fact_8827_range__abs__Nats,axiom,
    image_int_int(abs_abs_int,top_top_set_int) = semiring_1_Nats_int ).

% range_abs_Nats
tff(fact_8828_inj__sgn__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_set_real_o(inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_xi,Nb)),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_8829_inj__Suc,axiom,
    ! [N4: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N4) ).

% inj_Suc
tff(fact_8830_inj__on__diff__nat,axiom,
    ! [N4: set_nat,K: nat] :
      ( ! [N: nat] :
          ( aa_set_nat_o(member_nat2(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_xg,K)),N4) ) ).

% inj_on_diff_nat
tff(fact_8831_pred__nat__def,axiom,
    pred_nat = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_yl)) ).

% pred_nat_def
tff(fact_8832_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xb),Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( ( Deg = Xaa )
                & $let(
                    n: nat,
                    n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))),
                    $let(
                      m2: nat,
                      m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                      ( ! [X3: vEBT_VEBT] :
                          ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X3),n) )
                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary),m2)
                      & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),m2) )
                      & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X6)
                            & ! [X3: vEBT_VEBT] :
                                ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeList))
                               => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ym(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_8833_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xb),Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ~ ( ( Deg = Xaa )
                  & $let(
                      n: nat,
                      n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))),
                      $let(
                        m2: nat,
                        m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                        ( ! [X3: vEBT_VEBT] :
                            ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X3),n) )
                        & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary),m2)
                        & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),m2) )
                        & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X6)
                              & ! [X3: vEBT_VEBT] :
                                  ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ym(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_8834_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(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,vEBT_Node(Mima2,Dega,TreeLista,Summarya)),Deg3)
    <=> ( ( Dega = Deg3 )
        & $let(
            n: nat,
            n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one))),
            $let(
              m2: nat,
              m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Dega),n),
              ( ! [X3: vEBT_VEBT] :
                  ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeLista))
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X3),n) )
              & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summarya),m2)
              & ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),m2) )
              & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summarya),X6)
                    & ! [X3: vEBT_VEBT] :
                        ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ym(Dega),TreeLista),Summarya),n),m2))),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_8835_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xb),Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( (Y)
          <=> ( Xaa != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( (Y)
              <=> ~ ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                          ( ! [X3: vEBT_VEBT] :
                              ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X3),n) )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary),m2)
                          & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),m2) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X6)
                                & ! [X3: vEBT_VEBT] :
                                    ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ym(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_8836_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xb),Xaa)
      <=> (Y) )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( (Y)
                <=> ( Xaa = one_one_nat ) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Leaf((Uu2),(Uv2))),Xaa)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( ( (Y)
                  <=> ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                            ( ! [X3: vEBT_VEBT] :
                                ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X3),n) )
                            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary),m2)
                            & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),m2) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X6)
                                  & ! [X3: vEBT_VEBT] :
                                      ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeList))
                                     => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ym(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(Mima,Deg,TreeList,Summary)),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_8837_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Leaf((Uu2),(Uv2))),Xaa))
               => ( Xaa != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(Mima,Deg,TreeList,Summary)),Xaa))
                 => ~ ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                            ( ! [X3: vEBT_VEBT] :
                                ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X3),n) )
                            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary),m2)
                            & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),m2) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X6)
                                  & ! [X3: vEBT_VEBT] :
                                      ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeList))
                                     => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ym(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_8838_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Leaf((Uu2),(Uv2))),Xaa))
               => ( Xaa = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,vEBT_Node(Mima,Deg,TreeList,Summary)),Xaa))
                 => ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                          ( ! [X3: vEBT_VEBT] :
                              ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X3),n) )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary),m2)
                          & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),m2) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X6)
                                & ! [X3: vEBT_VEBT] :
                                    ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ym(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_8839_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_8840_take__bit__num__simps_I2_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(aa_nat_nat(suc,Nb),one) = some_num(one) ).

% take_bit_num_simps(2)
tff(fact_8841_take__bit__num__simps_I3_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),bit0(Mb)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_yn,bit_take_bit_num(Nb,Mb)) ).

% take_bit_num_simps(3)
tff(fact_8842_take__bit__num__simps_I4_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit1,Mb)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Nb,Mb))) ).

% take_bit_num_simps(4)
tff(fact_8843_take__bit__num__def,axiom,
    ! [Nb: nat,Mb: num] :
      bit_take_bit_num(Nb,Mb) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Mb)) = zero_zero_nat,none_num,some_num(num_of_nat(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Mb))))) ).

% take_bit_num_def
tff(fact_8844_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_yo) ).

% Rats_eq_int_div_nat
tff(fact_8845_min__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)) ).

% min_Suc_Suc
tff(fact_8846_min__0L,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% min_0L
tff(fact_8847_min__0R,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_8848_min__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),Nb)) ).

% min_numeral_Suc
tff(fact_8849_min__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),aa_num_nat(pred_numeral,K))) ).

% min_Suc_numeral
tff(fact_8850_nat__mult__min__left,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ).

% nat_mult_min_left
tff(fact_8851_nat__mult__min__right,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q2)) ).

% nat_mult_min_right
tff(fact_8852_min__diff,axiom,
    ! [Mb: nat,Ia: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Ia)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ia)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)),Ia) ).

% min_diff
tff(fact_8853_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_8854_atLeastLessThan__add__Un,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Ia),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = sup_sup_set_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Ia),J),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_8855_min__Suc1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Nb)),Mb) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_yp,Nb)),Mb) ).

% min_Suc1
tff(fact_8856_min__Suc2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_yq,Nb)),Mb) ).

% min_Suc2
tff(fact_8857_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
tff(fact_8858_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_8859_inf__int__def,axiom,
    inf_inf_int = ord_min_int ).

% inf_int_def
tff(fact_8860_binomial__def,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_yr(Nb),K))) ).

% binomial_def
tff(fact_8861_less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mb),Nb),transi6264000038957366511cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% less_eq
tff(fact_8862_pred__nat__trancl__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mb),Nb),transi2905341329935302413cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% pred_nat_trancl_eq_le
tff(fact_8863_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,K)),cons_nat(K,nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_8864_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,K)),cons_nat(aa_nat_nat(suc,K),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_8865_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Ia: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Ia)),J)
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or6659071591806873216st_nat,Ia),J)) = cons_nat(aa_nat_nat(suc,Ia),linord2614967742042102400et_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or6659071591806873216st_nat,aa_nat_nat(suc,Ia)),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_8866_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Ia: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Ia)),J)
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or5834768355832116004an_nat,Ia),J)) = cons_nat(aa_nat_nat(suc,Ia),linord2614967742042102400et_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or5834768355832116004an_nat,aa_nat_nat(suc,Ia)),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_8867_list__encode_Oelims,axiom,
    ! [Xb: list_nat,Y: nat] :
      ( ( nat_list_encode(Xb) = Y )
     => ( ( ( Xb = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X4: nat,Xs2: list_nat] :
              ( ( Xb = cons_nat(X4,Xs2) )
             => ( Y != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),nat_list_encode(Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_8868_list__encode_Osimps_I1_J,axiom,
    nat_list_encode(nil_nat) = zero_zero_nat ).

% list_encode.simps(1)
tff(fact_8869_list__encode_Osimps_I2_J,axiom,
    ! [Xb: nat,Xs: list_nat] : nat_list_encode(cons_nat(Xb,Xs)) = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),nat_list_encode(Xs)))) ).

% list_encode.simps(2)
tff(fact_8870_upto__aux__rec,axiom,
    ! [Ia: int,J: int,Js: list_int] :
      upto_aux(Ia,J,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),Ia),Js,upto_aux(Ia,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js))) ).

% upto_aux_rec
tff(fact_8871_upto_Opsimps,axiom,
    ! [Ia: int,J: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Ia),J))
     => ( upto(Ia,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),J),cons_int(Ia,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ia),one_one_int),J)),nil_int) ) ) ).

% upto.psimps
tff(fact_8872_upto_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: list_int] :
      ( ( upto(Xb,Xaa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa))
       => ~ ( ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa),cons_int(Xb,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int),Xaa)),nil_int) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa)) ) ) ) ).

% upto.pelims
tff(fact_8873_upto__Nil,axiom,
    ! [Ia: int,J: int] :
      ( ( upto(Ia,J) = nil_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),Ia) ) ).

% upto_Nil
tff(fact_8874_upto__Nil2,axiom,
    ! [Ia: int,J: int] :
      ( ( nil_int = upto(Ia,J) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),Ia) ) ).

% upto_Nil2
tff(fact_8875_upto__empty,axiom,
    ! [J: int,Ia: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,J),Ia)
     => ( upto(Ia,J) = nil_int ) ) ).

% upto_empty
tff(fact_8876_upto__single,axiom,
    ! [Ia: int] : upto(Ia,Ia) = cons_int(Ia,nil_int) ).

% upto_single
tff(fact_8877_nth__upto,axiom,
    ! [Ia: 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,Ia),aa_nat_int(semiri1314217659103216013at_int,K))),J)
     => ( aa_nat_int(nth_int(upto(Ia,J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Ia),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_8878_length__upto,axiom,
    ! [Ia: int,J: int] : size_size_list_int(upto(Ia,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),Ia)),one_one_int)) ).

% length_upto
tff(fact_8879_upto__rec__numeral_I1_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(numeral_numeral_int(Mb),numeral_numeral_int(Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(Mb)),numeral_numeral_int(Nb)),cons_int(numeral_numeral_int(Mb),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(Mb)),one_one_int),numeral_numeral_int(Nb))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_8880_upto__rec__numeral_I4_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,numeral_numeral_int(Mb)),aa_int_int(uminus_uminus_int,numeral_numeral_int(Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(Mb))),aa_int_int(uminus_uminus_int,numeral_numeral_int(Nb))),cons_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Mb)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(Mb))),one_one_int),aa_int_int(uminus_uminus_int,numeral_numeral_int(Nb)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_8881_upto__rec__numeral_I3_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,numeral_numeral_int(Mb)),numeral_numeral_int(Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(Mb))),numeral_numeral_int(Nb)),cons_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(Mb)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(Mb))),one_one_int),numeral_numeral_int(Nb))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_8882_upto__rec__numeral_I2_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(numeral_numeral_int(Mb),aa_int_int(uminus_uminus_int,numeral_numeral_int(Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(Mb)),aa_int_int(uminus_uminus_int,numeral_numeral_int(Nb))),cons_int(numeral_numeral_int(Mb),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(Mb)),one_one_int),aa_int_int(uminus_uminus_int,numeral_numeral_int(Nb)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_8883_distinct__upto,axiom,
    ! [Ia: int,J: int] : aa_list_int_o(distinct_int,upto(Ia,J)) ).

% distinct_upto
tff(fact_8884_atLeastAtMost__upto,axiom,
    ! [Ia: int,J: int] : set_or1266510415728281911st_int(Ia,J) = set_int2(upto(Ia,J)) ).

% atLeastAtMost_upto
tff(fact_8885_atLeastLessThan__upto,axiom,
    ! [Ia: int,J: int] : set_or4662586982721622107an_int(Ia,J) = set_int2(upto(Ia,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_8886_greaterThanAtMost__upto,axiom,
    ! [Ia: int,J: int] : set_or6656581121297822940st_int(Ia,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ia),one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_8887_upto__aux__def,axiom,
    ! [Ia: int,J: int,Js: list_int] : upto_aux(Ia,J,Js) = append_int(upto(Ia,J),Js) ).

% upto_aux_def
tff(fact_8888_upto__code,axiom,
    ! [Ia: int,J: int] : upto(Ia,J) = upto_aux(Ia,J,nil_int) ).

% upto_code
tff(fact_8889_upto__split2,axiom,
    ! [Ia: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(Ia,K) = append_int(upto(Ia,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_8890_upto__split1,axiom,
    ! [Ia: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(Ia,K) = append_int(upto(Ia,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_8891_greaterThanLessThan__upto,axiom,
    ! [Ia: int,J: int] : set_or5832277885323065728an_int(Ia,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ia),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_8892_upto_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: list_int] :
      ( ( upto(Xb,Xaa) = Y )
     => ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa),cons_int(Xb,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int),Xaa)),nil_int) ) ) ).

% upto.elims
tff(fact_8893_upto_Osimps,axiom,
    ! [Ia: int,J: int] :
      upto(Ia,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),J),cons_int(Ia,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ia),one_one_int),J)),nil_int) ).

% upto.simps
tff(fact_8894_upto__rec1,axiom,
    ! [Ia: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),J)
     => ( upto(Ia,J) = cons_int(Ia,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ia),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_8895_upto__rec2,axiom,
    ! [Ia: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),J)
     => ( upto(Ia,J) = append_int(upto(Ia,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_8896_upto__split3,axiom,
    ! [Ia: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(Ia,K) = append_int(upto(Ia,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_8897_list__encode_Opelims,axiom,
    ! [Xb: list_nat,Y: nat] :
      ( ( nat_list_encode(Xb) = Y )
     => ( aa_list_nat_o(accp_list_nat(nat_list_encode_rel),Xb)
       => ( ( ( Xb = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),nil_nat) ) )
         => ~ ! [X4: nat,Xs2: list_nat] :
                ( ( Xb = cons_nat(X4,Xs2) )
               => ( ( Y = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),nat_list_encode(Xs2)))) )
                 => ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),cons_nat(X4,Xs2)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_8898_remdups__upt,axiom,
    ! [Mb: nat,Nb: nat] : remdups_nat(upt(Mb,Nb)) = upt(Mb,Nb) ).

% remdups_upt
tff(fact_8899_hd__upt,axiom,
    ! [Ia: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => ( hd_nat(upt(Ia,J)) = Ia ) ) ).

% hd_upt
tff(fact_8900_drop__upt,axiom,
    ! [Mb: nat,Ia: nat,J: nat] : drop_nat(Mb,upt(Ia,J)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),Mb),J) ).

% drop_upt
tff(fact_8901_length__upt,axiom,
    ! [Ia: nat,J: nat] : size_size_list_nat(upt(Ia,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Ia) ).

% length_upt
tff(fact_8902_take__upt,axiom,
    ! [Ia: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),Mb)),Nb)
     => ( take_nat(Mb,upt(Ia,Nb)) = upt(Ia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),Mb)) ) ) ).

% take_upt
tff(fact_8903_upt__conv__Nil,axiom,
    ! [J: nat,Ia: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),Ia)
     => ( upt(Ia,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_8904_sorted__list__of__set__range,axiom,
    ! [Mb: nat,Nb: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) = upt(Mb,Nb) ).

% sorted_list_of_set_range
tff(fact_8905_upt__eq__Nil__conv,axiom,
    ! [Ia: nat,J: nat] :
      ( ( upt(Ia,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),Ia) ) ) ).

% upt_eq_Nil_conv
tff(fact_8906_nth__upt,axiom,
    ! [Ia: 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,Ia),K)),J)
     => ( aa_nat_nat(nth_nat(upt(Ia,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K) ) ) ).

% nth_upt
tff(fact_8907_upt__rec__numeral,axiom,
    ! [Mb: num,Nb: num] :
      upt(aa_num_nat(numeral_numeral_nat,Mb),aa_num_nat(numeral_numeral_nat,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)),cons_nat(aa_num_nat(numeral_numeral_nat,Mb),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),nil_nat) ).

% upt_rec_numeral
tff(fact_8908_map__add__upt,axiom,
    ! [Nb: nat,Mb: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ys,Nb),upt(zero_zero_nat,Mb)) = upt(Nb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% map_add_upt
tff(fact_8909_upt__0,axiom,
    ! [Ia: nat] : upt(Ia,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_8910_upt__conv__Cons__Cons,axiom,
    ! [Mb: nat,Nb: nat,Ns: list_nat,Q2: nat] :
      ( ( cons_nat(Mb,cons_nat(Nb,Ns)) = upt(Mb,Q2) )
    <=> ( cons_nat(Nb,Ns) = upt(aa_nat_nat(suc,Mb),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_8911_map__Suc__upt,axiom,
    ! [Mb: nat,Nb: nat] : map_nat_nat(suc,upt(Mb,Nb)) = upt(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb)) ).

% map_Suc_upt
tff(fact_8912_greaterThanAtMost__upt,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or6659071591806873216st_nat,Nb),Mb) = set_nat2(upt(aa_nat_nat(suc,Nb),aa_nat_nat(suc,Mb))) ).

% greaterThanAtMost_upt
tff(fact_8913_distinct__upt,axiom,
    ! [Ia: nat,J: nat] : aa_list_nat_o(distinct_nat,upt(Ia,J)) ).

% distinct_upt
tff(fact_8914_atLeast__upt,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,Nb) = set_nat2(upt(zero_zero_nat,Nb)) ).

% atLeast_upt
tff(fact_8915_atLeastLessThan__upt,axiom,
    ! [Ia: nat,J: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Ia),J) = set_nat2(upt(Ia,J)) ).

% atLeastLessThan_upt
tff(fact_8916_atLeastAtMost__upt,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,Nb),Mb) = set_nat2(upt(Nb,aa_nat_nat(suc,Mb))) ).

% atLeastAtMost_upt
tff(fact_8917_greaterThanLessThan__upt,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or5834768355832116004an_nat,Nb),Mb) = set_nat2(upt(aa_nat_nat(suc,Nb),Mb)) ).

% greaterThanLessThan_upt
tff(fact_8918_atMost__upto,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,Nb) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,Nb))) ).

% atMost_upto
tff(fact_8919_upt__conv__Cons,axiom,
    ! [Ia: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => ( upt(Ia,J) = cons_nat(Ia,upt(aa_nat_nat(suc,Ia),J)) ) ) ).

% upt_conv_Cons
tff(fact_8920_map__decr__upt,axiom,
    ! [Mb: nat,Nb: nat] : map_nat_nat(aTP_Lamm_yt,upt(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb))) = upt(Mb,Nb) ).

% map_decr_upt
tff(fact_8921_upt__add__eq__append,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => ( upt(Ia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(upt(Ia,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_8922_upt__eq__Cons__conv,axiom,
    ! [Ia: nat,J: nat,Xb: nat,Xs: list_nat] :
      ( ( upt(Ia,J) = cons_nat(Xb,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
        & ( Ia = Xb )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_8923_upt__rec,axiom,
    ! [Ia: nat,J: nat] :
      upt(Ia,J) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J),cons_nat(Ia,upt(aa_nat_nat(suc,Ia),J)),nil_nat) ).

% upt_rec
tff(fact_8924_upt__Suc__append,axiom,
    ! [Ia: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => ( upt(Ia,aa_nat_nat(suc,J)) = append_nat(upt(Ia,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_8925_upt__Suc,axiom,
    ! [Ia: nat,J: nat] :
      upt(Ia,aa_nat_nat(suc,J)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J),append_nat(upt(Ia,J),cons_nat(J,nil_nat)),nil_nat) ).

% upt_Suc
tff(fact_8926_sum__list__upt,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups4561878855575611511st_nat(upt(Mb,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Mb),Nb)) ) ) ).

% sum_list_upt
tff(fact_8927_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_yu(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_yv(Mb),N4)))),finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_yw(Mb),N4)))) ) ) ).

% card_length_sum_list_rec
tff(fact_8928_card__length__sum__list,axiom,
    ! [Mb: nat,N4: nat] : finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_yu(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_8929_tl__upt,axiom,
    ! [Mb: nat,Nb: nat] : tl_nat(upt(Mb,Nb)) = upt(aa_nat_nat(suc,Mb),Nb) ).

% tl_upt
tff(fact_8930_sorted__wrt__upt,axiom,
    ! [Mb: nat,Nb: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_nat),upt(Mb,Nb)) ).

% sorted_wrt_upt
tff(fact_8931_sorted__upt,axiom,
    ! [Mb: nat,Nb: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),upt(Mb,Nb)) ).

% sorted_upt
tff(fact_8932_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,Ia: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Ns)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_nat(Ns))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),aa_nat_nat(nth_nat(Ns),Ia)) ) ) ).

% sorted_wrt_less_idx
tff(fact_8933_sorted__wrt__upto,axiom,
    ! [Ia: int,J: int] : aa_list_int_o(sorted_wrt_int(ord_less_int),upto(Ia,J)) ).

% sorted_wrt_upto
tff(fact_8934_sorted__upto,axiom,
    ! [Mb: int,Nb: int] : aa_list_int_o(sorted_wrt_int(ord_less_eq_int),upto(Mb,Nb)) ).

% sorted_upto
tff(fact_8935_pairs__le__eq__Sigma,axiom,
    ! [Mb: nat] : collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_nd(Mb))) = produc457027306803732586at_nat(aa_nat_set_nat(set_ord_atMost_nat,Mb),aa_nat1818925681583257441et_nat(aTP_Lamm_yx,Mb)) ).

% pairs_le_eq_Sigma
tff(fact_8936_vimage__Suc__insert__Suc,axiom,
    ! [Nb: nat,A2: set_nat] : vimage_nat_nat(suc,insert_nat2(aa_nat_nat(suc,Nb),A2)) = insert_nat2(Nb,vimage_nat_nat(suc,A2)) ).

% vimage_Suc_insert_Suc
tff(fact_8937_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_8938_vimage__Suc__insert__0,axiom,
    ! [A2: set_nat] : vimage_nat_nat(suc,insert_nat2(zero_zero_nat,A2)) = vimage_nat_nat(suc,A2) ).

% vimage_Suc_insert_0
tff(fact_8939_set__decode__div__2,axiom,
    ! [Xb: nat] : aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,bit0(one)))) = vimage_nat_nat(suc,aa_nat_set_nat(nat_set_decode,Xb)) ).

% set_decode_div_2
tff(fact_8940_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,bit0(one))) ).

% set_encode_vimage_Suc
tff(fact_8941_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_8942_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_8943_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_o,B: nat] :
      ( ? [X_1: nat] : aa_nat_o(P,X_1)
     => ( ! [Y3: nat] :
            ( aa_nat_o(P,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_ex_nat
tff(fact_8944_Field__natLeq__on,axiom,
    ! [Nb: nat] : field_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_yy(Nb)))) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aj,Nb)) ).

% Field_natLeq_on
tff(fact_8945_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_nat)) ).

% natLess_def
tff(fact_8946_Restr__natLeq,axiom,
    ! [Nb: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aj,Nb)),aa_nat1818925681583257441et_nat(aTP_Lamm_yz,Nb))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_yy(Nb))) ).

% Restr_natLeq
tff(fact_8947_wf__int__ge__less__than2,axiom,
    ! [D: int] : wf_int(int_ge_less_than2(D)) ).

% wf_int_ge_less_than2
tff(fact_8948_wf__int__ge__less__than,axiom,
    ! [D: int] : wf_int(int_ge_less_than(D)) ).

% wf_int_ge_less_than
tff(fact_8949_wf__less,axiom,
    wf_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_nat))) ).

% wf_less
tff(fact_8950_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_eq_nat)) ).

% natLeq_def
tff(fact_8951_Restr__natLeq2,axiom,
    ! [Nb: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),Nb),aa_nat1818925681583257441et_nat(aTP_Lamm_za,Nb))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_yy(Nb))) ).

% Restr_natLeq2
tff(fact_8952_natLeq__underS__less,axiom,
    ! [Nb: nat] : aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),Nb) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aj,Nb)) ).

% natLeq_underS_less
tff(fact_8953_bij__betw__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_set_complex_o(bij_betw_nat_complex(aTP_Lamm_zb(Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)),collect_complex(aTP_Lamm_cd(Nb))) ) ).

% bij_betw_roots_unity
tff(fact_8954_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_8955_bij__betw__nth__root__unity,axiom,
    ! [C: complex,Nb: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_set_complex_o(bij_be1856998921033663316omplex(times_times_complex(aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_nat_fun_real_real(root,Nb),aa_complex_real(real_V1022390504157884413omplex,C)))),aa_real_complex(cis,aa_real_real(divide_divide_real(aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,Nb))))),collect_complex(aTP_Lamm_cd(Nb))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_wt(C),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_8956_sort__upt,axiom,
    ! [Mb: nat,Nb: nat] : linord738340561235409698at_nat(aTP_Lamm_bc,upt(Mb,Nb)) = upt(Mb,Nb) ).

% sort_upt
tff(fact_8957_sort__upto,axiom,
    ! [Ia: int,J: int] : linord1735203802627413978nt_int(aTP_Lamm_bd,upto(Ia,J)) = upto(Ia,J) ).

% sort_upto
tff(fact_8958_strict__mono__imp__increasing,axiom,
    ! [F: fun_nat_nat,Nb: nat] :
      ( order_5726023648592871131at_nat(F)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(F,Nb)) ) ).

% strict_mono_imp_increasing
tff(fact_8959_le__enumerate,axiom,
    ! [S3: set_nat,Nb: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(infini8530281810654367211te_nat(S3),Nb)) ) ).

% le_enumerate
tff(fact_8960_finite__le__enumerate,axiom,
    ! [S3: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_set_nat_nat(finite_card_nat,S3))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(infini8530281810654367211te_nat(S3),Nb)) ) ) ).

% finite_le_enumerate
tff(fact_8961_Least__eq__0,axiom,
    ! [P: fun_nat_o] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ord_Least_nat(P) = zero_zero_nat ) ) ).

% Least_eq_0
tff(fact_8962_Least__Suc,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_yf(P))) ) ) ) ).

% Least_Suc
tff(fact_8963_Least__Suc2,axiom,
    ! [P: fun_nat_o,Nb: nat,Q: fun_nat_o,Mb: nat] :
      ( aa_nat_o(P,Nb)
     => ( aa_nat_o(Q,Mb)
       => ( ~ aa_nat_o(P,zero_zero_nat)
         => ( ! [K3: nat] :
                ( aa_nat_o(P,aa_nat_nat(suc,K3))
              <=> aa_nat_o(Q,K3) )
           => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_8964_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(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,N3),M5)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_8965_last__upt,axiom,
    ! [Ia: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => ( last_nat(upt(Ia,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),one_one_nat) ) ) ).

% last_upt
tff(fact_8966_natLeq__on__wo__rel,axiom,
    ! [Nb: nat] : bNF_We3818239936649020644el_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_yy(Nb)))) ).

% natLeq_on_wo_rel
tff(fact_8967_gcd__nat_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(dvd_dvd_nat,aTP_Lamm_zc),zero_zero_nat) ).

% gcd_nat.ordering_top_axioms
tff(fact_8968_bot__nat__0_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(aTP_Lamm_ak,aTP_Lamm_aj),zero_zero_nat) ).

% bot_nat_0.ordering_top_axioms
tff(fact_8969_less__eq__enat__def,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Mb),Nb)
    <=> aa_Extended_enat_o(extended_case_enat_o(aTP_Lamm_zd(Mb),$true),Nb) ) ).

% less_eq_enat_def
tff(fact_8970_less__enat__def,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Mb),Nb)
    <=> aa_Extended_enat_o(extended_case_enat_o(aTP_Lamm_ze(Nb),$false),Mb) ) ).

% less_enat_def
tff(fact_8971_natLeq__on__well__order__on,axiom,
    ! [Nb: nat] : order_2888998067076097458on_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aj,Nb)),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_yy(Nb)))) ).

% natLeq_on_well_order_on
tff(fact_8972_natLeq__on__Well__order,axiom,
    ! [Nb: nat] : order_2888998067076097458on_nat(field_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_yy(Nb)))),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_yy(Nb)))) ).

% natLeq_on_Well_order
tff(fact_8973_prod__decode__triangle__add,axiom,
    ! [K: nat,Mb: nat] : aa_nat7603221163087113075at_nat(nat_prod_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K)),Mb)) = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),Mb) ).

% prod_decode_triangle_add
tff(fact_8974_prod__decode__def,axiom,
    nat_prod_decode = nat_prod_decode_aux(zero_zero_nat) ).

% prod_decode_def
tff(fact_8975_list__decode_Opinduct,axiom,
    ! [A0: nat,P: fun_nat_o] :
      ( aa_nat_o(accp_nat(nat_list_decode_rel),A0)
     => ( ( aa_nat_o(accp_nat(nat_list_decode_rel),zero_zero_nat)
         => aa_nat_o(P,zero_zero_nat) )
       => ( ! [N: nat] :
              ( aa_nat_o(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N))
             => ( ! [X: nat,Y4: nat] :
                    ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y4) = aa_nat7603221163087113075at_nat(nat_prod_decode,N) )
                   => aa_nat_o(P,Y4) )
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,A0) ) ) ) ).

% list_decode.pinduct
tff(fact_8976_list__decode_Oelims,axiom,
    ! [Xb: nat,Y: list_nat] :
      ( ( nat_list_decode(Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != nil_nat ) )
       => ~ ! [N: nat] :
              ( ( Xb = aa_nat_nat(suc,N) )
             => ( Y != produc2761476792215241774st_nat(aTP_Lamm_zf,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) ) ) ) ) ).

% list_decode.elims
tff(fact_8977_list__decode_Opsimps_I1_J,axiom,
    ( aa_nat_o(accp_nat(nat_list_decode_rel),zero_zero_nat)
   => ( nat_list_decode(zero_zero_nat) = nil_nat ) ) ).

% list_decode.psimps(1)
tff(fact_8978_list__decode_Osimps_I1_J,axiom,
    nat_list_decode(zero_zero_nat) = nil_nat ).

% list_decode.simps(1)
tff(fact_8979_list__decode_Opsimps_I2_J,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,Nb))
     => ( nat_list_decode(aa_nat_nat(suc,Nb)) = produc2761476792215241774st_nat(aTP_Lamm_zf,aa_nat7603221163087113075at_nat(nat_prod_decode,Nb)) ) ) ).

% list_decode.psimps(2)
tff(fact_8980_list__decode_Osimps_I2_J,axiom,
    ! [Nb: nat] : nat_list_decode(aa_nat_nat(suc,Nb)) = produc2761476792215241774st_nat(aTP_Lamm_zf,aa_nat7603221163087113075at_nat(nat_prod_decode,Nb)) ).

% list_decode.simps(2)
tff(fact_8981_list__decode_Opelims,axiom,
    ! [Xb: nat,Y: list_nat] :
      ( ( nat_list_decode(Xb) = Y )
     => ( aa_nat_o(accp_nat(nat_list_decode_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = nil_nat )
             => ~ aa_nat_o(accp_nat(nat_list_decode_rel),zero_zero_nat) ) )
         => ~ ! [N: nat] :
                ( ( Xb = aa_nat_nat(suc,N) )
               => ( ( Y = produc2761476792215241774st_nat(aTP_Lamm_zf,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) )
                 => ~ aa_nat_o(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N)) ) ) ) ) ) ).

% list_decode.pelims
tff(fact_8982_inj__on__char__of__nat,axiom,
    aa_set_nat_o(inj_on_nat_char(unique3096191561947761185of_nat),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one))))))))))) ).

% inj_on_char_of_nat
tff(fact_8983_UNIV__char__of__nat,axiom,
    top_top_set_char = image_nat_char(unique3096191561947761185of_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one))))))))))) ).

% UNIV_char_of_nat
tff(fact_8984_range__nat__of__char,axiom,
    image_char_nat(comm_s629917340098488124ar_nat,top_top_set_char) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one)))))))))) ).

% range_nat_of_char
tff(fact_8985_nat__of__char__less__256,axiom,
    ! [C: char] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C)),aa_num_nat(numeral_numeral_nat,bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one)))))))))) ).

% nat_of_char_less_256
tff(fact_8986_one__int_Otransfer,axiom,
    aa_int_o(aa_Pro2912763917461432128_int_o(pcr_int,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,one_one_nat),zero_zero_nat)),one_one_int) ).

% one_int.transfer
tff(fact_8987_zero__int_Otransfer,axiom,
    aa_int_o(aa_Pro2912763917461432128_int_o(pcr_int,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),zero_zero_nat)),zero_zero_int) ).

% zero_int.transfer
tff(fact_8988_Rats__abs__nat__div__natE,axiom,
    ! [Xb: real] :
      ( aa_set_real_o(member_real2(Xb),field_5140801741446780682s_real)
     => ~ ! [M: nat,N: nat] :
            ( ( N != zero_zero_nat )
           => ( ( aa_real_real(abs_abs_real,Xb) = aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,M),N) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_8989_minus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_xe),minus_minus_int) ).

% minus_int.transfer
tff(fact_8990_plus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_xc),plus_plus_int) ).

% plus_int.transfer
tff(fact_8991_less__int_Otransfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_wy),ord_less_int) ).

% less_int.transfer
tff(fact_8992_less__eq__int_Otransfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_xa),ord_less_eq_int) ).

% less_eq_int.transfer
tff(fact_8993_coprime__Suc__0__left,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) ).

% coprime_Suc_0_left
tff(fact_8994_coprime__Suc__0__right,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) ).

% coprime_Suc_0_right
tff(fact_8995_coprime__Suc__left__nat,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(suc,Nb)),Nb) ).

% coprime_Suc_left_nat
tff(fact_8996_coprime__Suc__right__nat,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Nb),aa_nat_nat(suc,Nb)) ).

% coprime_Suc_right_nat
tff(fact_8997_times__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_wv),times_times_int) ).

% times_int.transfer
tff(fact_8998_coprime__diff__one__right__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ).

% coprime_diff_one_right_nat
tff(fact_8999_coprime__diff__one__left__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),Nb) ) ).

% coprime_diff_one_left_nat
tff(fact_9000_uminus__int_Otransfer,axiom,
    aa_fun_int_int_o(aa_fun856177901615811547_int_o(bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc2626176000494625587at_nat(aTP_Lamm_ww)),uminus_uminus_int) ).

% uminus_int.transfer
tff(fact_9001_nat_Otransfer,axiom,
    bNF_re4555766996558763186at_nat(pcr_int,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),nat2) ).

% nat.transfer
tff(fact_9002_int__transfer,axiom,
    bNF_re6830278522597306478at_int(fequal_nat,pcr_int,aTP_Lamm_zg,semiri1314217659103216013at_int) ).

% int_transfer
tff(fact_9003_times__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_wv),produc4528391901023682385at_nat(aTP_Lamm_wv)) ).

% times_int.rsp
tff(fact_9004_intrel__iff,axiom,
    ! [Xb: nat,Y: nat,U: nat,V: nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Y)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,U),V))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),V) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,U),Y) ) ) ).

% intrel_iff
tff(fact_9005_nat_Orsp,axiom,
    bNF_re8246922863344978751at_nat(intrel,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat.rsp
tff(fact_9006_uminus__int_Orsp,axiom,
    aa_fun3538214411680374423_nat_o(aa_fun6920661532540581549_nat_o(bNF_re2241393799969408733at_nat(intrel,intrel),produc2626176000494625587at_nat(aTP_Lamm_ww)),produc2626176000494625587at_nat(aTP_Lamm_ww)) ).

% uminus_int.rsp
tff(fact_9007_zero__int_Orsp,axiom,
    aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),zero_zero_nat)) ).

% zero_int.rsp
tff(fact_9008_int_Oabs__eq__iff,axiom,
    ! [Xb: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( aa_Pro4240683299535347697at_int(abs_Integ,Xb) = aa_Pro4240683299535347697at_int(abs_Integ,Y) )
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,Xb),Y) ) ).

% int.abs_eq_iff
tff(fact_9009_one__int_Orsp,axiom,
    aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,one_one_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,one_one_nat),zero_zero_nat)) ).

% one_int.rsp
tff(fact_9010_intrel__def,axiom,
    intrel = produc6395135316500328296_nat_o(aTP_Lamm_zi) ).

% intrel_def
tff(fact_9011_less__int_Orsp,axiom,
    bNF_re4103222659376994899_nat_o(intrel,bNF_re3666534408544137501at_o_o(intrel,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_wy),produc6395135316500328296_nat_o(aTP_Lamm_wy)) ).

% less_int.rsp
tff(fact_9012_less__eq__int_Orsp,axiom,
    bNF_re4103222659376994899_nat_o(intrel,bNF_re3666534408544137501at_o_o(intrel,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_xa),produc6395135316500328296_nat_o(aTP_Lamm_xa)) ).

% less_eq_int.rsp
tff(fact_9013_int_Orel__eq__transfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),intrel,fequal_int) ).

% int.rel_eq_transfer
tff(fact_9014_plus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_xc),produc4528391901023682385at_nat(aTP_Lamm_xc)) ).

% plus_int.rsp
tff(fact_9015_minus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_xe),produc4528391901023682385at_nat(aTP_Lamm_xe)) ).

% minus_int.rsp
tff(fact_9016_int_Obi__total,axiom,
    bi_tot896582865486249351at_int(pcr_int) ).

% int.bi_total
tff(fact_9017_less__than__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Y),less_than)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ).

% less_than_iff
tff(fact_9018_elimnum,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_Node(Info,Dega,TreeLista,Summarya)),Nb)
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info,Dega,TreeLista,Summarya),aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Nb))) = vEBT_Node(Info,Dega,TreeLista,Summarya) ) ) ).

% elimnum
tff(fact_9019_idiff__enat__0__right,axiom,
    ! [Nb: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(Nb),aa_nat_Extended_enat(extended_enat2,zero_zero_nat)) = Nb ).

% idiff_enat_0_right
tff(fact_9020_idiff__enat__0,axiom,
    ! [Nb: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(aa_nat_Extended_enat(extended_enat2,zero_zero_nat)),Nb) = aa_nat_Extended_enat(extended_enat2,zero_zero_nat) ).

% idiff_enat_0
tff(fact_9021_plus__enat__simps_I1_J,axiom,
    ! [Mb: nat,Nb: nat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(extended_enat2,Mb)),aa_nat_Extended_enat(extended_enat2,Nb)) = aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% plus_enat_simps(1)
tff(fact_9022_max__enat__simps_I1_J,axiom,
    ! [Mb: nat,Nb: nat] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(aa_nat_Extended_enat(extended_enat2,Mb)),aa_nat_Extended_enat(extended_enat2,Nb)) = aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)) ).

% max_enat_simps(1)
tff(fact_9023_enat__ord__simps_I2_J,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(extended_enat2,Mb)),aa_nat_Extended_enat(extended_enat2,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% enat_ord_simps(2)
tff(fact_9024_enat__ord__simps_I1_J,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(extended_enat2,Mb)),aa_nat_Extended_enat(extended_enat2,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% enat_ord_simps(1)
tff(fact_9025_numeral__less__enat__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,numera1916890842035813515d_enat(Mb)),aa_nat_Extended_enat(extended_enat2,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb) ) ).

% numeral_less_enat_iff
tff(fact_9026_numeral__le__enat__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,numera1916890842035813515d_enat(Mb)),aa_nat_Extended_enat(extended_enat2,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb) ) ).

% numeral_le_enat_iff
tff(fact_9027_zero__enat__def,axiom,
    zero_z5237406670263579293d_enat = aa_nat_Extended_enat(extended_enat2,zero_zero_nat) ).

% zero_enat_def
tff(fact_9028_enat__0__iff_I1_J,axiom,
    ! [Xb: nat] :
      ( ( aa_nat_Extended_enat(extended_enat2,Xb) = zero_z5237406670263579293d_enat )
    <=> ( Xb = zero_zero_nat ) ) ).

% enat_0_iff(1)
tff(fact_9029_enat__0__iff_I2_J,axiom,
    ! [Xb: nat] :
      ( ( zero_z5237406670263579293d_enat = aa_nat_Extended_enat(extended_enat2,Xb) )
    <=> ( Xb = zero_zero_nat ) ) ).

% enat_0_iff(2)
tff(fact_9030_less__enatE,axiom,
    ! [Nb: extended_enat,Mb: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Nb),aa_nat_Extended_enat(extended_enat2,Mb))
     => ~ ! [K3: nat] :
            ( ( Nb = aa_nat_Extended_enat(extended_enat2,K3) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Mb) ) ) ).

% less_enatE
tff(fact_9031_Suc__ile__eq,axiom,
    ! [Mb: nat,Nb: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(extended_enat2,aa_nat_nat(suc,Mb))),Nb)
    <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(extended_enat2,Mb)),Nb) ) ).

% Suc_ile_eq
tff(fact_9032_iadd__le__enat__iff,axiom,
    ! [Xb: extended_enat,Y: extended_enat,Nb: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Xb),Y)),aa_nat_Extended_enat(extended_enat2,Nb))
    <=> ? [Y5: nat,X7: nat] :
          ( ( Xb = aa_nat_Extended_enat(extended_enat2,X7) )
          & ( Y = aa_nat_Extended_enat(extended_enat2,Y5) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X7),Y5)),Nb) ) ) ).

% iadd_le_enat_iff
tff(fact_9033_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,L: nat] : vEBT_VEBT_elim_dead(vEBT_Node(Info,Dega,TreeLista,Summarya),aa_nat_Extended_enat(extended_enat2,L)) = vEBT_Node(Info,Dega,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_zj(Dega),TreeLista)),vEBT_VEBT_elim_dead(Summarya,aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,bit0(one)))))))) ).

% VEBT_internal.elim_dead.simps(3)
tff(fact_9034_VEBT__internal_Oelim__dead_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: extended_enat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(Xb,Xaa) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => ( Y != vEBT_Leaf((A3),(B3)) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info2,Deg,TreeList,Summary) )
             => ( ( Xaa = extend5688581933313929465d_enat )
               => ( Y != vEBT_Node(Info2,Deg,map_VE8901447254227204932T_VEBT(aTP_Lamm_zj(Deg),TreeList),vEBT_VEBT_elim_dead(Summary,extend5688581933313929465d_enat)) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,Deg,TreeList,Summary) )
               => ! [L3: nat] :
                    ( ( Xaa = aa_nat_Extended_enat(extended_enat2,L3) )
                   => ( Y != vEBT_Node(Info2,Deg,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_zj(Deg),TreeList)),vEBT_VEBT_elim_dead(Summary,aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))))) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.elims
tff(fact_9035_elimcomplete,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_Node(Info,Dega,TreeLista,Summarya)),Nb)
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info,Dega,TreeLista,Summarya),extend5688581933313929465d_enat) = vEBT_Node(Info,Dega,TreeLista,Summarya) ) ) ).

% elimcomplete
tff(fact_9036_times__enat__simps_I4_J,axiom,
    ! [Mb: nat] :
      aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(extended_enat2,Mb)),extend5688581933313929465d_enat) = $ite(Mb = zero_zero_nat,zero_z5237406670263579293d_enat,extend5688581933313929465d_enat) ).

% times_enat_simps(4)
tff(fact_9037_times__enat__simps_I3_J,axiom,
    ! [Nb: nat] :
      aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extend5688581933313929465d_enat),aa_nat_Extended_enat(extended_enat2,Nb)) = $ite(Nb = zero_zero_nat,zero_z5237406670263579293d_enat,extend5688581933313929465d_enat) ).

% times_enat_simps(3)
tff(fact_9038_VEBT__internal_Oelim__dead_Ocases,axiom,
    ! [Xb: produc7272778201969148633d_enat] :
      ( ! [A3: $o,B3: $o,Uu2: extended_enat] : Xb != aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,vEBT_Leaf((A3),(B3))),Uu2)
     => ( ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb != aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,vEBT_Node(Info2,Deg,TreeList,Summary)),extend5688581933313929465d_enat)
       => ~ ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,L3: nat] : Xb != aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,vEBT_Node(Info2,Deg,TreeList,Summary)),aa_nat_Extended_enat(extended_enat2,L3)) ) ) ).

% VEBT_internal.elim_dead.cases
tff(fact_9039_Sup__enat__def,axiom,
    ! [A2: set_Extended_enat] :
      comple4398354569131411667d_enat(A2) = $ite(
        A2 = bot_bo7653980558646680370d_enat,
        zero_z5237406670263579293d_enat,
        $ite(aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2),lattic921264341876707157d_enat(A2),extend5688581933313929465d_enat) ) ).

% Sup_enat_def
tff(fact_9040_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_9041_Inf__enat__def,axiom,
    ! [A2: set_Extended_enat] :
      comple2295165028678016749d_enat(A2) = $ite(A2 = bot_bo7653980558646680370d_enat,extend5688581933313929465d_enat,ord_Le1955565732374568822d_enat(aTP_Lamm_zk(A2))) ).

% Inf_enat_def
tff(fact_9042_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] : vEBT_VEBT_elim_dead(vEBT_Node(Info,Dega,TreeLista,Summarya),extend5688581933313929465d_enat) = vEBT_Node(Info,Dega,map_VE8901447254227204932T_VEBT(aTP_Lamm_zj(Dega),TreeLista),vEBT_VEBT_elim_dead(Summarya,extend5688581933313929465d_enat)) ).

% VEBT_internal.elim_dead.simps(2)
tff(fact_9043_VEBT__internal_Oelim__dead_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: extended_enat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(Xb,Xaa) = Y )
     => ( aa_Pro1441479085436938555enat_o(accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel),aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,Xb),Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = vEBT_Leaf((A3),(B3)) )
               => ~ aa_Pro1441479085436938555enat_o(accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel),aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,vEBT_Leaf((A3),(B3))),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,Deg,TreeList,Summary) )
               => ( ( Xaa = extend5688581933313929465d_enat )
                 => ( ( Y = vEBT_Node(Info2,Deg,map_VE8901447254227204932T_VEBT(aTP_Lamm_zj(Deg),TreeList),vEBT_VEBT_elim_dead(Summary,extend5688581933313929465d_enat)) )
                   => ~ aa_Pro1441479085436938555enat_o(accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel),aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,vEBT_Node(Info2,Deg,TreeList,Summary)),extend5688581933313929465d_enat)) ) ) )
           => ~ ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info2,Deg,TreeList,Summary) )
                 => ! [L3: nat] :
                      ( ( Xaa = aa_nat_Extended_enat(extended_enat2,L3) )
                     => ( ( Y = vEBT_Node(Info2,Deg,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_zj(Deg),TreeList)),vEBT_VEBT_elim_dead(Summary,aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))))) )
                       => ~ aa_Pro1441479085436938555enat_o(accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel),aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,vEBT_Node(Info2,Deg,TreeList,Summary)),aa_nat_Extended_enat(extended_enat2,L3))) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.pelims
tff(fact_9044_times__enat__def,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(Mb),Nb) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_zm(Nb),
          $ite(Nb = zero_z5237406670263579293d_enat,zero_z5237406670263579293d_enat,extend5688581933313929465d_enat)),Mb) ).

% times_enat_def
tff(fact_9045_plus__enat__def,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Mb),Nb) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_zo(Nb),extend5688581933313929465d_enat),Mb) ).

% plus_enat_def
tff(fact_9046_eSuc__Max,axiom,
    ! [A2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( aa_Ext7969882492699152700d_enat(extended_eSuc,lattic921264341876707157d_enat(A2)) = lattic921264341876707157d_enat(image_80655429650038917d_enat(extended_eSuc,A2)) ) ) ) ).

% eSuc_Max
tff(fact_9047_eSuc__def,axiom,
    ! [Ia: extended_enat] : aa_Ext7969882492699152700d_enat(extended_eSuc,Ia) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_zp,extend5688581933313929465d_enat),Ia) ).

% eSuc_def
tff(fact_9048_eSuc__enat,axiom,
    ! [Nb: nat] : aa_Ext7969882492699152700d_enat(extended_eSuc,aa_nat_Extended_enat(extended_enat2,Nb)) = aa_nat_Extended_enat(extended_enat2,aa_nat_nat(suc,Nb)) ).

% eSuc_enat
tff(fact_9049_eSuc__enat__iff,axiom,
    ! [Xb: extended_enat,Y: nat] :
      ( ( aa_Ext7969882492699152700d_enat(extended_eSuc,Xb) = aa_nat_Extended_enat(extended_enat2,Y) )
    <=> ? [N3: nat] :
          ( ( Y = aa_nat_nat(suc,N3) )
          & ( Xb = aa_nat_Extended_enat(extended_enat2,N3) ) ) ) ).

% eSuc_enat_iff
tff(fact_9050_enat__eSuc__iff,axiom,
    ! [Y: nat,Xb: extended_enat] :
      ( ( aa_nat_Extended_enat(extended_enat2,Y) = aa_Ext7969882492699152700d_enat(extended_eSuc,Xb) )
    <=> ? [N3: nat] :
          ( ( Y = aa_nat_nat(suc,N3) )
          & ( aa_nat_Extended_enat(extended_enat2,N3) = Xb ) ) ) ).

% enat_eSuc_iff
tff(fact_9051_eSuc__Sup,axiom,
    ! [A2: set_Extended_enat] :
      ( ( A2 != bot_bo7653980558646680370d_enat )
     => ( aa_Ext7969882492699152700d_enat(extended_eSuc,comple4398354569131411667d_enat(A2)) = comple4398354569131411667d_enat(image_80655429650038917d_enat(extended_eSuc,A2)) ) ) ).

% eSuc_Sup
tff(fact_9052_UNIV__bool,axiom,
    top_top_set_o = insert_o2($false,insert_o2($true,bot_bot_set_o)) ).

% UNIV_bool
tff(fact_9053_Rep__unit__induct,axiom,
    ! [Y: $o,P: fun_o_o] :
      ( aa_set_o_o(member_o2((Y)),insert_o2($true,bot_bot_set_o))
     => ( ! [X4: product_unit] : aa_o_o(P,aa_Product_unit_o(product_Rep_unit,X4))
       => aa_o_o(P,(Y)) ) ) ).

% Rep_unit_induct
tff(fact_9054_Rep__unit,axiom,
    ! [Xb: product_unit] : aa_set_o_o(member_o2(aa_Product_unit_o(product_Rep_unit,Xb)),insert_o2($true,bot_bot_set_o)) ).

% Rep_unit
tff(fact_9055_Rep__unit__cases,axiom,
    ! [Y: $o] :
      ( aa_set_o_o(member_o2((Y)),insert_o2($true,bot_bot_set_o))
     => ~ ! [X4: product_unit] :
            ( (Y)
          <=> ~ aa_Product_unit_o(product_Rep_unit,X4) ) ) ).

% Rep_unit_cases
tff(fact_9056_Abs__unit__inverse,axiom,
    ! [Y: $o] :
      ( aa_set_o_o(member_o2((Y)),insert_o2($true,bot_bot_set_o))
     => ( aa_Product_unit_o(product_Rep_unit,aa_o_Product_unit(product_Abs_unit,(Y)))
      <=> (Y) ) ) ).

% Abs_unit_inverse
tff(fact_9057_Abs__unit__inject,axiom,
    ! [Xb: $o,Y: $o] :
      ( aa_set_o_o(member_o2((Xb)),insert_o2($true,bot_bot_set_o))
     => ( aa_set_o_o(member_o2((Y)),insert_o2($true,bot_bot_set_o))
       => ( ( aa_o_Product_unit(product_Abs_unit,(Xb)) = aa_o_Product_unit(product_Abs_unit,(Y)) )
        <=> ( (Xb)
          <=> (Y) ) ) ) ) ).

% Abs_unit_inject
tff(fact_9058_Abs__unit__induct,axiom,
    ! [P: fun_Product_unit_o,Xb: product_unit] :
      ( ! [Y3: $o] :
          ( aa_set_o_o(member_o2((Y3)),insert_o2($true,bot_bot_set_o))
         => aa_Product_unit_o(P,aa_o_Product_unit(product_Abs_unit,(Y3))) )
     => aa_Product_unit_o(P,Xb) ) ).

% Abs_unit_induct
tff(fact_9059_Abs__unit__cases,axiom,
    ! [Xb: product_unit] :
      ~ ! [Y3: $o] :
          ( ( Xb = aa_o_Product_unit(product_Abs_unit,(Y3)) )
         => ~ aa_set_o_o(member_o2((Y3)),insert_o2($true,bot_bot_set_o)) ) ).

% Abs_unit_cases
tff(fact_9060_type__definition__unit,axiom,
    aa_set_o_o(type_d6188575255521822967unit_o(product_Rep_unit,product_Abs_unit),insert_o2($true,bot_bot_set_o)) ).

% type_definition_unit
tff(fact_9061_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_9062_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_9063_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_9064_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_9065_gcd__0__nat,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),zero_zero_nat) = Xb ).

% gcd_0_nat
tff(fact_9066_gcd__0__left__nat,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),Xb) = Xb ).

% gcd_0_left_nat
tff(fact_9067_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_9068_gcd__pos__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb))
    <=> ( ( Mb != zero_zero_nat )
        | ( Nb != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9069_Gcd__in,axiom,
    ! [A2: set_nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_set_nat_o(member_nat2(A3),A2)
         => ( aa_set_nat_o(member_nat2(B3),A2)
           => aa_set_nat_o(member_nat2(aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A3),B3)),A2) ) )
     => ( ( A2 != bot_bot_set_nat )
       => aa_set_nat_o(member_nat2(aa_set_nat_nat(gcd_Gcd_nat,A2)),A2) ) ) ).

% Gcd_in
tff(fact_9070_gcd__diff2__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) ) ) ).

% gcd_diff2_nat
tff(fact_9071_gcd__diff1__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) ) ) ).

% gcd_diff1_nat
tff(fact_9072_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_9073_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_9074_gcd__nat_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero_nat,Xb,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa))) ) ) ).

% gcd_nat.elims
tff(fact_9075_gcd__nat_Osimps,axiom,
    ! [Xb: nat,Y: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Y) = $ite(Y = zero_zero_nat,Xb,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y))) ).

% gcd_nat.simps
tff(fact_9076_gcd__non__0__nat,axiom,
    ! [Y: nat,Xb: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)) ) ) ).

% gcd_non_0_nat
tff(fact_9077_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X4: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = 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_9078_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X4: 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),X4))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4)),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),X4))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4)),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_9079_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] : aa_set_nat_nat(gcd_Gcd_nat,set_nat2(Xs)) = aa_nat_nat(fold_nat_nat(gcd_gcd_nat,Xs),zero_zero_nat) ).

% Gcd_nat_set_eq_fold
tff(fact_9080_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_zc) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_9081_gcd__is__Max__divisors__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_zq(Nb),Mb))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_9082_gcd__nat_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Xaa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero_nat,Xb,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xaa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_9083_cr__int__def,axiom,
    ! [X: product_prod_nat_nat] : aa_Pro2912763917461432128_int_o(cr_int,X) = aa_int_fun_int_o(fequal_int,aa_Pro4240683299535347697at_int(abs_Integ,X)) ).

% cr_int_def
tff(fact_9084_int_Opcr__cr__eq,axiom,
    pcr_int = cr_int ).

% int.pcr_cr_eq
tff(fact_9085_Quotient__int,axiom,
    quotie1194848508323700631at_int(intrel,abs_Integ,rep_Integ,cr_int) ).

% Quotient_int
tff(fact_9086_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_mk,Uu) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),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,bit0(one)))),one_one_nat))) ).

% ATP.lambda_1
tff(fact_9087_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_le,Uu) = aa_nat_real(power_power_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_nat(suc,Uu)) ).

% ATP.lambda_2
tff(fact_9088_ATP_Olambda__3,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_jr,Uu) = aa_real_real(times_times_real(aa_nat_real(cos_coeff,Uu)),aa_nat_real(power_power_real(zero_zero_real),Uu)) ).

% ATP.lambda_3
tff(fact_9089_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_yt,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_9090_ATP_Olambda__5,axiom,
    ! [Uu: extended_enat] : aa_Ext7969882492699152700d_enat(aTP_Lamm_in,Uu) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Uu),one_on7984719198319812577d_enat) ).

% ATP.lambda_5
tff(fact_9091_ATP_Olambda__6,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_im,Uu) = aa_complex_complex(plus_plus_complex(Uu),one_one_complex) ).

% ATP.lambda_6
tff(fact_9092_ATP_Olambda__7,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_io,Uu) = aa_real_real(plus_plus_real(Uu),one_one_real) ).

% ATP.lambda_7
tff(fact_9093_ATP_Olambda__8,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_iq,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_8
tff(fact_9094_ATP_Olambda__9,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ip,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).

% ATP.lambda_9
tff(fact_9095_ATP_Olambda__10,axiom,
    ! [Uu: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_zg,Uu) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uu),zero_zero_nat) ).

% ATP.lambda_10
tff(fact_9096_ATP_Olambda__11,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_xp,Uu) = suminf_real(aTP_Lamm_rs(Uu)) ).

% ATP.lambda_11
tff(fact_9097_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_yj,Uu) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bj,Uu)) ).

% ATP.lambda_12
tff(fact_9098_ATP_Olambda__13,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ya,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).

% ATP.lambda_13
tff(fact_9099_ATP_Olambda__14,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_yn,Uu) = some_num(bit0(Uu)) ).

% ATP.lambda_14
tff(fact_9100_ATP_Olambda__15,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_dn,Uu) = aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(suc,Uu)) ).

% ATP.lambda_15
tff(fact_9101_ATP_Olambda__16,axiom,
    ! [Uu: nat] : aa_nat_Extended_enat(aTP_Lamm_zp,Uu) = aa_nat_Extended_enat(extended_enat2,aa_nat_nat(suc,Uu)) ).

% ATP.lambda_16
tff(fact_9102_ATP_Olambda__17,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_yo,Uu)
    <=> ? [I4: int,N3: nat] :
          ( ( Uu = aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,I4)),aa_nat_real(semiri5074537144036343181t_real,N3)) )
          & ( N3 != zero_zero_nat ) ) ) ).

% ATP.lambda_17
tff(fact_9103_ATP_Olambda__18,axiom,
    ! [Uu: complex,Uua: nat] :
      aa_nat_complex(aTP_Lamm_sr(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua),aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_complex(power_power_complex(Uu),Uua)),zero_zero_complex) ).

% ATP.lambda_18
tff(fact_9104_ATP_Olambda__19,axiom,
    ! [Uu: real,Uua: nat] :
      aa_nat_real(aTP_Lamm_sq(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua),aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(power_power_real(Uu),Uua)),zero_zero_real) ).

% ATP.lambda_19
tff(fact_9105_ATP_Olambda__20,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aTP_Lamm_pj(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_complex) ).

% ATP.lambda_20
tff(fact_9106_ATP_Olambda__21,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_pl,Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_real) ).

% ATP.lambda_21
tff(fact_9107_ATP_Olambda__22,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_pk,Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_int) ).

% ATP.lambda_22
tff(fact_9108_ATP_Olambda__23,axiom,
    ! [Uu: complex,Uua: nat] :
      aa_nat_complex(aTP_Lamm_sx(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua),zero_zero_complex,aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_complex(power_power_complex(Uu),Uua))) ).

% ATP.lambda_23
tff(fact_9109_ATP_Olambda__24,axiom,
    ! [Uu: real,Uua: nat] :
      aa_nat_real(aTP_Lamm_sw(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua),zero_zero_real,aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(power_power_real(Uu),Uua))) ).

% ATP.lambda_24
tff(fact_9110_ATP_Olambda__25,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      aa_nat_Extended_enat(aTP_Lamm_zm(Uu),Uua) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_zl(Uua),
          $ite(Uua = zero_zero_nat,zero_z5237406670263579293d_enat,extend5688581933313929465d_enat)),Uu) ).

% ATP.lambda_25
tff(fact_9111_ATP_Olambda__26,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_zo(Uu),Uua) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_zn(Uua),extend5688581933313929465d_enat),Uu) ).

% ATP.lambda_26
tff(fact_9112_ATP_Olambda__27,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_zd(Uu),Uua)
    <=> aa_Extended_enat_o(extended_case_enat_o(aa_nat_fun_nat_o(aTP_Lamm_ak,Uua),$false),Uu) ) ).

% ATP.lambda_27
tff(fact_9113_ATP_Olambda__28,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aTP_Lamm_pm(Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_complex) ).

% ATP.lambda_28
tff(fact_9114_ATP_Olambda__29,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_po,Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_real) ).

% ATP.lambda_29
tff(fact_9115_ATP_Olambda__30,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_pn,Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_int) ).

% ATP.lambda_30
tff(fact_9116_ATP_Olambda__31,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ze(Uu),Uua)
    <=> aa_Extended_enat_o(extended_case_enat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),$true),Uu) ) ).

% ATP.lambda_31
tff(fact_9117_ATP_Olambda__32,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_nj(Uu),Uua) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ni(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_32
tff(fact_9118_ATP_Olambda__33,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_nk(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ne(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_33
tff(fact_9119_ATP_Olambda__34,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nh(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ng(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_34
tff(fact_9120_ATP_Olambda__35,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_nf(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ne(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_35
tff(fact_9121_ATP_Olambda__36,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_mf(Uu),Uua) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_36
tff(fact_9122_ATP_Olambda__37,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_pw(Uu),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)),one_one_nat)))),aa_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,bit0(one))),Uua)),one_one_nat))) ).

% ATP.lambda_37
tff(fact_9123_ATP_Olambda__38,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ql(Uu),Uua) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)),aa_nat_complex(semiri8010041392384452111omplex,Uua))),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_38
tff(fact_9124_ATP_Olambda__39,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_qn,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_39
tff(fact_9125_ATP_Olambda__40,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qm,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Uua)),aa_nat_int(semiri1314217659103216013at_int,Uua))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_40
tff(fact_9126_ATP_Olambda__41,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ot(Uu),Uua) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_complex(power_power_complex(numera6690914467698888265omplex(bit0(one))),Uua)) ).

% ATP.lambda_41
tff(fact_9127_ATP_Olambda__42,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ou,Uu),Uua) = aa_real_real(divide_divide_real(aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_real(power_power_real(numeral_numeral_real(bit0(one))),Uua)) ).

% ATP.lambda_42
tff(fact_9128_ATP_Olambda__43,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Extended_enat(aa_nat2025370463774033377d_enat(aTP_Lamm_ie,Uu),Uua) = $let(
        m3: extended_enat,
        m3:= aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(bit0(one))),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(m3),one_on7984719198319812577d_enat)) ) ).

% ATP.lambda_43
tff(fact_9129_ATP_Olambda__44,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_id,Uu),Uua) = $let(
        m3: complex,
        m3:= aa_complex_complex(times_times_complex(numera6690914467698888265omplex(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_44
tff(fact_9130_ATP_Olambda__45,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_if,Uu),Uua) = $let(
        m3: real,
        m3:= aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_real_real(plus_plus_real(m3),one_one_real)) ) ).

% ATP.lambda_45
tff(fact_9131_ATP_Olambda__46,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ih,Uu),Uua) = $let(
        m3: nat,
        m3:= aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,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_46
tff(fact_9132_ATP_Olambda__47,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ig,Uu),Uua) = $let(
        m3: int,
        m3:= aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(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_47
tff(fact_9133_ATP_Olambda__48,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_rs(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_nat)))),aa_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,bit0(one)))),one_one_nat)))) ).

% ATP.lambda_48
tff(fact_9134_ATP_Olambda__49,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_xq(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% ATP.lambda_49
tff(fact_9135_ATP_Olambda__50,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_xv(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_50
tff(fact_9136_ATP_Olambda__51,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_qr(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_51
tff(fact_9137_ATP_Olambda__52,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_qt,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_52
tff(fact_9138_ATP_Olambda__53,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qs,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Uua)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_53
tff(fact_9139_ATP_Olambda__54,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_nb(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))),Uua) ).

% ATP.lambda_54
tff(fact_9140_ATP_Olambda__55,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_nc(Uu),Uua) = aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))),Uua) ).

% ATP.lambda_55
tff(fact_9141_ATP_Olambda__56,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_mg(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(bit0(one)))),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_56
tff(fact_9142_ATP_Olambda__57,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_mh(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(gbinomial_real(Uu),Uua)),aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(Uu),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_57
tff(fact_9143_ATP_Olambda__58,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_zc,Uu),Uua)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_58
tff(fact_9144_ATP_Olambda__59,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_sj(Uu),Uua) = aa_real_real(divide_divide_real(aa_nat_real(power_power_real(Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_59
tff(fact_9145_ATP_Olambda__60,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yi,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_60
tff(fact_9146_ATP_Olambda__61,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pt,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_61
tff(fact_9147_ATP_Olambda__62,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ps,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_62
tff(fact_9148_ATP_Olambda__63,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_cd(Uu),Uua)
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_63
tff(fact_9149_ATP_Olambda__64,axiom,
    ! [Uu: nat,Uua: real] :
      ( aa_real_o(aa_nat_fun_real_o(aTP_Lamm_cc,Uu),Uua)
    <=> ( aa_nat_real(power_power_real(Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_64
tff(fact_9150_ATP_Olambda__65,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_si(Uu),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_nat)))),aa_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,bit0(one)))),one_one_nat))) ).

% ATP.lambda_65
tff(fact_9151_ATP_Olambda__66,axiom,
    ! [Uu: fun_na8797796813480665199d_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_kw(Uu),Uua) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua))),aa_nat_Extended_enat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)))) ).

% ATP.lambda_66
tff(fact_9152_ATP_Olambda__67,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_kv(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua))),aa_nat_complex(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)))) ).

% ATP.lambda_67
tff(fact_9153_ATP_Olambda__68,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ku(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one))),Uua)))) ).

% ATP.lambda_68
tff(fact_9154_ATP_Olambda__69,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ky(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one))),Uua)))) ).

% ATP.lambda_69
tff(fact_9155_ATP_Olambda__70,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_kx(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,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,bit0(one))),Uua)))) ).

% ATP.lambda_70
tff(fact_9156_ATP_Olambda__71,axiom,
    ! [Uu: fun_na8797796813480665199d_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_hr(Uu),Uua) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua))),aa_nat_Extended_enat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)))) ).

% ATP.lambda_71
tff(fact_9157_ATP_Olambda__72,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ht(Uu),Uua) = aa_real_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,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,bit0(one))),Uua)))) ).

% ATP.lambda_72
tff(fact_9158_ATP_Olambda__73,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_hs(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,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,bit0(one))),Uua)))) ).

% ATP.lambda_73
tff(fact_9159_ATP_Olambda__74,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_hq(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,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,bit0(one))),Uua)))) ).

% ATP.lambda_74
tff(fact_9160_ATP_Olambda__75,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hk(Uu),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu,aa_nat_nat(suc,Uua))),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_75
tff(fact_9161_ATP_Olambda__76,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_hj(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_76
tff(fact_9162_ATP_Olambda__77,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_un(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu,Uua)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Uua)) ).

% ATP.lambda_77
tff(fact_9163_ATP_Olambda__78,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_um(Uu),Uua) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uua)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Uua)) ).

% ATP.lambda_78
tff(fact_9164_ATP_Olambda__79,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_uk(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(Uu,Uua)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Uua)) ).

% ATP.lambda_79
tff(fact_9165_ATP_Olambda__80,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_uj(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uua)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Uua)) ).

% ATP.lambda_80
tff(fact_9166_ATP_Olambda__81,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_eu(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_81
tff(fact_9167_ATP_Olambda__82,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ev(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(Uu,Uua)),aa_nat_real(power_power_real(zero_zero_real),Uua)) ).

% ATP.lambda_82
tff(fact_9168_ATP_Olambda__83,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_rl(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_83
tff(fact_9169_ATP_Olambda__84,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hp(Uu),Uua) = aa_real_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_84
tff(fact_9170_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ho(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_85
tff(fact_9171_ATP_Olambda__86,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_hl(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_86
tff(fact_9172_ATP_Olambda__87,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hn(Uu),Uua) = aa_real_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_87
tff(fact_9173_ATP_Olambda__88,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_hm(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_88
tff(fact_9174_ATP_Olambda__89,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_iy(Uu),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_89
tff(fact_9175_ATP_Olambda__90,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ix(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_90
tff(fact_9176_ATP_Olambda__91,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_mz(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_91
tff(fact_9177_ATP_Olambda__92,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_mw(Uu),Uua) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_92
tff(fact_9178_ATP_Olambda__93,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mt(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(Uu,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_93
tff(fact_9179_ATP_Olambda__94,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_mq(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_94
tff(fact_9180_ATP_Olambda__95,axiom,
    ! [Uu: fun_Ex1451620480498224165t_real,Uua: extended_enat] :
      ( aa_Extended_enat_o(aTP_Lamm_gz(Uu),Uua)
    <=> ( aa_Ext2541572767299617110t_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_95
tff(fact_9181_ATP_Olambda__96,axiom,
    ! [Uu: fun_Ex7660298398204468809at_nat,Uua: extended_enat] :
      ( aa_Extended_enat_o(aTP_Lamm_gv(Uu),Uua)
    <=> ( aa_Extended_enat_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_96
tff(fact_9182_ATP_Olambda__97,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_gx(Uu),Uua)
    <=> ( aa_complex_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_97
tff(fact_9183_ATP_Olambda__98,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_gt(Uu),Uua)
    <=> ( aa_complex_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_98
tff(fact_9184_ATP_Olambda__99,axiom,
    ! [Uu: fun_complex_int,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_hb(Uu),Uua)
    <=> ( aa_complex_int(Uu,Uua) = zero_zero_int ) ) ).

% ATP.lambda_99
tff(fact_9185_ATP_Olambda__100,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_gw(Uu),Uua)
    <=> ( aa_real_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_100
tff(fact_9186_ATP_Olambda__101,axiom,
    ! [Uu: fun_real_nat,Uua: real] :
      ( aa_real_o(aTP_Lamm_gs(Uu),Uua)
    <=> ( aa_real_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_101
tff(fact_9187_ATP_Olambda__102,axiom,
    ! [Uu: fun_real_int,Uua: real] :
      ( aa_real_o(aTP_Lamm_ha(Uu),Uua)
    <=> ( aa_real_int(Uu,Uua) = zero_zero_int ) ) ).

% ATP.lambda_102
tff(fact_9188_ATP_Olambda__103,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( aa_int_o(aTP_Lamm_gy(Uu),Uua)
    <=> ( aa_int_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_103
tff(fact_9189_ATP_Olambda__104,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_gu(Uu),Uua)
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_104
tff(fact_9190_ATP_Olambda__105,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_xw(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xv(Uu)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)),one_one_nat))) ).

% ATP.lambda_105
tff(fact_9191_ATP_Olambda__106,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ye(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xv(Uu)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua))) ).

% ATP.lambda_106
tff(fact_9192_ATP_Olambda__107,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xf,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_107
tff(fact_9193_ATP_Olambda__108,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_sl(Uu),Uua) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% ATP.lambda_108
tff(fact_9194_ATP_Olambda__109,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_sk(Uu),Uua) = aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% ATP.lambda_109
tff(fact_9195_ATP_Olambda__110,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_sv(Uu),Uua) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Uua)))),aa_nat_complex(power_power_complex(Uu),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_110
tff(fact_9196_ATP_Olambda__111,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_su(Uu),Uua) = aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Uua)))),aa_nat_real(power_power_real(Uu),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_111
tff(fact_9197_ATP_Olambda__112,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_sm(Uu),Uua) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_complex(power_power_complex(Uu),Uua)) ).

% ATP.lambda_112
tff(fact_9198_ATP_Olambda__113,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_so(Uu),Uua) = aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_113
tff(fact_9199_ATP_Olambda__114,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ka(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_114
tff(fact_9200_ATP_Olambda__115,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_jz(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_115
tff(fact_9201_ATP_Olambda__116,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_xi,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_116
tff(fact_9202_ATP_Olambda__117,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_qu(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(sin_coeff,Uua)),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_117
tff(fact_9203_ATP_Olambda__118,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_qk(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(cos_coeff,Uua)),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_118
tff(fact_9204_ATP_Olambda__119,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_mi(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),Uu) ).

% ATP.lambda_119
tff(fact_9205_ATP_Olambda__120,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_mj,Uu),Uua) = aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,Uua)),Uu) ).

% ATP.lambda_120
tff(fact_9206_ATP_Olambda__121,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_dh,Uu),Uua) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_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,bit0(one))),Uua)),one_one_nat)) ).

% ATP.lambda_121
tff(fact_9207_ATP_Olambda__122,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_dg,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,numeral_numeral_int(bit0(one))),Uua)),one_one_int)) ).

% ATP.lambda_122
tff(fact_9208_ATP_Olambda__123,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_df,Uu),Uua) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)) ).

% ATP.lambda_123
tff(fact_9209_ATP_Olambda__124,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_de,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),Uua)) ).

% ATP.lambda_124
tff(fact_9210_ATP_Olambda__125,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_js(Uu),Uua) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),numera6690914467698888265omplex(bit0(one)))) ).

% ATP.lambda_125
tff(fact_9211_ATP_Olambda__126,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_jt(Uu),Uua) = aa_real_real(plus_plus_real(Uu),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Uua)),numeral_numeral_real(bit0(one)))) ).

% ATP.lambda_126
tff(fact_9212_ATP_Olambda__127,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_yd(Uu),Uua) = aa_real_real(times_times_real(Uu),aa_real_real(plus_plus_real(one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_127
tff(fact_9213_ATP_Olambda__128,axiom,
    ! [Uu: nat,Uua: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(aTP_Lamm_zj(Uu),Uua) = vEBT_VEBT_elim_dead(Uua,aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% ATP.lambda_128
tff(fact_9214_ATP_Olambda__129,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vs,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Uua),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Uu),Uua))) ).

% ATP.lambda_129
tff(fact_9215_ATP_Olambda__130,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_vt(Uu),Uua) = aa_int_int(bit_se545348938243370406it_int(Uua),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(Uu),Uua))) ).

% ATP.lambda_130
tff(fact_9216_ATP_Olambda__131,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_yc(Uu),Uua) = aa_real_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_131
tff(fact_9217_ATP_Olambda__132,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_yb(Uu),Uua) = aa_real_real(plus_plus_real(Uu),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_132
tff(fact_9218_ATP_Olambda__133,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_uu(Uu),Uua) = aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua)) ).

% ATP.lambda_133
tff(fact_9219_ATP_Olambda__134,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_qo(Uu),Uua) = aa_real_real(minus_minus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_134
tff(fact_9220_ATP_Olambda__135,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qq,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_135
tff(fact_9221_ATP_Olambda__136,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_qp(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_136
tff(fact_9222_ATP_Olambda__137,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_km(Uu),Uua) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Uu),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uua)) ).

% ATP.lambda_137
tff(fact_9223_ATP_Olambda__138,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_kn(Uu),Uua) = aa_real_real(plus_plus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_138
tff(fact_9224_ATP_Olambda__139,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kp,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_139
tff(fact_9225_ATP_Olambda__140,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_ko(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_140
tff(fact_9226_ATP_Olambda__141,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_list_nat(aa_nat6157064778417537467st_nat(aTP_Lamm_zf,Uu),Uua) = cons_nat(Uu,nat_list_decode(Uua)) ).

% ATP.lambda_141
tff(fact_9227_ATP_Olambda__142,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_yl,Uu),Uua)
    <=> ( Uua = aa_nat_nat(suc,Uu) ) ) ).

% ATP.lambda_142
tff(fact_9228_ATP_Olambda__143,axiom,
    ! [Uu: filter_nat,Uua: filter_nat] :
      ( aa_filter_nat_o(aTP_Lamm_ml(Uu),Uua)
    <=> ord_le2510731241096832064er_nat(Uua,Uu) ) ).

% ATP.lambda_143
tff(fact_9229_ATP_Olambda__144,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_mm,Uu),Uua)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),Uu) ) ).

% ATP.lambda_144
tff(fact_9230_ATP_Olambda__145,axiom,
    ! [Uu: set_Extended_enat,Uua: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(aTP_Lamm_ah(Uu),Uua)
    <=> aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(Uua),Uu) ) ).

% ATP.lambda_145
tff(fact_9231_ATP_Olambda__146,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( aa_set_complex_o(aTP_Lamm_af(Uu),Uua)
    <=> aa_set_complex_o(ord_le211207098394363844omplex(Uua),Uu) ) ).

% ATP.lambda_146
tff(fact_9232_ATP_Olambda__147,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_ai,Uu),Uua)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Uua),Uu) ) ).

% ATP.lambda_147
tff(fact_9233_ATP_Olambda__148,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aTP_Lamm_ag(Uu),Uua)
    <=> aa_set_int_o(ord_less_eq_set_int(Uua),Uu) ) ).

% ATP.lambda_148
tff(fact_9234_ATP_Olambda__149,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ak,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_149
tff(fact_9235_ATP_Olambda__150,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_mn,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uua),Uu) ) ).

% ATP.lambda_150
tff(fact_9236_ATP_Olambda__151,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xh,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_151
tff(fact_9237_ATP_Olambda__152,axiom,
    ! [Uu: extended_enat,Uua: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(aTP_Lamm_is,Uu),Uua)
    <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Uua),Uu) ) ).

% ATP.lambda_152
tff(fact_9238_ATP_Olambda__153,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_it,Uu),Uua)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),Uu) ) ).

% ATP.lambda_153
tff(fact_9239_ATP_Olambda__154,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_ir,Uu),Uua)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Uua),Uu) ) ).

% ATP.lambda_154
tff(fact_9240_ATP_Olambda__155,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aj,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_155
tff(fact_9241_ATP_Olambda__156,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_iu,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uu) ) ).

% ATP.lambda_156
tff(fact_9242_ATP_Olambda__157,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xx,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_157
tff(fact_9243_ATP_Olambda__158,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xg,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_158
tff(fact_9244_ATP_Olambda__159,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_xr,Uu),Uua) = aa_nat_real(power_power_real(Uua),Uu) ).

% ATP.lambda_159
tff(fact_9245_ATP_Olambda__160,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ys,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_160
tff(fact_9246_ATP_Olambda__161,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_bh,Uu),Uua)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Uua),Uu) ) ).

% ATP.lambda_161
tff(fact_9247_ATP_Olambda__162,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bj,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_162
tff(fact_9248_ATP_Olambda__163,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bi,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu) ) ).

% ATP.lambda_163
tff(fact_9249_ATP_Olambda__164,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ww,Uu),Uua) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uu) ).

% ATP.lambda_164
tff(fact_9250_ATP_Olambda__165,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pr,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_165
tff(fact_9251_ATP_Olambda__166,axiom,
    ! [Uu: set_Extended_enat,Uua: extended_enat] :
      ( aa_Extended_enat_o(aTP_Lamm_zk(Uu),Uua)
    <=> aa_set3337841544853333262enat_o(member_Extended_enat(Uua),Uu) ) ).

% ATP.lambda_166
tff(fact_9252_ATP_Olambda__167,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_aa(Uu),Uua)
    <=> aa_set_real_o(member_real2(Uua),Uu) ) ).

% ATP.lambda_167
tff(fact_9253_ATP_Olambda__168,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ab(Uu),Uua)
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_168
tff(fact_9254_ATP_Olambda__169,axiom,
    ! [Uu: set_o,Uua: $o] :
      ( aa_o_o(aTP_Lamm_a(Uu),(Uua))
    <=> aa_set_o_o(member_o2((Uua)),Uu) ) ).

% ATP.lambda_169
tff(fact_9255_ATP_Olambda__170,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ac(Uu),Uua)
    <=> aa_set_set_nat_o(member_set_nat2(Uua),Uu) ) ).

% ATP.lambda_170
tff(fact_9256_ATP_Olambda__171,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ad(Uu),Uua)
    <=> aa_set_nat_o(member_nat2(Uua),Uu) ) ).

% ATP.lambda_171
tff(fact_9257_ATP_Olambda__172,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ae(Uu),Uua)
    <=> aa_set_int_o(member_int2(Uua),Uu) ) ).

% ATP.lambda_172
tff(fact_9258_ATP_Olambda__173,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_du,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_173
tff(fact_9259_ATP_Olambda__174,axiom,
    ! [Uu: list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ds(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_174
tff(fact_9260_ATP_Olambda__175,axiom,
    ! [Uu: $o,Uua: $o] :
      ( aa_o_o(aa_o_fun_o_o(aTP_Lamm_dv,(Uu)),(Uua))
    <=> ( (Uua)
      <=> (Uu) ) ) ).

% ATP.lambda_175
tff(fact_9261_ATP_Olambda__176,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_dt,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_176
tff(fact_9262_ATP_Olambda__177,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_dw,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_177
tff(fact_9263_ATP_Olambda__178,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_dx,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_178
tff(fact_9264_ATP_Olambda__179,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(aTP_Lamm_za,Uu),Uua) = aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),Uu) ).

% ATP.lambda_179
tff(fact_9265_ATP_Olambda__180,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_jq(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,bit0(one))),Uua)),one_one_nat)) ).

% ATP.lambda_180
tff(fact_9266_ATP_Olambda__181,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_jp(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)) ).

% ATP.lambda_181
tff(fact_9267_ATP_Olambda__182,axiom,
    ! [Uu: fun_na8797796813480665199d_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_hi(Uu),Uua) = aa_nat_Extended_enat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_182
tff(fact_9268_ATP_Olambda__183,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ki(Uu),Uua) = aa_nat_complex(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_183
tff(fact_9269_ATP_Olambda__184,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_gp(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_184
tff(fact_9270_ATP_Olambda__185,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_yf(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_185
tff(fact_9271_ATP_Olambda__186,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_go(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_186
tff(fact_9272_ATP_Olambda__187,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_hh(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_187
tff(fact_9273_ATP_Olambda__188,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_zi,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_zh(Uu),Uua)) ).

% ATP.lambda_188
tff(fact_9274_ATP_Olambda__189,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_xe,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_xd(Uu),Uua)) ).

% ATP.lambda_189
tff(fact_9275_ATP_Olambda__190,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_xc,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_xb(Uu),Uua)) ).

% ATP.lambda_190
tff(fact_9276_ATP_Olambda__191,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_xa,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_wz(Uu),Uua)) ).

% ATP.lambda_191
tff(fact_9277_ATP_Olambda__192,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_wy,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_wx(Uu),Uua)) ).

% ATP.lambda_192
tff(fact_9278_ATP_Olambda__193,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_wv,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_wu(Uu),Uua)) ).

% ATP.lambda_193
tff(fact_9279_ATP_Olambda__194,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_xu(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_xt(Uu),Uua)) ).

% ATP.lambda_194
tff(fact_9280_ATP_Olambda__195,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_zb(Uu),Uua) = aa_real_complex(cis,aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu))) ).

% ATP.lambda_195
tff(fact_9281_ATP_Olambda__196,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_pp(Uu),Uua) = times_times_complex(aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_196
tff(fact_9282_ATP_Olambda__197,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_pq(Uu),Uua) = times_times_real(aa_real_real(minus_minus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_197
tff(fact_9283_ATP_Olambda__198,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa_nat9102221060835343569d_enat(aTP_Lamm_jv(Uu),Uua) = times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Uu),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uua))) ).

% ATP.lambda_198
tff(fact_9284_ATP_Olambda__199,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_ju(Uu),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_199
tff(fact_9285_ATP_Olambda__200,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_jw(Uu),Uua) = times_times_real(aa_real_real(plus_plus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_200
tff(fact_9286_ATP_Olambda__201,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_jy(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_201
tff(fact_9287_ATP_Olambda__202,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_jx(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_202
tff(fact_9288_ATP_Olambda__203,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(aTP_Lamm_yx,Uu),Uua) = aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uua)) ).

% ATP.lambda_203
tff(fact_9289_ATP_Olambda__204,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_zl(Uu),Uua) = aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uua)) ).

% ATP.lambda_204
tff(fact_9290_ATP_Olambda__205,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_zn(Uu),Uua) = aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)) ).

% ATP.lambda_205
tff(fact_9291_ATP_Olambda__206,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yp,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_206
tff(fact_9292_ATP_Olambda__207,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yq,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_207
tff(fact_9293_ATP_Olambda__208,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vq,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_208
tff(fact_9294_ATP_Olambda__209,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vr,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_209
tff(fact_9295_ATP_Olambda__210,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_rq(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_210
tff(fact_9296_ATP_Olambda__211,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_xz(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_211
tff(fact_9297_ATP_Olambda__212,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] : aa_complex_complex(aTP_Lamm_hw(Uu),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_complex_nat(Uu,Uua)) ).

% ATP.lambda_212
tff(fact_9298_ATP_Olambda__213,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_real(aTP_Lamm_hz(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_213
tff(fact_9299_ATP_Olambda__214,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_hy(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_214
tff(fact_9300_ATP_Olambda__215,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_hx(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_215
tff(fact_9301_ATP_Olambda__216,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_kb(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_216
tff(fact_9302_ATP_Olambda__217,axiom,
    ! [Uu: fun_complex_int,Uua: complex] : aa_complex_complex(aTP_Lamm_dy(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_complex_int(Uu,Uua)) ).

% ATP.lambda_217
tff(fact_9303_ATP_Olambda__218,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_real(aTP_Lamm_dz(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_218
tff(fact_9304_ATP_Olambda__219,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_kc(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_219
tff(fact_9305_ATP_Olambda__220,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_real(aTP_Lamm_kd(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_int_int(Uu,Uua)) ).

% ATP.lambda_220
tff(fact_9306_ATP_Olambda__221,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_ke(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_221
tff(fact_9307_ATP_Olambda__222,axiom,
    ! [Uu: fun_na8797796813480665199d_enat,Uua: nat] : aa_nat9102221060835343569d_enat(aTP_Lamm_lb(Uu),Uua) = plus_p3455044024723400733d_enat(aa_nat_Extended_enat(Uu,Uua)) ).

% ATP.lambda_222
tff(fact_9308_ATP_Olambda__223,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_la(Uu),Uua) = plus_plus_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_223
tff(fact_9309_ATP_Olambda__224,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_ld(Uu),Uua) = plus_plus_real(aa_nat_real(Uu,Uua)) ).

% ATP.lambda_224
tff(fact_9310_ATP_Olambda__225,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_lc(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_225
tff(fact_9311_ATP_Olambda__226,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_kz(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_226
tff(fact_9312_ATP_Olambda__227,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_rx(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_227
tff(fact_9313_ATP_Olambda__228,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(aTP_Lamm_yz,Uu),Uua) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aj,Uu)) ).

% ATP.lambda_228
tff(fact_9314_ATP_Olambda__229,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_jo(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uub),aa_nat_real(Uu,Uub),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_229
tff(fact_9315_ATP_Olambda__230,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_dd(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(aa_nat8803788312118491388at_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,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(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)),Uub)) ).

% ATP.lambda_230
tff(fact_9316_ATP_Olambda__231,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_dc(Uu),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_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,numeral_numeral_int(bit0(one))),Uua)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),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,numeral_numeral_int(bit0(one))),Uua)),Uub)) ).

% ATP.lambda_231
tff(fact_9317_ATP_Olambda__232,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lz(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat2(Uub),Uu),aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_232
tff(fact_9318_ATP_Olambda__233,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_mb(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat2(Uub),Uu),aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_233
tff(fact_9319_ATP_Olambda__234,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ma(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat2(Uub),Uu),aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_234
tff(fact_9320_ATP_Olambda__235,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ly(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat2(Uub),Uu),aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_235
tff(fact_9321_ATP_Olambda__236,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: extended_enat] :
      aa_Ext2541572767299617110t_real(aa_fun1749635019461158908t_real(aTP_Lamm_et(Uu),Uua),Uub) = $ite(Uu = Uub,aa_Ext2541572767299617110t_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_236
tff(fact_9322_ATP_Olambda__237,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex7660298398204468809at_nat,Uub: extended_enat] :
      aa_Extended_enat_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_eo(Uu),Uua),Uub) = $ite(Uu = Uub,aa_Extended_enat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_237
tff(fact_9323_ATP_Olambda__238,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_er(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_238
tff(fact_9324_ATP_Olambda__239,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: complex] :
      aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_em(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_239
tff(fact_9325_ATP_Olambda__240,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ep(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_240
tff(fact_9326_ATP_Olambda__241,axiom,
    ! [Uu: real,Uua: fun_real_nat,Uub: real] :
      aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ek(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_241
tff(fact_9327_ATP_Olambda__242,axiom,
    ! [Uu: $o,Uua: fun_o_real,Uub: $o] :
      aa_o_real(aa_fun688608109223240316o_real(aTP_Lamm_eq((Uu)),Uua),(Uub)) = $ite((Uu) = (Uub),aa_o_real(Uua,(Uub)),zero_zero_real) ).

% ATP.lambda_242
tff(fact_9328_ATP_Olambda__243,axiom,
    ! [Uu: $o,Uua: fun_o_nat,Uub: $o] :
      aa_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_el((Uu)),Uua),(Uub)) = $ite((Uu) = (Uub),aa_o_nat(Uua,(Uub)),zero_zero_nat) ).

% ATP.lambda_243
tff(fact_9329_ATP_Olambda__244,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_es(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_244
tff(fact_9330_ATP_Olambda__245,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] :
      aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_en(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_245
tff(fact_9331_ATP_Olambda__246,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: extended_enat] :
      aa_Ext2541572767299617110t_real(aa_fun1749635019461158908t_real(aTP_Lamm_ej(Uu),Uua),Uub) = $ite(Uub = Uu,aa_Ext2541572767299617110t_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_246
tff(fact_9332_ATP_Olambda__247,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex7660298398204468809at_nat,Uub: extended_enat] :
      aa_Extended_enat_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_ee(Uu),Uua),Uub) = $ite(Uub = Uu,aa_Extended_enat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_247
tff(fact_9333_ATP_Olambda__248,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_eh(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_248
tff(fact_9334_ATP_Olambda__249,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: complex] :
      aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_ec(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_249
tff(fact_9335_ATP_Olambda__250,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ef(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_250
tff(fact_9336_ATP_Olambda__251,axiom,
    ! [Uu: real,Uua: fun_real_nat,Uub: real] :
      aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ea(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_251
tff(fact_9337_ATP_Olambda__252,axiom,
    ! [Uu: $o,Uua: fun_o_real,Uub: $o] :
      aa_o_real(aa_fun688608109223240316o_real(aTP_Lamm_eg((Uu)),Uua),(Uub)) = $ite((Uub) = (Uu),aa_o_real(Uua,(Uub)),zero_zero_real) ).

% ATP.lambda_252
tff(fact_9338_ATP_Olambda__253,axiom,
    ! [Uu: $o,Uua: fun_o_nat,Uub: $o] :
      aa_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_eb((Uu)),Uua),(Uub)) = $ite((Uub) = (Uu),aa_o_nat(Uua,(Uub)),zero_zero_nat) ).

% ATP.lambda_253
tff(fact_9339_ATP_Olambda__254,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lm(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_254
tff(fact_9340_ATP_Olambda__255,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_lk(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_255
tff(fact_9341_ATP_Olambda__256,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_lj(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_256
tff(fact_9342_ATP_Olambda__257,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ll(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_257
tff(fact_9343_ATP_Olambda__258,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_ei(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_258
tff(fact_9344_ATP_Olambda__259,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] :
      aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ed(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_259
tff(fact_9345_ATP_Olambda__260,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lv(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_260
tff(fact_9346_ATP_Olambda__261,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_lx(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_261
tff(fact_9347_ATP_Olambda__262,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_lw(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_262
tff(fact_9348_ATP_Olambda__263,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_lu(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_263
tff(fact_9349_ATP_Olambda__264,axiom,
    ! [Uu: fun_Ex1451620480498224165t_real,Uua: fun_Extended_enat_o,Uub: extended_enat] :
      aa_Ext2541572767299617110t_real(aa_fun6735295395714377522t_real(aTP_Lamm_gi(Uu),Uua),Uub) = $ite(aa_Extended_enat_o(Uua,Uub),aa_Ext2541572767299617110t_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_264
tff(fact_9350_ATP_Olambda__265,axiom,
    ! [Uu: fun_Ex7660298398204468809at_nat,Uua: fun_Extended_enat_o,Uub: extended_enat] :
      aa_Extended_enat_nat(aa_fun1905096444444700630at_nat(aTP_Lamm_gd(Uu),Uua),Uub) = $ite(aa_Extended_enat_o(Uua,Uub),aa_Extended_enat_nat(Uu,Uub),zero_zero_nat) ).

% ATP.lambda_265
tff(fact_9351_ATP_Olambda__266,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_o,Uub: complex] :
      aa_complex_real(aa_fun8261057609414086622x_real(aTP_Lamm_gg(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_266
tff(fact_9352_ATP_Olambda__267,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_o,Uub: complex] :
      aa_complex_nat(aa_fun7518244829075205762ex_nat(aTP_Lamm_ga(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_nat(Uu,Uub),zero_zero_nat) ).

% ATP.lambda_267
tff(fact_9353_ATP_Olambda__268,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_o,Uub: real] :
      aa_real_real(aa_fun4448911596306054750l_real(aTP_Lamm_gf(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_268
tff(fact_9354_ATP_Olambda__269,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_o,Uub: real] :
      aa_real_nat(aa_fun5513133441921015042al_nat(aTP_Lamm_fy(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_nat(Uu,Uub),zero_zero_nat) ).

% ATP.lambda_269
tff(fact_9355_ATP_Olambda__270,axiom,
    ! [Uu: fun_o_real,Uua: fun_o_o,Uub: $o] :
      aa_o_real(aa_fun1851579542500809110o_real(aTP_Lamm_ge(Uu),Uua),(Uub)) = $ite(aa_o_o(Uua,(Uub)),aa_o_real(Uu,(Uub)),zero_zero_real) ).

% ATP.lambda_270
tff(fact_9356_ATP_Olambda__271,axiom,
    ! [Uu: fun_o_nat,Uua: fun_o_o,Uub: $o] :
      aa_o_nat(aa_fun_o_o_fun_o_nat(aTP_Lamm_fx(Uu),Uua),(Uub)) = $ite(aa_o_o(Uua,(Uub)),aa_o_nat(Uu,(Uub)),zero_zero_nat) ).

% ATP.lambda_271
tff(fact_9357_ATP_Olambda__272,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_o,Uub: int] :
      aa_int_real(aa_fun7506669883404309598t_real(aTP_Lamm_gh(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_272
tff(fact_9358_ATP_Olambda__273,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_o,Uub: int] :
      aa_int_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_gb(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_nat(Uu,Uub),zero_zero_nat) ).

% ATP.lambda_273
tff(fact_9359_ATP_Olambda__274,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ng(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)) ).

% ATP.lambda_274
tff(fact_9360_ATP_Olambda__275,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ne(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)) ).

% ATP.lambda_275
tff(fact_9361_ATP_Olambda__276,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ni(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)) ).

% ATP.lambda_276
tff(fact_9362_ATP_Olambda__277,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_mu(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_277
tff(fact_9363_ATP_Olambda__278,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mr(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_278
tff(fact_9364_ATP_Olambda__279,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_mx(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_279
tff(fact_9365_ATP_Olambda__280,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rg(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_rf(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_280
tff(fact_9366_ATP_Olambda__281,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_re(Uu),Uua),Uub) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_rd(Uu),Uua),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_281
tff(fact_9367_ATP_Olambda__282,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rc(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_rb(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_282
tff(fact_9368_ATP_Olambda__283,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ra(Uu),Uua),Uub) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_qz(Uu),Uua),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_283
tff(fact_9369_ATP_Olambda__284,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qy(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qx(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_284
tff(fact_9370_ATP_Olambda__285,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_qw(Uu),Uua),Uub) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_qv(Uu),Uua),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_285
tff(fact_9371_ATP_Olambda__286,axiom,
    ! [Uu: nat,Uua: complex,Uub: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_me(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_286
tff(fact_9372_ATP_Olambda__287,axiom,
    ! [Uu: nat,Uua: real,Uub: nat] :
      aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_md(Uu),Uua),Uub) = aa_real_real(times_times_real($ite(Uub = Uu,one_one_real,zero_zero_real)),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_287
tff(fact_9373_ATP_Olambda__288,axiom,
    ! [Uu: nat,Uua: int,Uub: nat] :
      aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_mc(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_288
tff(fact_9374_ATP_Olambda__289,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_my(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_mx(Uu),Uub)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_289
tff(fact_9375_ATP_Olambda__290,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_na(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mr(Uu),Uub)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_290
tff(fact_9376_ATP_Olambda__291,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_mv(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_mu(Uu),Uub)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_291
tff(fact_9377_ATP_Olambda__292,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ms(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mr(Uu),Uub)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_292
tff(fact_9378_ATP_Olambda__293,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xj(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_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(power_power_real(Uua),Uub)) ).

% ATP.lambda_293
tff(fact_9379_ATP_Olambda__294,axiom,
    ! [Uu: real,Uua: fun_na2978037997607195054t_real,Uub: nat] : aa_nat_real(aa_fun2748096013305118955t_real(aTP_Lamm_qj(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_Ext2541572767299617110t_real(aa_nat6736956760766371991t_real(Uua,Uub),zero_z5237406670263579293d_enat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_294
tff(fact_9380_ATP_Olambda__295,axiom,
    ! [Uu: real,Uua: fun_na5723442673449079972x_real,Uub: nat] : aa_nat_real(aa_fun5746040925223427547t_real(aTP_Lamm_qi(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_complex_real(aa_nat480766339200631653x_real(Uua,Uub),zero_zero_complex)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_295
tff(fact_9381_ATP_Olambda__296,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_qg(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_296
tff(fact_9382_ATP_Olambda__297,axiom,
    ! [Uu: real,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_qf(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(aa_nat_fun_nat_real(Uua,Uub),zero_zero_nat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_297
tff(fact_9383_ATP_Olambda__298,axiom,
    ! [Uu: real,Uua: fun_nat_fun_int_real,Uub: nat] : aa_nat_real(aa_fun241848180832411225t_real(aTP_Lamm_qh(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_int_real(aa_nat_fun_int_real(Uua,Uub),zero_zero_int)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_298
tff(fact_9384_ATP_Olambda__299,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real] :
      ( aa_real_o(aa_nat_fun_real_o(aTP_Lamm_nm(Uu),Uua),Uub)
    <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mp(Uu),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_real ) ) ).

% ATP.lambda_299
tff(fact_9385_ATP_Olambda__300,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_nl(Uu),Uua),Uub)
    <=> ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mo(Uu),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_complex ) ) ).

% ATP.lambda_300
tff(fact_9386_ATP_Olambda__301,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ew(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_301
tff(fact_9387_ATP_Olambda__302,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ex(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(power_power_real(zero_zero_real),Uub))),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_302
tff(fact_9388_ATP_Olambda__303,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xs(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_303
tff(fact_9389_ATP_Olambda__304,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_yv(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),one_one_nat) )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_304
tff(fact_9390_ATP_Olambda__305,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ru(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_305
tff(fact_9391_ATP_Olambda__306,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rw(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Uub)),aa_nat_real(Uu,Uub))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_306
tff(fact_9392_ATP_Olambda__307,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_yw(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_307
tff(fact_9393_ATP_Olambda__308,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_vw(Uu),Uua),Uub)
    <=> ( ( size_s6755466524823107622T_VEBT(Uub) = Uua )
        & aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Uub)
        & ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu) ) ) ).

% ATP.lambda_308
tff(fact_9394_ATP_Olambda__309,axiom,
    ! [Uu: set_Extended_enat,Uua: nat,Uub: list_Extended_enat] :
      ( aa_lis6083379149953029096enat_o(aa_nat4163650157883470461enat_o(aTP_Lamm_vv(Uu),Uua),Uub)
    <=> ( ( size_s3941691890525107288d_enat(Uub) = Uua )
        & distin4523846830085650399d_enat(Uub)
        & aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(set_Extended_enat2(Uub)),Uu) ) ) ).

% ATP.lambda_309
tff(fact_9395_ATP_Olambda__310,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_vu(Uu),Uua),Uub)
    <=> ( ( size_s3451745648224563538omplex(Uub) = Uua )
        & distinct_complex(Uub)
        & aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu) ) ) ).

% ATP.lambda_310
tff(fact_9396_ATP_Olambda__311,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_vy(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uua )
        & aa_list_nat_o(distinct_nat,Uub)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Uub)),Uu) ) ) ).

% ATP.lambda_311
tff(fact_9397_ATP_Olambda__312,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_vx(Uu),Uua),Uub)
    <=> ( ( size_size_list_int(Uub) = Uua )
        & aa_list_int_o(distinct_int,Uub)
        & aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu) ) ) ).

% ATP.lambda_312
tff(fact_9398_ATP_Olambda__313,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_lis764731566332388511VEBT_o(aTP_Lamm_vi(Uu),Uua),Uub)
    <=> ( ( size_s6755466524823107622T_VEBT(Uub) = Uu )
        & ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),set_VEBT_VEBT2(Uua)) ) ) ).

% ATP.lambda_313
tff(fact_9399_ATP_Olambda__314,axiom,
    ! [Uu: nat,Uua: list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(aTP_Lamm_vk(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Uub)),set_nat2(Uua)) ) ) ).

% ATP.lambda_314
tff(fact_9400_ATP_Olambda__315,axiom,
    ! [Uu: nat,Uua: list_int,Uub: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(aTP_Lamm_vj(Uu),Uua),Uub)
    <=> ( ( size_size_list_int(Uub) = Uu )
        & aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),set_int2(Uua)) ) ) ).

% ATP.lambda_315
tff(fact_9401_ATP_Olambda__316,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_by(Uu),Uua),Uub)
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Uub)),Uua) ) ) ).

% ATP.lambda_316
tff(fact_9402_ATP_Olambda__317,axiom,
    ! [Uu: set_Extended_enat,Uua: nat,Uub: list_Extended_enat] :
      ( aa_lis6083379149953029096enat_o(aa_nat4163650157883470461enat_o(aTP_Lamm_bx(Uu),Uua),Uub)
    <=> ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(set_Extended_enat2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3941691890525107288d_enat(Uub)),Uua) ) ) ).

% ATP.lambda_317
tff(fact_9403_ATP_Olambda__318,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_bw(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_318
tff(fact_9404_ATP_Olambda__319,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_ca(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,size_size_list_nat(Uub)),Uua) ) ) ).

% ATP.lambda_319
tff(fact_9405_ATP_Olambda__320,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_bz(Uu),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Uub)),Uua) ) ) ).

% ATP.lambda_320
tff(fact_9406_ATP_Olambda__321,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_bt(Uu),Uua),Uub)
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_321
tff(fact_9407_ATP_Olambda__322,axiom,
    ! [Uu: set_Extended_enat,Uua: nat,Uub: list_Extended_enat] :
      ( aa_lis6083379149953029096enat_o(aa_nat4163650157883470461enat_o(aTP_Lamm_bs(Uu),Uua),Uub)
    <=> ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(set_Extended_enat2(Uub)),Uu)
        & ( size_s3941691890525107288d_enat(Uub) = Uua ) ) ) ).

% ATP.lambda_322
tff(fact_9408_ATP_Olambda__323,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_br(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu)
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_323
tff(fact_9409_ATP_Olambda__324,axiom,
    ! [Uu: set_list_nat,Uua: nat,Uub: list_list_nat] :
      ( aa_list_list_nat_o(aa_nat158606821896287091_nat_o(aTP_Lamm_wq(Uu),Uua),Uub)
    <=> ( ord_le6045566169113846134st_nat(set_list_nat2(Uub),Uu)
        & ( size_s3023201423986296836st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_324
tff(fact_9410_ATP_Olambda__325,axiom,
    ! [Uu: set_set_nat,Uua: nat,Uub: list_set_nat] :
      ( aa_list_set_nat_o(aa_nat2247512698059773005_nat_o(aTP_Lamm_wp(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_set_nat2(Uub)),Uu)
        & ( size_s3254054031482475050et_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_325
tff(fact_9411_ATP_Olambda__326,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_bv(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Uub)),Uu)
        & ( size_size_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_326
tff(fact_9412_ATP_Olambda__327,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_bu(Uu),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu)
        & ( size_size_list_int(Uub) = Uua ) ) ) ).

% ATP.lambda_327
tff(fact_9413_ATP_Olambda__328,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_yu(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_328
tff(fact_9414_ATP_Olambda__329,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_wr(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat2(aa_nat_nat(suc,Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_329
tff(fact_9415_ATP_Olambda__330,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_yr(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat2(Uub),pow_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Uu)))
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_330
tff(fact_9416_ATP_Olambda__331,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ut(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uub))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_331
tff(fact_9417_ATP_Olambda__332,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_yy(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_332
tff(fact_9418_ATP_Olambda__333,axiom,
    ! [Uu: set_Extended_enat,Uua: nat,Uub: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(aa_nat9166527414068455075enat_o(aTP_Lamm_wn(Uu),Uua),Uub)
    <=> ( aa_set3337841544853333262enat_o(ord_le7203529160286727270d_enat(Uub),Uu)
        & ( finite121521170596916366d_enat(Uub) = Uua ) ) ) ).

% ATP.lambda_333
tff(fact_9419_ATP_Olambda__334,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: set_complex] :
      ( aa_set_complex_o(aa_nat5471291301243885951plex_o(aTP_Lamm_wl(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(Uub),Uu)
        & ( finite_card_complex(Uub) = Uua ) ) ) ).

% ATP.lambda_334
tff(fact_9420_ATP_Olambda__335,axiom,
    ! [Uu: set_list_nat,Uua: nat,Uub: set_list_nat] :
      ( aa_set_list_nat_o(aa_nat7721107121239158477_nat_o(aTP_Lamm_wk(Uu),Uua),Uub)
    <=> ( ord_le6045566169113846134st_nat(Uub,Uu)
        & ( finite_card_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_335
tff(fact_9421_ATP_Olambda__336,axiom,
    ! [Uu: set_set_nat,Uua: nat,Uub: set_set_nat] :
      ( aa_set_set_nat_o(aa_nat5160323483289836711_nat_o(aTP_Lamm_wj(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(Uub),Uu)
        & ( finite_card_set_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_336
tff(fact_9422_ATP_Olambda__337,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_wo(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Uub),Uu)
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_337
tff(fact_9423_ATP_Olambda__338,axiom,
    ! [Uu: set_int,Uua: nat,Uub: set_int] :
      ( aa_set_int_o(aa_nat_fun_set_int_o(aTP_Lamm_wm(Uu),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(Uub),Uu)
        & ( finite_card_int(Uub) = Uua ) ) ) ).

% ATP.lambda_338
tff(fact_9424_ATP_Olambda__339,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ws(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),aa_nat_nat(suc,Uua)) ) ) ).

% ATP.lambda_339
tff(fact_9425_ATP_Olambda__340,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_rt(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_340
tff(fact_9426_ATP_Olambda__341,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rv(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(diffs_real(Uu),Uub)),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_341
tff(fact_9427_ATP_Olambda__342,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pu(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_342
tff(fact_9428_ATP_Olambda__343,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_dp(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_343
tff(fact_9429_ATP_Olambda__344,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_cy(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_344
tff(fact_9430_ATP_Olambda__345,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_do(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_345
tff(fact_9431_ATP_Olambda__346,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_da(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_346
tff(fact_9432_ATP_Olambda__347,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_db(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_347
tff(fact_9433_ATP_Olambda__348,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_cz(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_348
tff(fact_9434_ATP_Olambda__349,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_zq(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_349
tff(fact_9435_ATP_Olambda__350,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Extended_enat_o,Uub: extended_enat] :
      ( aa_Extended_enat_o(aa_fun3141604562678283816enat_o(aTP_Lamm_gc(Uu),Uua),Uub)
    <=> ( aa_set3337841544853333262enat_o(member_Extended_enat(Uub),Uu)
        & aa_Extended_enat_o(Uua,Uub) ) ) ).

% ATP.lambda_350
tff(fact_9436_ATP_Olambda__351,axiom,
    ! [Uu: set_complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_fz(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_351
tff(fact_9437_ATP_Olambda__352,axiom,
    ! [Uu: set_real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_am(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real2(Uub),Uu)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_352
tff(fact_9438_ATP_Olambda__353,axiom,
    ! [Uu: set_list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_an(Uu),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu)
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_353
tff(fact_9439_ATP_Olambda__354,axiom,
    ! [Uu: set_o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_al(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o2((Uub)),Uu)
        & aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_354
tff(fact_9440_ATP_Olambda__355,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ao(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat2(Uub),Uu)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_355
tff(fact_9441_ATP_Olambda__356,axiom,
    ! [Uu: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_aq(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat2(Uub),Uu)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_356
tff(fact_9442_ATP_Olambda__357,axiom,
    ! [Uu: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ap(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int2(Uub),Uu)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_357
tff(fact_9443_ATP_Olambda__358,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_fj(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_358
tff(fact_9444_ATP_Olambda__359,axiom,
    ! [Uu: list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_fh(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_359
tff(fact_9445_ATP_Olambda__360,axiom,
    ! [Uu: $o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_fk((Uu)),Uua),(Uub))
    <=> ( ( (Uu)
        <=> (Uub) )
        & aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_360
tff(fact_9446_ATP_Olambda__361,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_fi(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_361
tff(fact_9447_ATP_Olambda__362,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_fl(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_362
tff(fact_9448_ATP_Olambda__363,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_fm(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_363
tff(fact_9449_ATP_Olambda__364,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_fd(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_364
tff(fact_9450_ATP_Olambda__365,axiom,
    ! [Uu: list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_fb(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_365
tff(fact_9451_ATP_Olambda__366,axiom,
    ! [Uu: $o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_fe((Uu)),Uua),(Uub))
    <=> ( ( (Uub)
        <=> (Uu) )
        & aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_366
tff(fact_9452_ATP_Olambda__367,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_fc(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_367
tff(fact_9453_ATP_Olambda__368,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ff(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_368
tff(fact_9454_ATP_Olambda__369,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_fg(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_369
tff(fact_9455_ATP_Olambda__370,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex3741802288643809319omplex,Uub: extended_enat] :
      ( aa_Extended_enat_o(aa_fun807264594178108256enat_o(aTP_Lamm_sz(Uu),Uua),Uub)
    <=> ( aa_set3337841544853333262enat_o(member_Extended_enat(Uub),Uu)
        & ( aa_Ext8748583412541189208omplex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_370
tff(fact_9456_ATP_Olambda__371,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex7660298398204468809at_nat,Uub: extended_enat] :
      ( aa_Extended_enat_o(aa_fun3064223910257113474enat_o(aTP_Lamm_co(Uu),Uua),Uub)
    <=> ( aa_set3337841544853333262enat_o(member_Extended_enat(Uub),Uu)
        & ( aa_Extended_enat_nat(Uua,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_371
tff(fact_9457_ATP_Olambda__372,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex3482447378695272101at_int,Uub: extended_enat] :
      ( aa_Extended_enat_o(aa_fun3959557771777301982enat_o(aTP_Lamm_te(Uu),Uua),Uub)
    <=> ( aa_set3337841544853333262enat_o(member_Extended_enat(Uub),Uu)
        & ( aa_Extended_enat_int(Uua,Uub) != zero_zero_int ) ) ) ).

% ATP.lambda_372
tff(fact_9458_ATP_Olambda__373,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aTP_Lamm_sy(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_373
tff(fact_9459_ATP_Olambda__374,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aTP_Lamm_cw(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_374
tff(fact_9460_ATP_Olambda__375,axiom,
    ! [Uu: set_complex,Uua: fun_complex_nat,Uub: complex] :
      ( aa_complex_o(aa_fun8723867720292580170plex_o(aTP_Lamm_ck(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_nat(Uua,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_375
tff(fact_9461_ATP_Olambda__376,axiom,
    ! [Uu: set_complex,Uua: fun_complex_int,Uub: complex] :
      ( aa_complex_o(aa_fun8831066929681621998plex_o(aTP_Lamm_td(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_int(Uua,Uub) != zero_zero_int ) ) ) ).

% ATP.lambda_376
tff(fact_9462_ATP_Olambda__377,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aTP_Lamm_ta(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real2(Uub),Uu)
        & ( aa_real_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_377
tff(fact_9463_ATP_Olambda__378,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_cs(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real2(Uub),Uu)
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_378
tff(fact_9464_ATP_Olambda__379,axiom,
    ! [Uu: set_real,Uua: fun_real_nat,Uub: real] :
      ( aa_real_o(aa_fun7775777430665565006real_o(aTP_Lamm_cg(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real2(Uub),Uu)
        & ( aa_real_nat(Uua,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_379
tff(fact_9465_ATP_Olambda__380,axiom,
    ! [Uu: set_real,Uua: fun_real_int,Uub: real] :
      ( aa_real_o(aa_fun429334026828149234real_o(aTP_Lamm_tf(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real2(Uub),Uu)
        & ( aa_real_int(Uua,Uub) != zero_zero_int ) ) ) ).

% ATP.lambda_380
tff(fact_9466_ATP_Olambda__381,axiom,
    ! [Uu: set_o,Uua: fun_o_complex,Uub: $o] :
      ( aa_o_o(aa_fun8807775440140275936un_o_o(aTP_Lamm_tb(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o2((Uub)),Uu)
        & ( aa_o_complex(Uua,(Uub)) != zero_zero_complex ) ) ) ).

% ATP.lambda_381
tff(fact_9467_ATP_Olambda__382,axiom,
    ! [Uu: set_o,Uua: fun_o_real,Uub: $o] :
      ( aa_o_o(aa_fun1386971751305049950un_o_o(aTP_Lamm_cq(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o2((Uub)),Uu)
        & ( aa_o_real(Uua,(Uub)) != zero_zero_real ) ) ) ).

% ATP.lambda_382
tff(fact_9468_ATP_Olambda__383,axiom,
    ! [Uu: set_o,Uua: fun_o_nat,Uub: $o] :
      ( aa_o_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_ce(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o2((Uub)),Uu)
        & ( aa_o_nat(Uua,(Uub)) != zero_zero_nat ) ) ) ).

% ATP.lambda_383
tff(fact_9469_ATP_Olambda__384,axiom,
    ! [Uu: set_o,Uua: fun_o_int,Uub: $o] :
      ( aa_o_o(aa_fun_o_int_fun_o_o(aTP_Lamm_tg(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o2((Uub)),Uu)
        & ( aa_o_int(Uua,(Uub)) != zero_zero_int ) ) ) ).

% ATP.lambda_384
tff(fact_9470_ATP_Olambda__385,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_cu(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat2(Uub),Uu)
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_385
tff(fact_9471_ATP_Olambda__386,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_ci(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat2(Uub),Uu)
        & ( aa_nat_nat(Uua,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_386
tff(fact_9472_ATP_Olambda__387,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aTP_Lamm_tc(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int2(Uub),Uu)
        & ( aa_int_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_387
tff(fact_9473_ATP_Olambda__388,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aTP_Lamm_cm(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int2(Uub),Uu)
        & ( aa_int_nat(Uua,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_388
tff(fact_9474_ATP_Olambda__389,axiom,
    ! [Uu: set_int,Uua: fun_int_int,Uub: int] :
      ( aa_int_o(aa_fun8155382727227203058_int_o(aTP_Lamm_th(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int2(Uub),Uu)
        & ( aa_int_int(Uua,Uub) != zero_zero_int ) ) ) ).

% ATP.lambda_389
tff(fact_9475_ATP_Olambda__390,axiom,
    ! [Uu: fun_Ex1451620480498224165t_real,Uua: set_Extended_enat,Uub: extended_enat] :
      ( aa_Extended_enat_o(aa_set3572156908926473983enat_o(aTP_Lamm_ug(Uu),Uua),Uub)
    <=> ( aa_set3337841544853333262enat_o(member_Extended_enat(Uub),Uua)
        & ( aa_Ext2541572767299617110t_real(Uu,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_390
tff(fact_9476_ATP_Olambda__391,axiom,
    ! [Uu: fun_Ex7660298398204468809at_nat,Uua: set_Extended_enat,Uub: extended_enat] :
      ( aa_Extended_enat_o(aa_set3572156908926473983enat_o(aTP_Lamm_ue(Uu),Uua),Uub)
    <=> ( aa_set3337841544853333262enat_o(member_Extended_enat(Uub),Uua)
        & ( aa_Extended_enat_nat(Uu,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_391
tff(fact_9477_ATP_Olambda__392,axiom,
    ! [Uu: fun_complex_real,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_uf(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uua)
        & ( aa_complex_real(Uu,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_392
tff(fact_9478_ATP_Olambda__393,axiom,
    ! [Uu: fun_complex_nat,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_ud(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uua)
        & ( aa_complex_nat(Uu,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_393
tff(fact_9479_ATP_Olambda__394,axiom,
    ! [Uu: fun_real_real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_tn(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real2(Uub),Uua)
        & ( aa_real_real(Uu,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_394
tff(fact_9480_ATP_Olambda__395,axiom,
    ! [Uu: fun_real_nat,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_tj(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real2(Uub),Uua)
        & ( aa_real_nat(Uu,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_395
tff(fact_9481_ATP_Olambda__396,axiom,
    ! [Uu: fun_real_int,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_tr(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real2(Uub),Uua)
        & ( aa_real_int(Uu,Uub) != zero_zero_int ) ) ) ).

% ATP.lambda_396
tff(fact_9482_ATP_Olambda__397,axiom,
    ! [Uu: fun_o_real,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_tm(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o2((Uub)),Uua)
        & ( aa_o_real(Uu,(Uub)) != zero_zero_real ) ) ) ).

% ATP.lambda_397
tff(fact_9483_ATP_Olambda__398,axiom,
    ! [Uu: fun_o_nat,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_ti(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o2((Uub)),Uua)
        & ( aa_o_nat(Uu,(Uub)) != zero_zero_nat ) ) ) ).

% ATP.lambda_398
tff(fact_9484_ATP_Olambda__399,axiom,
    ! [Uu: fun_o_int,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_tq(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o2((Uub)),Uua)
        & ( aa_o_int(Uu,(Uub)) != zero_zero_int ) ) ) ).

% ATP.lambda_399
tff(fact_9485_ATP_Olambda__400,axiom,
    ! [Uu: fun_nat_real,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_to(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat2(Uub),Uua)
        & ( aa_nat_real(Uu,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_400
tff(fact_9486_ATP_Olambda__401,axiom,
    ! [Uu: fun_nat_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_tk(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat2(Uub),Uua)
        & ( aa_nat_nat(Uu,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_401
tff(fact_9487_ATP_Olambda__402,axiom,
    ! [Uu: fun_int_real,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_tp(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int2(Uub),Uua)
        & ( aa_int_real(Uu,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_402
tff(fact_9488_ATP_Olambda__403,axiom,
    ! [Uu: fun_int_nat,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_tl(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int2(Uub),Uua)
        & ( aa_int_nat(Uu,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_403
tff(fact_9489_ATP_Olambda__404,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_yk(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_404
tff(fact_9490_ATP_Olambda__405,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_nd(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_405
tff(fact_9491_ATP_Olambda__406,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_no(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu) ) ).

% ATP.lambda_406
tff(fact_9492_ATP_Olambda__407,axiom,
    ! [Uu: set_Pr2457182780427864761d_enat,Uua: vEBT_VEBT,Uub: extended_enat] :
      ( aa_Extended_enat_o(aa_VEB2705759856535948141enat_o(aTP_Lamm_bo(Uu),Uua),Uub)
    <=> member38198578724832770d_enat(aa_Ext2517513142808584623d_enat(aa_VEB1142851967252899118d_enat(produc581526299967858633d_enat,Uua),Uub),Uu) ) ).

% ATP.lambda_407
tff(fact_9493_ATP_Olambda__408,axiom,
    ! [Uu: set_Pr7556676689462069481BT_nat,Uua: vEBT_VEBT,Uub: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(aTP_Lamm_bm(Uu),Uua),Uub)
    <=> member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(aa_VEB7867754424810990968BT_nat(produc738532404422230701BT_nat,Uua),Uub),Uu) ) ).

% ATP.lambda_408
tff(fact_9494_ATP_Olambda__409,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: product_prod_nat_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aTP_Lamm_bl(Uu),Uua),Uub)
    <=> member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(aa_Pro5145766267327615662at_nat(produc6161850002892822231at_nat,Uua),Uub),Uu) ) ).

% ATP.lambda_409
tff(fact_9495_ATP_Olambda__410,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bk(Uu),Uua),Uub)
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub),Uu) ) ).

% ATP.lambda_410
tff(fact_9496_ATP_Olambda__411,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bn(Uu),Uua),Uub)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub),Uu) ) ).

% ATP.lambda_411
tff(fact_9497_ATP_Olambda__412,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_cb(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_412
tff(fact_9498_ATP_Olambda__413,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_wt(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_413
tff(fact_9499_ATP_Olambda__414,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_rn(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_414
tff(fact_9500_ATP_Olambda__415,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rm(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,aa_nat_nat(suc,Uub))),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_415
tff(fact_9501_ATP_Olambda__416,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xt(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(power_power_real(Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_416
tff(fact_9502_ATP_Olambda__417,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_mo(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_417
tff(fact_9503_ATP_Olambda__418,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mp(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_418
tff(fact_9504_ATP_Olambda__419,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nv(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_419
tff(fact_9505_ATP_Olambda__420,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_nn(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uub)),aa_nat_int(power_power_int(Uua),Uub)) ).

% ATP.lambda_420
tff(fact_9506_ATP_Olambda__421,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bp(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_421
tff(fact_9507_ATP_Olambda__422,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_vg(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_422
tff(fact_9508_ATP_Olambda__423,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_vh(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_423
tff(fact_9509_ATP_Olambda__424,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_vf(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_424
tff(fact_9510_ATP_Olambda__425,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ry(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_425
tff(fact_9511_ATP_Olambda__426,axiom,
    ! [Uu: fun_real_int,Uua: fun_real_int,Uub: real] : aa_real_int(aa_fun8270430325945277160al_int(aTP_Lamm_sc(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_426
tff(fact_9512_ATP_Olambda__427,axiom,
    ! [Uu: fun_o_real,Uua: fun_o_real,Uub: $o] : aa_o_real(aa_fun688608109223240316o_real(aTP_Lamm_rz(Uu),Uua),(Uub)) = aa_real_real(times_times_real(aa_o_real(Uua,(Uub))),aa_o_real(Uu,(Uub))) ).

% ATP.lambda_427
tff(fact_9513_ATP_Olambda__428,axiom,
    ! [Uu: fun_o_int,Uua: fun_o_int,Uub: $o] : aa_o_int(aa_fun4727766026823762812_o_int(aTP_Lamm_sd(Uu),Uua),(Uub)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(Uua,(Uub))),aa_o_int(Uu,(Uub))) ).

% ATP.lambda_428
tff(fact_9514_ATP_Olambda__429,axiom,
    ! [Uu: fun_set_nat_real,Uua: fun_set_nat_real,Uub: set_nat] : aa_set_nat_real(aa_fun3569254276315353500t_real(aTP_Lamm_sa(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_set_nat_real(Uua,Uub)),aa_set_nat_real(Uu,Uub)) ).

% ATP.lambda_429
tff(fact_9515_ATP_Olambda__430,axiom,
    ! [Uu: fun_set_nat_int,Uua: fun_set_nat_int,Uub: set_nat] : aa_set_nat_int(aa_fun3600621996379952540at_int(aTP_Lamm_se(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(Uua,Uub)),aa_set_nat_int(Uu,Uub)) ).

% ATP.lambda_430
tff(fact_9516_ATP_Olambda__431,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_sh(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uua,Uub)),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_431
tff(fact_9517_ATP_Olambda__432,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_sf(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_432
tff(fact_9518_ATP_Olambda__433,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_sb(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_int_real(Uua,Uub)),aa_int_real(Uu,Uub)) ).

% ATP.lambda_433
tff(fact_9519_ATP_Olambda__434,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_sg(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uua,Uub)),aa_int_int(Uu,Uub)) ).

% ATP.lambda_434
tff(fact_9520_ATP_Olambda__435,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_xy(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_435
tff(fact_9521_ATP_Olambda__436,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_gj(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_436
tff(fact_9522_ATP_Olambda__437,axiom,
    ! [Uu: fun_o_nat,Uua: fun_o_nat,Uub: $o] : aa_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_gk(Uu),Uua),(Uub)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_o_nat(Uua,(Uub))),aa_o_nat(Uu,(Uub))) ).

% ATP.lambda_437
tff(fact_9523_ATP_Olambda__438,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_gl(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_438
tff(fact_9524_ATP_Olambda__439,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_lh(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uua,Uub)),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_439
tff(fact_9525_ATP_Olambda__440,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_gn(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_440
tff(fact_9526_ATP_Olambda__441,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_gm(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_441
tff(fact_9527_ATP_Olambda__442,axiom,
    ! [Uu: fun_Ex7660298398204468809at_nat,Uua: fun_Ex7660298398204468809at_nat,Uub: extended_enat] : aa_Extended_enat_nat(aa_fun1810565495048290044at_nat(aTP_Lamm_tu(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Extended_enat_nat(Uu,Uub)),aa_Extended_enat_nat(Uua,Uub)) ).

% ATP.lambda_442
tff(fact_9528_ATP_Olambda__443,axiom,
    ! [Uu: fun_Ex3482447378695272101at_int,Uua: fun_Ex3482447378695272101at_int,Uub: extended_enat] : aa_Extended_enat_int(aa_fun1695976788104708604at_int(aTP_Lamm_tx(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Extended_enat_int(Uu,Uub)),aa_Extended_enat_int(Uua,Uub)) ).

% ATP.lambda_443
tff(fact_9529_ATP_Olambda__444,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_ez(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_444
tff(fact_9530_ATP_Olambda__445,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_ty(Uu),Uua),Uub) = aa_real_real(plus_plus_real(aa_complex_real(Uu,Uub)),aa_complex_real(Uua,Uub)) ).

% ATP.lambda_445
tff(fact_9531_ATP_Olambda__446,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_ts(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(Uu,Uub)),aa_complex_nat(Uua,Uub)) ).

% ATP.lambda_446
tff(fact_9532_ATP_Olambda__447,axiom,
    ! [Uu: fun_complex_int,Uua: fun_complex_int,Uub: complex] : aa_complex_int(aa_fun4286351638413746156ex_int(aTP_Lamm_tv(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(Uu,Uub)),aa_complex_int(Uua,Uub)) ).

% ATP.lambda_447
tff(fact_9533_ATP_Olambda__448,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_uc(Uu),Uua),Uub) = aa_real_real(plus_plus_real(aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_448
tff(fact_9534_ATP_Olambda__449,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ua(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(Uu,Uub)),aa_real_nat(Uua,Uub)) ).

% ATP.lambda_449
tff(fact_9535_ATP_Olambda__450,axiom,
    ! [Uu: fun_o_real,Uua: fun_o_real,Uub: $o] : aa_o_real(aa_fun688608109223240316o_real(aTP_Lamm_ub(Uu),Uua),(Uub)) = aa_real_real(plus_plus_real(aa_o_real(Uu,(Uub))),aa_o_real(Uua,(Uub))) ).

% ATP.lambda_450
tff(fact_9536_ATP_Olambda__451,axiom,
    ! [Uu: fun_o_nat,Uua: fun_o_nat,Uub: $o] : aa_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_tz(Uu),Uua),(Uub)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(Uu,(Uub))),aa_o_nat(Uua,(Uub))) ).

% ATP.lambda_451
tff(fact_9537_ATP_Olambda__452,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_fa(Uu),Uua),Uub) = aa_real_real(plus_plus_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_452
tff(fact_9538_ATP_Olambda__453,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ey(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_453
tff(fact_9539_ATP_Olambda__454,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ln(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_454
tff(fact_9540_ATP_Olambda__455,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_tt(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(Uu,Uub)),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_455
tff(fact_9541_ATP_Olambda__456,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_tw(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_456
tff(fact_9542_ATP_Olambda__457,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bq(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_457
tff(fact_9543_ATP_Olambda__458,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_rh(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_458
tff(fact_9544_ATP_Olambda__459,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ri(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_459
tff(fact_9545_ATP_Olambda__460,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lp(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uub)),Uu) ).

% ATP.lambda_460
tff(fact_9546_ATP_Olambda__461,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_lo(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uua,Uub)),Uu) ).

% ATP.lambda_461
tff(fact_9547_ATP_Olambda__462,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yh(Uu),Uua),Uub) = aa_nat_real(power_power_real(aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_462
tff(fact_9548_ATP_Olambda__463,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_us(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uub))) ).

% ATP.lambda_463
tff(fact_9549_ATP_Olambda__464,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_sn(Uu),Uua),Uub) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)))),aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ).

% ATP.lambda_464
tff(fact_9550_ATP_Olambda__465,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_sp(Uu),Uua),Uub) = aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)))),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ).

% ATP.lambda_465
tff(fact_9551_ATP_Olambda__466,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ur(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(Uu),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uua))) ).

% ATP.lambda_466
tff(fact_9552_ATP_Olambda__467,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_uq(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(Uu),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uua))) ).

% ATP.lambda_467
tff(fact_9553_ATP_Olambda__468,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_up(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(Uu),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uua))) ).

% ATP.lambda_468
tff(fact_9554_ATP_Olambda__469,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uo(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(Uu),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uua))) ).

% ATP.lambda_469
tff(fact_9555_ATP_Olambda__470,axiom,
    ! [Uu: extended_enat,Uua: extended_enat,Uub: nat] : aa_nat_Extended_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_ib(Uu),Uua),Uub) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Uu),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uub)),Uua)) ).

% ATP.lambda_470
tff(fact_9556_ATP_Olambda__471,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ia(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_471
tff(fact_9557_ATP_Olambda__472,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ic(Uu),Uua),Uub) = aa_real_real(plus_plus_real(Uu),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Uub)),Uua)) ).

% ATP.lambda_472
tff(fact_9558_ATP_Olambda__473,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hv(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_473
tff(fact_9559_ATP_Olambda__474,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_hu(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_474
tff(fact_9560_ATP_Olambda__475,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_jm(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_475
tff(fact_9561_ATP_Olambda__476,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_jn(Uu),Uua),Uub) = aa_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_476
tff(fact_9562_ATP_Olambda__477,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_jl(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_477
tff(fact_9563_ATP_Olambda__478,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_hd(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_478
tff(fact_9564_ATP_Olambda__479,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_he(Uu),Uua),Uub) = aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_479
tff(fact_9565_ATP_Olambda__480,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_hc(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_480
tff(fact_9566_ATP_Olambda__481,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dq(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_481
tff(fact_9567_ATP_Olambda__482,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lr(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_482
tff(fact_9568_ATP_Olambda__483,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_lq(Uu),Uua),Uub) = aa_real_real(times_times_real(Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_483
tff(fact_9569_ATP_Olambda__484,axiom,
    ! [Uu: extended_enat,Uua: nat,Uub: nat] : aa_nat_Extended_enat(aa_nat2025370463774033377d_enat(aTP_Lamm_kq(Uu),Uua),Uub) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Uu),aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_484
tff(fact_9570_ATP_Olambda__485,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_kr(Uu),Uua),Uub) = aa_real_real(plus_plus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_485
tff(fact_9571_ATP_Olambda__486,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kt(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_486
tff(fact_9572_ATP_Olambda__487,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ks(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_487
tff(fact_9573_ATP_Olambda__488,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_iw(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_488
tff(fact_9574_ATP_Olambda__489,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_iv(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_489
tff(fact_9575_ATP_Olambda__490,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_kg(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_490
tff(fact_9576_ATP_Olambda__491,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_rr(Uu),Uua),Uub) = aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_491
tff(fact_9577_ATP_Olambda__492,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_gr(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_492
tff(fact_9578_ATP_Olambda__493,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_yg(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_493
tff(fact_9579_ATP_Olambda__494,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_gq(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_494
tff(fact_9580_ATP_Olambda__495,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_kf(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_495
tff(fact_9581_ATP_Olambda__496,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_di(Uu),Uua),Uub)
    <=> aa_Pro6150637604445964947_nat_o(Uu,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub)) ) ).

% ATP.lambda_496
tff(fact_9582_ATP_Olambda__497,axiom,
    ! [Uu: fun_Pr2602117396471556196at_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dj(Uu),Uua),Uub) = aa_Pro4243173770044397973at_nat(Uu,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub)) ).

% ATP.lambda_497
tff(fact_9583_ATP_Olambda__498,axiom,
    ! [Uu: fun_Pr2477239776880934807at_nat,Uua: nat,Uub: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_dk(Uu),Uua),Uub) = aa_Pro8023792847356852296at_nat(Uu,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub)) ).

% ATP.lambda_498
tff(fact_9584_ATP_Olambda__499,axiom,
    ! [Uu: fun_Pr2028404973800837934_nat_o,Uua: nat,Uub: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_dl(Uu),Uua),Uub) = aa_Pro989595756709598047_nat_o(Uu,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub)) ).

% ATP.lambda_499
tff(fact_9585_ATP_Olambda__500,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_dm(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ) ).

% ATP.lambda_500
tff(fact_9586_ATP_Olambda__501,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lt(Uu),Uua),Uub) = aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_501
tff(fact_9587_ATP_Olambda__502,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ls(Uu),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_502
tff(fact_9588_ATP_Olambda__503,axiom,
    ! [Uu: fun_o_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_vl(Uu),Uua),Uub)
    <=> aa_o_o(Uu,aa_nat_o(Uua,Uub)) ) ).

% ATP.lambda_503
tff(fact_9589_ATP_Olambda__504,axiom,
    ! [Uu: fun_o_nat,Uua: fun_nat_o,Uub: nat] : aa_nat_nat(aa_fun4204860950123276802at_nat(aTP_Lamm_vm(Uu),Uua),Uub) = aa_o_nat(Uu,aa_nat_o(Uua,Uub)) ).

% ATP.lambda_504
tff(fact_9590_ATP_Olambda__505,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_vn(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(Uua,Uub)) ) ).

% ATP.lambda_505
tff(fact_9591_ATP_Olambda__506,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_vo(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_506
tff(fact_9592_ATP_Olambda__507,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_rd(Uu),Uua),Uub),Uuc) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uuc) ),
        aa_complex_complex(times_times_complex(aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,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)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),
        zero_zero_complex ) ).

% ATP.lambda_507
tff(fact_9593_ATP_Olambda__508,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_rf(Uu),Uua),Uub),Uuc) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uuc) ),
        aa_real_real(times_times_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,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)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),
        zero_zero_real ) ).

% ATP.lambda_508
tff(fact_9594_ATP_Olambda__509,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_qv(Uu),Uua),Uub),Uuc) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uub)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uuc) ),
        aa_complex_complex(times_times_complex(aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,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)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub)))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),
        zero_zero_complex ) ).

% ATP.lambda_509
tff(fact_9595_ATP_Olambda__510,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qx(Uu),Uua),Uub),Uuc) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uub)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uuc) ),
        aa_real_real(times_times_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,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)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub)))),aa_nat_real(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),
        zero_zero_real ) ).

% ATP.lambda_510
tff(fact_9596_ATP_Olambda__511,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_qz(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uub),aa_complex_complex(times_times_complex(aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,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)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),zero_zero_complex) ).

% ATP.lambda_511
tff(fact_9597_ATP_Olambda__512,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_rb(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uub),aa_real_real(times_times_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,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)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),zero_zero_real) ).

% ATP.lambda_512
tff(fact_9598_ATP_Olambda__513,axiom,
    ! [Uu: nat,Uua: fun_na8797796813480665199d_enat,Uub: fun_na8797796813480665199d_enat,Uuc: nat] :
      aa_nat_Extended_enat(aa_fun2512298231148344956d_enat(aa_fun1703792696854984886d_enat(aTP_Lamm_oc(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_Extended_enat(Uua,Uuc),
        $ite(Uuc = Uu,zero_z5237406670263579293d_enat,aa_nat_Extended_enat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_513
tff(fact_9599_ATP_Olambda__514,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_oa(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_complex(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_complex,aa_nat_complex(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_514
tff(fact_9600_ATP_Olambda__515,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_og(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_real(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_real,aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_515
tff(fact_9601_ATP_Olambda__516,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_oe(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_nat(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_nat,aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_516
tff(fact_9602_ATP_Olambda__517,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ny(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_int(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_int,aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_517
tff(fact_9603_ATP_Olambda__518,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_oj(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_complex(Uua,Uuc),
        $ite(Uuc = Uu,one_one_complex,aa_nat_complex(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_518
tff(fact_9604_ATP_Olambda__519,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_oi(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_real(Uua,Uuc),
        $ite(Uuc = Uu,one_one_real,aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_519
tff(fact_9605_ATP_Olambda__520,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_ol(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_nat(Uua,Uuc),
        $ite(Uuc = Uu,one_one_nat,aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_520
tff(fact_9606_ATP_Olambda__521,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ok(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_int(Uua,Uuc),
        $ite(Uuc = Uu,one_one_int,aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_521
tff(fact_9607_ATP_Olambda__522,axiom,
    ! [Uu: nat,Uua: fun_na8797796813480665199d_enat,Uub: fun_na8797796813480665199d_enat,Uuc: nat] :
      aa_nat_Extended_enat(aa_fun2512298231148344956d_enat(aa_fun1703792696854984886d_enat(aTP_Lamm_od(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_Extended_enat(Uua,Uuc),aa_nat_Extended_enat(Uub,Uuc)) ).

% ATP.lambda_522
tff(fact_9608_ATP_Olambda__523,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_ob(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_complex(Uua,Uuc),aa_nat_complex(Uub,Uuc)) ).

% ATP.lambda_523
tff(fact_9609_ATP_Olambda__524,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_oh(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_real(Uua,Uuc),aa_nat_real(Uub,Uuc)) ).

% ATP.lambda_524
tff(fact_9610_ATP_Olambda__525,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_of(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_nat(Uua,Uuc),aa_nat_nat(Uub,Uuc)) ).

% ATP.lambda_525
tff(fact_9611_ATP_Olambda__526,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_nz(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_int(Uua,Uuc),aa_nat_int(Uub,Uuc)) ).

% ATP.lambda_526
tff(fact_9612_ATP_Olambda__527,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_li(Uu),Uua),Uub),Uuc) = $ite(aa_set_nat_o(member_nat2(Uuc),Uua),aa_nat_real(Uub,Uuc),aa_nat_real(Uu,Uuc)) ).

% ATP.lambda_527
tff(fact_9613_ATP_Olambda__528,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex5852502975177633731d_enat,Uub: fun_Ex5852502975177633731d_enat,Uuc: extended_enat] :
      aa_Ext7969882492699152700d_enat(aa_fun7865826685190571204d_enat(aa_fun6128326792922203780d_enat(aTP_Lamm_fu(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_Ext7969882492699152700d_enat(Uua,Uuc),aa_Ext7969882492699152700d_enat(Uub,Uuc)) ).

% ATP.lambda_528
tff(fact_9614_ATP_Olambda__529,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: fun_Ex1451620480498224165t_real,Uuc: extended_enat] :
      aa_Ext2541572767299617110t_real(aa_fun1749635019461158908t_real(aa_fun967516747568145260t_real(aTP_Lamm_fs(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_Ext2541572767299617110t_real(Uua,Uuc),aa_Ext2541572767299617110t_real(Uub,Uuc)) ).

% ATP.lambda_529
tff(fact_9615_ATP_Olambda__530,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex7660298398204468809at_nat,Uub: fun_Ex7660298398204468809at_nat,Uuc: extended_enat] :
      aa_Extended_enat_nat(aa_fun1810565495048290044at_nat(aa_fun5812849712608808208at_nat(aTP_Lamm_fo(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_Extended_enat_nat(Uua,Uuc),aa_Extended_enat_nat(Uub,Uuc)) ).

% ATP.lambda_530
tff(fact_9616_ATP_Olambda__531,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex3482447378695272101at_int,Uub: fun_Ex3482447378695272101at_int,Uuc: extended_enat] :
      aa_Extended_enat_int(aa_fun1695976788104708604at_int(aa_fun8556764835550362732at_int(aTP_Lamm_fq(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_Extended_enat_int(Uua,Uuc),aa_Extended_enat_int(Uub,Uuc)) ).

% ATP.lambda_531
tff(fact_9617_ATP_Olambda__532,axiom,
    ! [Uu: complex,Uua: fun_co6496365337113880337d_enat,Uub: fun_co6496365337113880337d_enat,Uuc: complex] :
      aa_com3683005661447327866d_enat(aa_fun4954905782667315452d_enat(aa_fun2568869125039355352d_enat(aTP_Lamm_ft(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_com3683005661447327866d_enat(Uua,Uuc),aa_com3683005661447327866d_enat(Uub,Uuc)) ).

% ATP.lambda_532
tff(fact_9618_ATP_Olambda__533,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_fr(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_real(Uua,Uuc),aa_complex_real(Uub,Uuc)) ).

% ATP.lambda_533
tff(fact_9619_ATP_Olambda__534,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] :
      aa_complex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_fn(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_nat(Uua,Uuc),aa_complex_nat(Uub,Uuc)) ).

% ATP.lambda_534
tff(fact_9620_ATP_Olambda__535,axiom,
    ! [Uu: complex,Uua: fun_complex_int,Uub: fun_complex_int,Uuc: complex] :
      aa_complex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_fp(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_int(Uua,Uuc),aa_complex_int(Uub,Uuc)) ).

% ATP.lambda_535
tff(fact_9621_ATP_Olambda__536,axiom,
    ! [Uu: real,Uua: fun_real_nat,Uub: fun_real_nat,Uuc: real] :
      aa_real_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_fv(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_real_nat(Uua,Uuc),aa_real_nat(Uub,Uuc)) ).

% ATP.lambda_536
tff(fact_9622_ATP_Olambda__537,axiom,
    ! [Uu: real,Uua: fun_real_int,Uub: fun_real_int,Uuc: real] :
      aa_real_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_fw(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_real_int(Uua,Uuc),aa_real_int(Uub,Uuc)) ).

% ATP.lambda_537
tff(fact_9623_ATP_Olambda__538,axiom,
    ! [Uu: $o,Uua: fun_o_nat,Uub: fun_o_nat,Uuc: $o] :
      aa_o_nat(aa_fun6176012476734143100_o_nat(aa_fun6251207968994965954_o_nat(aTP_Lamm_kj((Uu)),Uua),Uub),(Uuc)) = $ite((Uuc) = (Uu),aa_o_nat(Uua,(Uuc)),aa_o_nat(Uub,(Uuc))) ).

% ATP.lambda_538
tff(fact_9624_ATP_Olambda__539,axiom,
    ! [Uu: $o,Uua: fun_o_int,Uub: fun_o_int,Uuc: $o] :
      aa_o_int(aa_fun4727766026823762812_o_int(aa_fun5648456818326528798_o_int(aTP_Lamm_kl((Uu)),Uua),Uub),(Uuc)) = $ite((Uuc) = (Uu),aa_o_int(Uua,(Uuc)),aa_o_int(Uub,(Uuc))) ).

% ATP.lambda_539
tff(fact_9625_ATP_Olambda__540,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: int] :
      aa_int_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_kk(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_int_nat(Uua,Uuc),aa_int_nat(Uub,Uuc)) ).

% ATP.lambda_540
tff(fact_9626_ATP_Olambda__541,axiom,
    ! [Uu: fun_VE3569950477627547253d_enat,Uua: extended_enat,Uub: list_VEBT_VEBT,Uuc: nat] : aa_nat_Extended_enat(aa_lis7361820021430418801d_enat(aa_Ext4914522244092902178d_enat(aTP_Lamm_wf(Uu),Uua),Uub),Uuc) = aa_Ext7969882492699152700d_enat(compow2216990816977393616d_enat(Uuc,times_7803423173614009249d_enat(Uua)),aa_VEB2793163758658435054d_enat(Uu,nth_VEBT_VEBT(Uub,Uuc))) ).

% ATP.lambda_541
tff(fact_9627_ATP_Olambda__542,axiom,
    ! [Uu: fun_VE5569070535936029429omplex,Uua: complex,Uub: list_VEBT_VEBT,Uuc: nat] : aa_nat_complex(aa_lis8713343626058569977omplex(aa_com9025189225015940188omplex(aTP_Lamm_wc(Uu),Uua),Uub),Uuc) = aa_complex_complex(compow7134260122181328870omplex(Uuc,times_times_complex(Uua)),aa_VEBT_VEBT_complex(Uu,nth_VEBT_VEBT(Uub,Uuc))) ).

% ATP.lambda_542
tff(fact_9628_ATP_Olambda__543,axiom,
    ! [Uu: fun_VEBT_VEBT_nat,Uua: nat,Uub: list_VEBT_VEBT,Uuc: nat] : aa_nat_nat(aa_lis7811502719995853595at_nat(aa_nat265247640139305436at_nat(aTP_Lamm_wi(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_fun5791449513213591928at_nat(aa_nat2689442564459254213at_nat(compow_fun_nat_nat,Uuc),aa_nat_fun_nat_nat(times_times_nat,Uua)),aa_VEBT_VEBT_nat(Uu,nth_VEBT_VEBT(Uub,Uuc))) ).

% ATP.lambda_543
tff(fact_9629_ATP_Olambda__544,axiom,
    ! [Uu: fun_VEBT_VEBT_int,Uua: int,Uub: list_VEBT_VEBT,Uuc: nat] : aa_nat_int(aa_lis3633651700486656887at_int(aa_int4104624148011801308at_int(aTP_Lamm_vz(Uu),Uua),Uub),Uuc) = aa_int_int(compow_fun_int_int(Uuc,aa_int_fun_int_int(times_times_int,Uua)),aa_VEBT_VEBT_int(Uu,nth_VEBT_VEBT(Uub,Uuc))) ).

% ATP.lambda_544
tff(fact_9630_ATP_Olambda__545,axiom,
    ! [Uu: fun_na8797796813480665199d_enat,Uua: extended_enat,Uub: list_nat,Uuc: nat] : aa_nat_Extended_enat(aa_lis8229176623098646865d_enat(aa_Ext1340118137906550078d_enat(aTP_Lamm_wh(Uu),Uua),Uub),Uuc) = aa_Ext7969882492699152700d_enat(compow2216990816977393616d_enat(Uuc,times_7803423173614009249d_enat(Uua)),aa_nat_Extended_enat(Uu,aa_nat_nat(nth_nat(Uub),Uuc))) ).

% ATP.lambda_545
tff(fact_9631_ATP_Olambda__546,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: list_nat,Uuc: nat] : aa_nat_complex(aa_lis3451821876164259097omplex(aa_com7804175510677644100omplex(aTP_Lamm_we(Uu),Uua),Uub),Uuc) = aa_complex_complex(compow7134260122181328870omplex(Uuc,times_times_complex(Uua)),aa_nat_complex(Uu,aa_nat_nat(nth_nat(Uub),Uuc))) ).

% ATP.lambda_546
tff(fact_9632_ATP_Olambda__547,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: list_nat,Uuc: nat] : aa_nat_int(aa_lis6345223608758921111at_int(aa_int7844622935458978624at_int(aTP_Lamm_wb(Uu),Uua),Uub),Uuc) = aa_int_int(compow_fun_int_int(Uuc,aa_int_fun_int_int(times_times_int,Uua)),aa_nat_int(Uu,aa_nat_nat(nth_nat(Uub),Uuc))) ).

% ATP.lambda_547
tff(fact_9633_ATP_Olambda__548,axiom,
    ! [Uu: fun_in5914218760682845459d_enat,Uua: extended_enat,Uub: list_int,Uuc: nat] : aa_nat_Extended_enat(aa_lis3069066898809486325d_enat(aa_Ext5403380450472165346d_enat(aTP_Lamm_wg(Uu),Uua),Uub),Uuc) = aa_Ext7969882492699152700d_enat(compow2216990816977393616d_enat(Uuc,times_7803423173614009249d_enat(Uua)),aa_int_Extended_enat(Uu,aa_nat_int(nth_int(Uub),Uuc))) ).

% ATP.lambda_548
tff(fact_9634_ATP_Olambda__549,axiom,
    ! [Uu: fun_int_complex,Uua: complex,Uub: list_int,Uuc: nat] : aa_nat_complex(aa_lis226970529952281973omplex(aa_com4579324164465666976omplex(aTP_Lamm_wd(Uu),Uua),Uub),Uuc) = aa_complex_complex(compow7134260122181328870omplex(Uuc,times_times_complex(Uua)),aa_int_complex(Uu,aa_nat_int(nth_int(Uub),Uuc))) ).

% ATP.lambda_549
tff(fact_9635_ATP_Olambda__550,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: list_int,Uuc: nat] : aa_nat_int(aa_lis1073296200319854579at_int(aa_int2572695527019912092at_int(aTP_Lamm_wa(Uu),Uua),Uub),Uuc) = aa_int_int(compow_fun_int_int(Uuc,aa_int_fun_int_int(times_times_int,Uua)),aa_int_int(Uu,aa_nat_int(nth_int(Uub),Uuc))) ).

% ATP.lambda_550
tff(fact_9636_ATP_Olambda__551,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_il(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ik(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_551
tff(fact_9637_ATP_Olambda__552,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ij(Uu),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ii(Uu),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_552
tff(fact_9638_ATP_Olambda__553,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_oq(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex($ite(
            Uuc = zero_zero_nat,
            aa_complex_complex(uminus1482373934393186551omplex,Uub),
            $ite(Uuc = Uu,one_one_complex,zero_zero_complex) )),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_553
tff(fact_9639_ATP_Olambda__554,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: nat] :
      aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_os(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real($ite(
            Uuc = zero_zero_nat,
            aa_real_real(uminus_uminus_real,Uub),
            $ite(Uuc = Uu,one_one_real,zero_zero_real) )),aa_nat_real(power_power_real(Uua),Uuc)) ).

% ATP.lambda_554
tff(fact_9640_ATP_Olambda__555,axiom,
    ! [Uu: nat,Uua: int,Uub: int,Uuc: nat] :
      aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_or(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,
          $ite(
            Uuc = zero_zero_nat,
            aa_int_int(uminus_uminus_int,Uub),
            $ite(Uuc = Uu,one_one_int,zero_zero_int) )),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_555
tff(fact_9641_ATP_Olambda__556,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_nx(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_nw(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uub),Uuc)) ).

% ATP.lambda_556
tff(fact_9642_ATP_Olambda__557,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_nu(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_nt(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_real(power_power_real(Uub),Uuc)) ).

% ATP.lambda_557
tff(fact_9643_ATP_Olambda__558,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_ns(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_nr(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_558
tff(fact_9644_ATP_Olambda__559,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_nq(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_fun6128006242533108349at_int(aTP_Lamm_np(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_int(power_power_int(Uub),Uuc)) ).

% ATP.lambda_559
tff(fact_9645_ATP_Olambda__560,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_xo(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_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(power_power_real(Uub),Uuc)) ).

% ATP.lambda_560
tff(fact_9646_ATP_Olambda__561,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_xm(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_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(power_power_real(Uub),Uuc)) ).

% ATP.lambda_561
tff(fact_9647_ATP_Olambda__562,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_xl(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(Uub),Uua)),Uuc)) ).

% ATP.lambda_562
tff(fact_9648_ATP_Olambda__563,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_xk(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(power_power_real(aa_real_real(minus_minus_real(Uua),Uub)),Uuc)) ).

% ATP.lambda_563
tff(fact_9649_ATP_Olambda__564,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_jb(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_564
tff(fact_9650_ATP_Olambda__565,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_jd(Uu),Uua),Uub),Uuc) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_565
tff(fact_9651_ATP_Olambda__566,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_iz(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_566
tff(fact_9652_ATP_Olambda__567,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_wu(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_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_567
tff(fact_9653_ATP_Olambda__568,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ii(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,bit0(one)))),Uuc))) ).

% ATP.lambda_568
tff(fact_9654_ATP_Olambda__569,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ik(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Uua),Uu)),Uuc)),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,bit0(one)))),Uuc))) ).

% ATP.lambda_569
tff(fact_9655_ATP_Olambda__570,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_qc(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_complex(comm_s2602460028002588243omplex(Uu),Uuc))),aa_nat_complex(comm_s2602460028002588243omplex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_570
tff(fact_9656_ATP_Olambda__571,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qe(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(comm_s7457072308508201937r_real(Uu),Uuc))),aa_nat_real(comm_s7457072308508201937r_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_571
tff(fact_9657_ATP_Olambda__572,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_qd(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_int(comm_s4660882817536571857er_int(Uu),Uuc))),aa_nat_int(comm_s4660882817536571857er_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_572
tff(fact_9658_ATP_Olambda__573,axiom,
    ! [Uu: extended_enat,Uua: extended_enat,Uub: nat,Uuc: nat] : aa_nat_Extended_enat(aa_nat2025370463774033377d_enat(aa_Ext8984936454148280526d_enat(aTP_Lamm_pz(Uu),Uua),Uub),Uuc) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_Extended_enat(power_8040749407984259932d_enat(Uu),Uuc))),aa_nat_Extended_enat(power_8040749407984259932d_enat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_573
tff(fact_9659_ATP_Olambda__574,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_py(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_574
tff(fact_9660_ATP_Olambda__575,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qb(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_575
tff(fact_9661_ATP_Olambda__576,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_px(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_576
tff(fact_9662_ATP_Olambda__577,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_qa(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_int(power_power_int(Uu),Uuc))),aa_nat_int(power_power_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_577
tff(fact_9663_ATP_Olambda__578,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ss(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_578
tff(fact_9664_ATP_Olambda__579,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_st(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(power_power_real(Uu),Uuc))),aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_579
tff(fact_9665_ATP_Olambda__580,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_jg(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_580
tff(fact_9666_ATP_Olambda__581,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_jh(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(power_power_real(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_real(power_power_real(Uu),Uuc)) ).

% ATP.lambda_581
tff(fact_9667_ATP_Olambda__582,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_jf(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_582
tff(fact_9668_ATP_Olambda__583,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_jc(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_583
tff(fact_9669_ATP_Olambda__584,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_je(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(power_power_real(Uu),Uuc)),aa_real_real(minus_minus_real(aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_584
tff(fact_9670_ATP_Olambda__585,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ja(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_585
tff(fact_9671_ATP_Olambda__586,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_jj(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_586
tff(fact_9672_ATP_Olambda__587,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_jk(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(power_power_real(Uu),Uuc)),aa_nat_real(power_power_real(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_587
tff(fact_9673_ATP_Olambda__588,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_ji(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_588
tff(fact_9674_ATP_Olambda__589,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_pv(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_589
tff(fact_9675_ATP_Olambda__590,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_wz(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_590
tff(fact_9676_ATP_Olambda__591,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_wx(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_591
tff(fact_9677_ATP_Olambda__592,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_xb(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_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_592
tff(fact_9678_ATP_Olambda__593,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_xd(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_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_593
tff(fact_9679_ATP_Olambda__594,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_zh(Uu),Uua),Uub),Uuc)
    <=> ( 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_594
tff(fact_9680_ATP_Olambda__595,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex7660298398204468809at_nat,Uub: fun_Ex7660298398204468809at_nat,Uuc: extended_enat] :
      ( aa_Extended_enat_o(aa_fun3064223910257113474enat_o(aa_fun5506575416149999492enat_o(aTP_Lamm_cp(Uu),Uua),Uub),Uuc)
    <=> ( aa_set3337841544853333262enat_o(member_Extended_enat(Uuc),Uu)
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Extended_enat_nat(Uua,Uuc)),aa_Extended_enat_nat(Uub,Uuc)) != zero_zero_nat ) ) ) ).

% ATP.lambda_595
tff(fact_9681_ATP_Olambda__596,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_cx(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_real_real(plus_plus_real(aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_596
tff(fact_9682_ATP_Olambda__597,axiom,
    ! [Uu: set_complex,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] :
      ( aa_complex_o(aa_fun8723867720292580170plex_o(aa_fun3914901917261573480plex_o(aTP_Lamm_cl(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(Uua,Uuc)),aa_complex_nat(Uub,Uuc)) != zero_zero_nat ) ) ) ).

% ATP.lambda_597
tff(fact_9683_ATP_Olambda__598,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_ct(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real2(Uuc),Uu)
        & ( aa_real_real(plus_plus_real(aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_598
tff(fact_9684_ATP_Olambda__599,axiom,
    ! [Uu: set_real,Uua: fun_real_nat,Uub: fun_real_nat,Uuc: real] :
      ( aa_real_o(aa_fun7775777430665565006real_o(aa_fun1158036852828689646real_o(aTP_Lamm_ch(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real2(Uuc),Uu)
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(Uua,Uuc)),aa_real_nat(Uub,Uuc)) != zero_zero_nat ) ) ) ).

% ATP.lambda_599
tff(fact_9685_ATP_Olambda__600,axiom,
    ! [Uu: set_o,Uua: fun_o_real,Uub: fun_o_real,Uuc: $o] :
      ( aa_o_o(aa_fun1386971751305049950un_o_o(aa_fun3246320876514483410un_o_o(aTP_Lamm_cr(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o2((Uuc)),Uu)
        & ( aa_real_real(plus_plus_real(aa_o_real(Uua,(Uuc))),aa_o_real(Uub,(Uuc))) != zero_zero_real ) ) ) ).

% ATP.lambda_600
tff(fact_9686_ATP_Olambda__601,axiom,
    ! [Uu: set_o,Uua: fun_o_nat,Uub: fun_o_nat,Uuc: $o] :
      ( aa_o_o(aa_fun_o_nat_fun_o_o(aa_fun5833554292436378194un_o_o(aTP_Lamm_cf(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o2((Uuc)),Uu)
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(Uua,(Uuc))),aa_o_nat(Uub,(Uuc))) != zero_zero_nat ) ) ) ).

% ATP.lambda_601
tff(fact_9687_ATP_Olambda__602,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_cv(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat2(Uuc),Uu)
        & ( aa_real_real(plus_plus_real(aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_602
tff(fact_9688_ATP_Olambda__603,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aa_fun6348600740168882050_nat_o(aTP_Lamm_cj(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat2(Uuc),Uu)
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uua,Uuc)),aa_nat_nat(Uub,Uuc)) != zero_zero_nat ) ) ) ).

% ATP.lambda_603
tff(fact_9689_ATP_Olambda__604,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aa_fun8703205213358249326_int_o(aTP_Lamm_cn(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int2(Uuc),Uu)
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(Uua,Uuc)),aa_int_nat(Uub,Uuc)) != zero_zero_nat ) ) ) ).

% ATP.lambda_604
tff(fact_9690_ATP_Olambda__605,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_rp(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_605
tff(fact_9691_ATP_Olambda__606,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ro(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_real(power_power_real(Uub),Uuc)) ).

% ATP.lambda_606
tff(fact_9692_ATP_Olambda__607,axiom,
    ! [Uu: fun_VE3569950477627547253d_enat,Uua: extended_enat,Uub: list_VEBT_VEBT,Uuc: nat] : aa_nat_Extended_enat(aa_lis7361820021430418801d_enat(aa_Ext4914522244092902178d_enat(aTP_Lamm_vb(Uu),Uua),Uub),Uuc) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_VEB2793163758658435054d_enat(Uu,nth_VEBT_VEBT(Uub,Uuc))),aa_nat_Extended_enat(power_8040749407984259932d_enat(Uua),Uuc)) ).

% ATP.lambda_607
tff(fact_9693_ATP_Olambda__608,axiom,
    ! [Uu: fun_VE5569070535936029429omplex,Uua: complex,Uub: list_VEBT_VEBT,Uuc: nat] : aa_nat_complex(aa_lis8713343626058569977omplex(aa_com9025189225015940188omplex(aTP_Lamm_uy(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_VEBT_VEBT_complex(Uu,nth_VEBT_VEBT(Uub,Uuc))),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_608
tff(fact_9694_ATP_Olambda__609,axiom,
    ! [Uu: fun_VEBT_VEBT_nat,Uua: nat,Uub: list_VEBT_VEBT,Uuc: nat] : aa_nat_nat(aa_lis7811502719995853595at_nat(aa_nat265247640139305436at_nat(aTP_Lamm_ve(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_VEBT_VEBT_nat(Uu,nth_VEBT_VEBT(Uub,Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uuc)) ).

% ATP.lambda_609
tff(fact_9695_ATP_Olambda__610,axiom,
    ! [Uu: fun_VEBT_VEBT_int,Uua: int,Uub: list_VEBT_VEBT,Uuc: nat] : aa_nat_int(aa_lis3633651700486656887at_int(aa_int4104624148011801308at_int(aTP_Lamm_uv(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_VEBT_VEBT_int(Uu,nth_VEBT_VEBT(Uub,Uuc))),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_610
tff(fact_9696_ATP_Olambda__611,axiom,
    ! [Uu: fun_na8797796813480665199d_enat,Uua: extended_enat,Uub: list_nat,Uuc: nat] : aa_nat_Extended_enat(aa_lis8229176623098646865d_enat(aa_Ext1340118137906550078d_enat(aTP_Lamm_vd(Uu),Uua),Uub),Uuc) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(Uu,aa_nat_nat(nth_nat(Uub),Uuc))),aa_nat_Extended_enat(power_8040749407984259932d_enat(Uua),Uuc)) ).

% ATP.lambda_611
tff(fact_9697_ATP_Olambda__612,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: list_nat,Uuc: nat] : aa_nat_complex(aa_lis3451821876164259097omplex(aa_com7804175510677644100omplex(aTP_Lamm_va(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(nth_nat(Uub),Uuc))),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_612
tff(fact_9698_ATP_Olambda__613,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: list_nat,Uuc: nat] : aa_nat_int(aa_lis6345223608758921111at_int(aa_int7844622935458978624at_int(aTP_Lamm_ux(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,aa_nat_nat(nth_nat(Uub),Uuc))),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_613
tff(fact_9699_ATP_Olambda__614,axiom,
    ! [Uu: fun_in5914218760682845459d_enat,Uua: extended_enat,Uub: list_int,Uuc: nat] : aa_nat_Extended_enat(aa_lis3069066898809486325d_enat(aa_Ext5403380450472165346d_enat(aTP_Lamm_vc(Uu),Uua),Uub),Uuc) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_int_Extended_enat(Uu,aa_nat_int(nth_int(Uub),Uuc))),aa_nat_Extended_enat(power_8040749407984259932d_enat(Uua),Uuc)) ).

% ATP.lambda_614
tff(fact_9700_ATP_Olambda__615,axiom,
    ! [Uu: fun_int_complex,Uua: complex,Uub: list_int,Uuc: nat] : aa_nat_complex(aa_lis226970529952281973omplex(aa_com4579324164465666976omplex(aTP_Lamm_uz(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_int_complex(Uu,aa_nat_int(nth_int(Uub),Uuc))),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_615
tff(fact_9701_ATP_Olambda__616,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: list_int,Uuc: nat] : aa_nat_int(aa_lis1073296200319854579at_int(aa_int2572695527019912092at_int(aTP_Lamm_uw(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,aa_nat_int(nth_int(Uub),Uuc))),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_616
tff(fact_9702_ATP_Olambda__617,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_pf(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(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,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_617
tff(fact_9703_ATP_Olambda__618,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ph(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(Uu,Uuc)),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_618
tff(fact_9704_ATP_Olambda__619,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_pd(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uuc)),aa_nat_int(power_power_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_619
tff(fact_9705_ATP_Olambda__620,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_nr(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uuc)),aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_620
tff(fact_9706_ATP_Olambda__621,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_nt(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(Uu,Uuc)),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_621
tff(fact_9707_ATP_Olambda__622,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_nw(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_622
tff(fact_9708_ATP_Olambda__623,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_fun6128006242533108349at_int(aTP_Lamm_np(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uuc)),aa_nat_int(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_623
tff(fact_9709_ATP_Olambda__624,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_ui(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)),aa_nat_nat(suc,Uuc))) ).

% ATP.lambda_624
tff(fact_9710_ATP_Olambda__625,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_uh(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)),aa_nat_nat(suc,Uuc))) ).

% ATP.lambda_625
tff(fact_9711_ATP_Olambda__626,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_ul(Uu),Uua),Uub),Uuc) = aa_nat_int(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)),aa_nat_nat(suc,Uuc))) ).

% ATP.lambda_626
tff(fact_9712_ATP_Olambda__627,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_hg(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_627
tff(fact_9713_ATP_Olambda__628,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_hf(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_628
tff(fact_9714_ATP_Olambda__629,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_kh(Uu),Uua),Uub),Uuc) = aa_nat_int(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_629
tff(fact_9715_ATP_Olambda__630,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_pc(Uu),Uua),Uub),Uuc),Uud) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_pb(Uua),Uub),Uuc),Uud)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_630
tff(fact_9716_ATP_Olambda__631,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_pa(Uu),Uua),Uub),Uuc),Uud) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_oz(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_631
tff(fact_9717_ATP_Olambda__632,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: int,Uuc: int,Uud: nat] : aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_oy(Uu),Uua),Uub),Uuc),Uud) = aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(aTP_Lamm_ox(Uua),Uub),Uuc),Uud)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_632
tff(fact_9718_ATP_Olambda__633,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_xn(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_xm(Uua),Uuc),Uud)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))),aa_real_real(times_times_real(Uub),aa_real_real(divide_divide_real(aa_nat_real(power_power_real(Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))))) ).

% ATP.lambda_633
tff(fact_9719_ATP_Olambda__634,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_pi(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(times_times_real(aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ph(Uua),Uuc),Uud)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Uud)),Uu))),aa_nat_real(power_power_real(Uub),Uud)) ).

% ATP.lambda_634
tff(fact_9720_ATP_Olambda__635,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_pg(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_pf(Uua),Uuc),Uud),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Uud)),Uu))),aa_nat_complex(power_power_complex(Uub),Uud)) ).

% ATP.lambda_635
tff(fact_9721_ATP_Olambda__636,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: int,Uuc: int,Uud: nat] : aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_pe(Uu),Uua),Uub),Uuc),Uud) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_pd(Uua),Uuc),Uud)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Uud)),Uu))),aa_nat_int(power_power_int(Uub),Uud)) ).

% ATP.lambda_636
tff(fact_9722_ATP_Olambda__637,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_ov(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Uud)),Uua)),one_one_complex)),Uud)),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_637
tff(fact_9723_ATP_Olambda__638,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_ow(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(gbinomial_real(aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Uud)),Uua)),one_one_real)),Uud)),aa_nat_real(power_power_real(Uub),Uud))),aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_638
tff(fact_9724_ATP_Olambda__639,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_om(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Uu)),Uua)),Uud)),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_639
tff(fact_9725_ATP_Olambda__640,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_oo(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(gbinomial_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Uu)),Uua)),Uud)),aa_nat_real(power_power_real(Uub),Uud))),aa_nat_real(power_power_real(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_640
tff(fact_9726_ATP_Olambda__641,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_on(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,Uua)),Uud)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Uub)),Uud))),aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_641
tff(fact_9727_ATP_Olambda__642,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_op(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(gbinomial_real(aa_real_real(uminus_uminus_real,Uua)),Uud)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,Uub)),Uud))),aa_nat_real(power_power_real(aa_real_real(plus_plus_real(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_642
tff(fact_9728_ATP_Olambda__643,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: complex,Uuc: nat,Uud: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_oz(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_643
tff(fact_9729_ATP_Olambda__644,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: real,Uuc: nat,Uud: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_pb(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_real(power_power_real(Uub),Uud))),aa_nat_real(power_power_real(Uua),Uuc)) ).

% ATP.lambda_644
tff(fact_9730_ATP_Olambda__645,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: int,Uuc: nat,Uud: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(aTP_Lamm_ox(Uu),Uua),Uub),Uuc),Uud) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_int(power_power_int(Uub),Uud))),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_645
tff(fact_9731_ATP_Olambda__646,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_ym(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,bit0(one))),Uu))
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uud))
           => ( ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(Uua,I4)),X_13)
            <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Uub),I4) ) )
        & $ite(
            Uue = Uuf,
            ! [X3: vEBT_VEBT] :
              ( member_VEBT_VEBT2(X3,set_VEBT_VEBT2(Uua))
             => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X6) ),
            ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),Uuf)
            & ! [X3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uu))
               => ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),X3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),X3)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_646
tff(fact_9732_ATP_Olambda__647,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_rk(Uu),Uua) = Uu ).

% ATP.lambda_647
tff(fact_9733_ATP_Olambda__648,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_rj(Uu),Uua) = Uu ).

% ATP.lambda_648
tff(fact_9734_ATP_Olambda__649,axiom,
    ! [Uu: extended_enat] : aa_Ext7969882492699152700d_enat(aTP_Lamm_bg,Uu) = Uu ).

% ATP.lambda_649
tff(fact_9735_ATP_Olambda__650,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_bf,Uu) = Uu ).

% ATP.lambda_650
tff(fact_9736_ATP_Olambda__651,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_be,Uu) = Uu ).

% ATP.lambda_651
tff(fact_9737_ATP_Olambda__652,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_bc,Uu) = Uu ).

% ATP.lambda_652
tff(fact_9738_ATP_Olambda__653,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_bd,Uu) = Uu ).

% ATP.lambda_653
tff(fact_9739_ATP_Olambda__654,axiom,
    ! [Uu: extended_enat] : aa_Ext7969882492699152700d_enat(aTP_Lamm_bb,Uu) = zero_z5237406670263579293d_enat ).

% ATP.lambda_654
tff(fact_9740_ATP_Olambda__655,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_ba,Uu) = zero_zero_complex ).

% ATP.lambda_655
tff(fact_9741_ATP_Olambda__656,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_az,Uu) = zero_zero_real ).

% ATP.lambda_656
tff(fact_9742_ATP_Olambda__657,axiom,
    ! [Uu: nat] : aa_nat_complex(aTP_Lamm_lg,Uu) = zero_zero_complex ).

% ATP.lambda_657
tff(fact_9743_ATP_Olambda__658,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_dr,Uu) = zero_zero_real ).

% ATP.lambda_658
tff(fact_9744_ATP_Olambda__659,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ax,Uu) = zero_zero_nat ).

% ATP.lambda_659
tff(fact_9745_ATP_Olambda__660,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_lf,Uu) = zero_zero_int ).

% ATP.lambda_660
tff(fact_9746_ATP_Olambda__661,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ay,Uu) = zero_zero_int ).

% ATP.lambda_661
tff(fact_9747_ATP_Olambda__662,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_at,Uu)
    <=> $false ) ).

% ATP.lambda_662
tff(fact_9748_ATP_Olambda__663,axiom,
    ! [Uu: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ar,Uu)
    <=> $false ) ).

% ATP.lambda_663
tff(fact_9749_ATP_Olambda__664,axiom,
    ! [Uu: $o] :
      ( aa_o_o(aTP_Lamm_au,(Uu))
    <=> $false ) ).

% ATP.lambda_664
tff(fact_9750_ATP_Olambda__665,axiom,
    ! [Uu: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_as,Uu)
    <=> $false ) ).

% ATP.lambda_665
tff(fact_9751_ATP_Olambda__666,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_av,Uu)
    <=> $false ) ).

% ATP.lambda_666
tff(fact_9752_ATP_Olambda__667,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_aw,Uu)
    <=> $false ) ).

% ATP.lambda_667
tff(fact_9753_ATP_Olambda__668,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_vp,Uu)
    <=> $true ) ).

% ATP.lambda_668

% Helper facts (12)
tff(help_fequal_2_1_fequal_001_Eo_T,axiom,
    ! [X8: $o,Y: $o] :
      ( ( (X8) != (Y) )
      | aa_o_o(aa_o_fun_o_o(fequal_o,(X8)),(Y)) ) ).

tff(help_fequal_1_1_fequal_001_Eo_T,axiom,
    ! [X8: $o,Y: $o] :
      ( ~ aa_o_o(aa_o_fun_o_o(fequal_o,(X8)),(Y))
      | ( (X8) = (Y) ) ) ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X8: int,Y: int] :
      ( ( X8 != Y )
      | aa_int_o(aa_int_fun_int_o(fequal_int,X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X8: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X8),Y)
      | ( X8 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X8: nat,Y: nat] :
      ( ( X8 != Y )
      | aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X8: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X8),Y)
      | ( X8 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X8: real,Y: real] :
      ( ( X8 != Y )
      | aa_real_o(aa_real_fun_real_o(fequal_real,X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X8: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(fequal_real,X8),Y)
      | ( X8 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X8: set_nat,Y: set_nat] :
      ( ( X8 != Y )
      | aa_set_nat_o(aa_set7861427066675100071_nat_o(fequal_set_nat,X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X8: set_nat,Y: set_nat] :
      ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(fequal_set_nat,X8),Y)
      | ( X8 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X8: list_nat,Y: list_nat] :
      ( ( X8 != Y )
      | aa_list_nat_o(fequal_list_nat(X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X8: list_nat,Y: list_nat] :
      ( ~ aa_list_nat_o(fequal_list_nat(X8),Y)
      | ( X8 = Y ) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    ? [Y4: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na),vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na))),i)),Y4) ).

%------------------------------------------------------------------------------